handbook of philosophical logic vol 6

Handbook of Philosophical Logic

2nd Edition

Volume 6

edited by Dov M. Gabbay and F. Guenthner

CONTENTS

viiEditorial PrefaceDov M. Gabbay

1Relevance LogicMike Dunn and Greg Restall

129Quantum LogicsMaria-Luisa Dalla Chiara and Roberto Giuntini

229Combinators, Proofs and Implicational LogicsMartin Bunder

287Paraconsistent LogicGraham Priest

395Index

PREFACE TO THE SECOND EDITION

It is with great pleasure that we are presenting to the community thesecond edition of this extraordinary handbook. It has been over 15 yearssince the publication of the �rst edition and there have been great changesin the landscape of philosophical logic since then.

The �rst edition has proved invaluable to generations of students andresearchers in formal philosophy and language, as well as to consumers oflogic in many applied areas. The main logic article in the EncyclopaediaBritannica 1999 has described the �rst edition as `the best starting pointfor exploring any of the topics in logic'. We are con�dent that the secondedition will prove to be just as good!

The �rst edition was the second handbook published for the logic commu-nity. It followed the North Holland one volume Handbook of MathematicalLogic, published in 1977, edited by the late Jon Barwise. The four volumeHandbook of Philosophical Logic, published 1983{1989 came at a fortunatetemporal junction at the evolution of logic. This was the time when logicwas gaining ground in computer science and arti�cial intelligence circles.

These areas were under increasing commercial pressure to provide deviceswhich help and/or replace the human in his daily activity. This pressurerequired the use of logic in the modelling of human activity and organisa-tion on the one hand and to provide the theoretical basis for the computerprogram constructs on the other. The result was that the Handbook ofPhilosophical Logic, which covered most of the areas needed from logic forthese active communities, became their bible.

The increased demand for philosophical logic from computer science andarti�cial intelligence and computational linguistics accelerated the devel-opment of the subject directly and indirectly. It directly pushed researchforward, stimulated by the needs of applications. New logic areas becameestablished and old areas were enriched and expanded. At the same time, itsocially provided employment for generations of logicians residing in com-puter science, linguistics and electrical engineering departments which ofcourse helped keep the logic community thriving. In addition to that, it sohappens (perhaps not by accident) that many of the Handbook contributorsbecame active in these application areas and took their place as time passedon, among the most famous leading �gures of applied philosophical logic ofour times. Today we have a handbook with a most extraordinary collectionof famous people as authors!

The table below will give our readers an idea of the landscape of logicand its relation to computer science and formal language and arti�cial in-telligence. It shows that the �rst edition is very close to the mark of whatwas needed. Two topics were not included in the �rst edition, even though

D. Gabbay and F. Guenthner (eds.),Handbook of Philosophical Logic, Volume 6, vii{ix.c 2002, Kluwer Academic Publishers. Printed in the Netherlands.

viii

they were extensively discussed by all authors in a 3-day Handbook meeting.These are:

� a chapter on non-monotonic logic

� a chapter on combinatory logic and �-calculus

We felt at the time (1979) that non-monotonic logic was not ready fora chapter yet and that combinatory logic and �-calculus was too far re-moved.1 Non-monotonic logic is now a very major area of philosophi-cal logic, alongside default logics, labelled deductive systems, �bring log-ics, multi-dimensional, multimodal and substructural logics. Intensive re-examinations of fragments of classical logic have produced fresh insights,including at time decision procedures and equivalence with non-classicalsystems.

Perhaps the most impressive achievement of philosophical logic as arisingin the past decade has been the e�ective negotiation of research partnershipswith fallacy theory, informal logic and argumentation theory, attested to bythe Amsterdam Conference in Logic and Argumentation in 1995, and thetwo Bonn Conferences in Practical Reasoning in 1996 and 1997.

These subjects are becoming more and more useful in agent theory andintelligent and reactive databases.

Finally, �fteen years after the start of the Handbook project, I wouldlike to take this opportunity to put forward my current views about logicin computer science, computational linguistics and arti�cial intelligence. Inthe early 1980s the perception of the role of logic in computer science wasthat of a speci�cation and reasoning tool and that of a basis for possiblyneat computer languages. The computer scientist was manipulating datastructures and the use of logic was one of his options.

My own view at the time was that there was an opportunity for logic toplay a key role in computer science and to exchange bene�ts with this richand important application area and thus enhance its own evolution. Therelationship between logic and computer science was perceived as very muchlike the relationship of applied mathematics to physics and engineering. Ap-plied mathematics evolves through its use as an essential tool, and so wehoped for logic. Today my view has changed. As computer science andarti�cial intelligence deal more and more with distributed and interactivesystems, processes, concurrency, agents, causes, transitions, communicationand control (to name a few), the researcher in this area is having more andmore in common with the traditional philosopher who has been analysing

1I am really sorry, in hindsight, about the omission of the non-monotonic logic chapter.I wonder how the subject would have developed, if the AI research community had hada theoretical model, in the form of a chapter, to look at. Perhaps the area would havedeveloped in a more streamlined way!

PREFACE TO THE SECOND EDITION ix

such questions for centuries (unrestricted by the capabilities of any hard-ware).

The principles governing the interaction of several processes, for example,are abstract an similar to principles governing the cooperation of two largeorganisation. A detailed rule based e�ective but rigid bureaucracy is verymuch similar to a complex computer program handling and manipulatingdata. My guess is that the principles underlying one are very much thesame as those underlying the other.

I believe the day is not far away in the future when the computer scientistwill wake up one morning with the realisation that he is actually a kind offormal philosopher!

The projected number of volumes for this Handbook is about 18. Thesubject has evolved and its areas have become interrelated to such an extentthat it no longer makes sense to dedicate volumes to topics. However, thevolumes do follow some natural groupings of chapters.

I would like to thank our authors are readers for their contributions andtheir commitment in making this Handbook a success. Thanks also toour publication administrator Mrs J. Spurr for her usual dedication andexcellence and to Kluwer Academic Publishers for their continuing supportfor the Handbook.

Dov GabbayKing's College London

x

Logic IT

Naturallanguageprocessing

Programcontrol spec-i�cation,veri�cation,concurrency

Arti�cial in-telligence

Logic pro-gramming

Temporallogic

Expressivepower of tenseoperators.Temporalindices. Sepa-ration of pastfrom future

Expressivepower for re-current events.Speci�cationof tempo-ral control.Decision prob-lems. Modelchecking.

Planning.Time depen-dent data.Event calculus.Persistencethrough time|the FrameProblem. Tem-poral querylanguage.temporaltransactions.

Extension ofHorn clausewith timecapability.Event calculus.Temporal logicprogramming.

Modal logic.Multi-modallogics

generalisedquanti�ers

Action logic Belief revision.Inferentialdatabases

Negation byfailure andmodality

Algorithmicproof

Discourse rep-resentation.Direct com-putation onlinguistic input

New logics.Generic theo-rem provers

General theoryof reasoning.Non-monotonicsystems

Procedural ap-proach to logic

Non-monotonicreasoning

Resolvingambigui-ties. Machinetranslation.Documentclassi�cation.Relevancetheory

Loop checking.Non-monotonicdecisions aboutloops. Faultsin systems.

Intrinsic logicaldiscipline forAI. Evolvingand com-municatingdatabases

Negation byfailure. Deduc-tive databases

Probabilisticand fuzzylogic

logical analysisof language

Real time sys-tems

Expert sys-tems. Machinelearning

Semantics forlogic programs

Intuitionisticlogic

Quanti�ers inlogic

Constructivereasoning andproof theoryabout speci�-cation design

Intuitionisticlogic is a betterlogical basisthan classicallogic

Horn clauselogic is reallyintuitionistic.Extension oflogic program-ming languages

Set theory,higher-orderlogic, �-calculus,types

Montaguesemantics.Situationsemantics

Non-well-founded sets

Hereditary �-nite predicates

�-calculus ex-tension to logicprograms

PREFACE TO THE SECOND EDITION xi

Imperativevs. declar-ative lan-guages

Databasetheory

Complexitytheory

Agent theory Special com-ments: Alook to thefuture

Temporal logicas a declarativeprogramminglanguage. Thechanging pastin databases.The imperativefuture

Temporaldatabasesand temporaltransactions

Complexityquestions ofdecision pro-cedures of thelogics involved

An essentialcomponent

Temporalsystems arebecoming moreand more so-phisticatedand extensivelyapplied

Dynamic logic Database up-dates andaction logic

Ditto Possible ac-tions

Multimodallogics areon the rise.Quanti�cationand contextbecoming veryactive

Types. Termrewrite sys-tems. Abstractinterpretation

Abduction, rel-evance

Ditto Agent'simplementationrely onproof theory.

Inferentialdatabases.Non-monotoniccoding ofdatabases

Ditto Agent's rea-soning isnon-monotonic

A major areanow. Impor-tant for formal-ising practicalreasoning

Fuzzy andprobabilisticdata

Ditto Connectionwith decisiontheory

Major areanow

Semantics forprogramminglanguages.Martin-L�oftheories

Databasetransactions.Inductivelearning

Ditto Agents con-structivereasoning

Still a majorcentral alterna-tive to classicallogic

Semantics forprogramminglanguages.Abstract in-terpretation.Domain recur-sion theory.

Ditto More centralthan ever!

xii

Classical logic.Classical frag-ments

Basic back-ground lan-guage

Program syn-thesis

A basic tool

Labelleddeductivesystems

Extremely use-ful in modelling

A unifyingframework.Contexttheory.

Annotatedlogic programs

Resource andsubstructurallogics

Lambek calcu-lus

Truthmaintenancesystems

Fibring andcombininglogics

Dynamic syn-tax

Modules.Combininglanguages

Logics of spaceand time

Combining fea-tures

Fallacytheory

LogicalDynamics

Widely appliedhere

Argumentationtheory games

Game seman-tics gainingground

Object level/metalevel

Extensivelyused in AI

Mechanisms:Abduction,defaultrelevance

ditto

Connectionwith neuralnets

Time-action-revision mod-els

ditto

PREFACE TO THE SECOND EDITION xiii

Relationaldatabases

Logical com-plexity classes

The workhorseof logic

The study offragments isvery active andpromising.

Labellingallows forcontextand control.

Essential tool. The new unify-ing frameworkfor logics

Linear logic Agentshavelimitedresources

Linkeddatabases.Reactivedatabases

Agents arebuilt up ofvarious �bredmechanisms

The notion ofself-�bring al-lows for self-reference

Fallacies arereally validmodes of rea-soning in theright context.

Potentially ap-plicable

A dynamicview of logic

On the risein all areas ofapplied logic.Promises agreat future

Important fea-ture of agents

Always centralin all areas

Very importantfor agents

Becoming partof the notion ofa logic

Of great im-portance to thefuture. Juststarting

A new theoryof logical agent

A new kind ofmodel

J. MICHAEL DUNN AND GREG RESTALL

RELEVANCE LOGIC

1 INTRODUCTION

1.1 Delimiting the topic

The title of this piece is not `A Survey of Relevance Logic'. Such a projectwas impossible in the mid 1980s when the �rst version of this article waspublished, due to the development of the �eld and even the space limitationsof the Handbook. The situation is if anything, more diÆcult now. Forexample Anderson and Belnap and Dunn's two volume [1975; 1992] workEntailment: The Logic of Relevance and Necessity, runs to over 1200 pages,and is their summary of just some of the work done by them and their co-workers up to about the late 1980s. Further, the comprehensive bibliography(prepared by R. G. Wolf) contains over 3000 entries in work on relevancelogic and related �elds.

So, we need some way of delimiting our topic. To be honest the fact thatwe are writing this is already a kind of delimitation. It is natural that youshall �nd emphasised here the work that we happen to know best. But stillrationality demands a less subjective rationale, and so we will proceed asfollows.

Anderson [1963] set forth some open problems for his and Belnap's sys-tem E that have given shape to much of the subsequent research in relevancelogic (even much of the earlier work can be seen as related to these openproblems, e.g. by giving rise to them). Anderson picks three of these prob-lems as major: (1) the admissibility of Ackermann's rule (the readershould not worry that he is expected to already know what this means),(2) the decision problems, (3) the providing of a semantics. Anderson alsolists additional problems which he calls `minor' because they have no `philo-sophical bite'. We will organise our remarks on relevance logic around threemajor problems of Anderson. The reader should be told in advance thateach of these problems are closed (but of course `closed' does not mean`�nished'|closing one problem invariably opens another related problem).This gives then three of our sections. It is obvious that to these we must addan introduction setting forth at least some of the motivations of relevancelogic and some syntactical speci�cations. To the end we will add a sectionwhich situates work in relevance logic in the wider context of study of otherlogical systems, since in the recent years it has become clear that relevancelogics �t well among a wider class of `resource-conscious' or `substructural'logics [Schroeder-Heister and Do�sen, 1993; Restall, 2000] [and cite the S{Harticle in this volume]. We thus have the following table of contents:

D. Gabbay and F. Guenthner (eds.),Handbook of Philosophical Logic, Volume 6, 1{128.c 2002, Kluwer Academic Publishers. Printed in the Netherlands.

2 J. MICHAEL DUNN AND GREG RESTALL

1. Introduction

2. The Admissibility of

3. Semantics

4. The Decision Problem

5. Looking About

We should add a word about the delimitation of our topic. There are by nowa host of formal systems that can be said with some justi�cation to be `rele-vance logics'. Some of these antedate the Anderson{Belnap approach, someare more recent. Some have been studied somewhat extensively, whereasothers have been discussed for only a few pages in some journal. It would beimpossible to describe all of these, let alone to assess in each and every casehow they compare with the Anderson{Belnap approach. It is clear that theAnderson{Belnap-style logics have been the most intensively studied. Sowe will concentrate on the research program of Anderson, Belnap and theirco-workers, and shall mention other approaches only insofar as they bearon this program. By way of minor recompense we mention that Ander-son and Belnap [1975] have been good about discussing related approaches,especially the older ones.

Finally, we should say that our paradigm of a relevance logic throughoutthis essay will be the Anderson{Belnap system R or relevant implication(�rst devised by Belnap|see [Belnap, 1967a; Belnap, 1967b] for its history)and not so much the Anderson{Belnap favourite, their system E of entail-ment. There will be more about each of these systems below (they are explic-itly formulated in Section 1.3), but let us simply say here that each of theseis concerned to formalise a species of implication (or the conditional|seeSection 1.2) in which the antecedent suÆces relevantly for the consequent.The system E di�ers from the system R primarily by adding necessity tothis relationship, and in this E is a modal logic as well as a relevance logic.This by itself gives good reason to consider R and not E as the paradigmof a relevance logic.1

1.2 Implication and the Conditional

Before turning to matters of logical substance, let us �rst introduce a frame-work for grammar and nomenclature that is helpful in understanding theways that writers on relevance logic often express themselves. We draw

1It should be entered in the record that there are some workers in relevance logicwho consider both R and E too strong for at least some purposes (see [Routley, 1977],[Routley et al., 1982], and more recently, [Brady, 1996]).

RELEVANCE LOGIC 3

heavily on the `Grammatical Propaedeutic' appendix of [Anderson and Bel-nap, 1975] and to a lesser extent on [Meyer, 1966], both of which are verymuch recommended to the reader for their wise heresy from logical tradition.

Thus logical tradition (think of [Quine, 1953]) makes much of the gram-matical distinction between ìf, then' (a connective), and ìmplies' or itsrough synonym èntails' (transitive verbs). This tradition opposes

1. If today is Tuesday, then this is Belgium

to the pair of sentences

2. `Today is Tuesday' implies `This is Belgium',

3. That today is Tuesday implies that this is Belgium.

And the tradition insists that (1) be called a conditional, and that (2) and(3) be called implications.

Sometimes much philosophical weight is made to rest on this distinction.It is said that since ìmplies' is a verb demanding nouns to ank it, thatimplication must then be a relation between the objects stood for by thosenouns, whereas it is said that ìf, then' is instead a connective combining thatimplication (unlike ìf, then') is really a metalinguistic notion, either overtlyas in (2) where the nouns are names of sentences, or else covertly as in (3)where the nouns are naming propositions (the `ghosts' of linguistic entities).This last is then felt to be especially bad because it involves ontologicalcommitment to propositions or some equally disreputable entities. The �rstis at least free of such questionable ontological commitments, but does raisereal complications about `nested implications', which would seem to take usinto a meta-metalanguage, etc.

The response of relevance logicians to this distinction has been largely oneof `What, me worry?' Sometime sympathetic outsiders have tried to apolo-gise for what might be quickly labelled a ùse{mention confusion' on the partof relevance logicians [Scott, 1971]. But `hard-core' relevance logicians oftenseem to luxuriate in this `confusion'. As Anderson and Belnap [1975, p. 473]

say of their `Grammatical Propaedeutic': \the principle aim of this pieceis to convince the reader that it is philosophically respectable to `confuse'implication or entailment with the conditional, and indeed philosophicallysuspect to harp on the dangers of such a `confusion'. (The suspicion isthat such harpists are plucking a metaphysical tune on merely grammaticalstrings.)"

The gist of the Anderson{Belnap position is that there is a genericconditional-implication notion, which can be carried into English by a va-riety of grammatical constructions. Implication itself can be viewed as aconnective requiring prenominalisation: `that implies that ', and assuch it nests. It is an incidental feature of English that it favours sentenceswith main subjects and verbs, and ìmplies' conforms to this reference by


the trick of disguising sentences as nouns by prenominalisation. But suchgrammatical prejudices need not be taken as enshrining ontological presup-positions.

Let us use the label `Correspondence Thesis' for the claim that Andersonand Belnap come close to making (but do not actually make), namely, thatin general there is nothing other than a purely grammatical distinctionbetween sentences of the forms

4. If A, then B, and

5. That A implies that B.

Now undoubtedly the Correspondence Thesis overstates matters. Thus, tobring in just one consideration, [Casta~neda, 1975, pp. 66 �.] distinguishesìf A then B' from À only if B' by virtue of an essentially pragmatic distinc-tion (frozen into grammar) of `thematic' emphases, which cuts across thelogical distinction of antecedent and consequent. Putting things quickly, ìf'introduces a suÆcient condition for something happening, something beingdone, etc. whereas ònly if' introduces a necessary condition. Thus ìf' (byitself or pre�xed with ònly') always introduces the state of a�airs thoughtof as a condition for something else, then something else being thus thefocus of attention. Since `that A implies that B' is devoid of such thematicindicators, it is not equivalent at every level of analysis to either ìf A thenB' or À only if B'.

It is worth remarking that since the formal logician's A ! B is equallydevoid of thematic indicators, `that A implies that B' would seem to makea better reading of it than either ìf A then B' or À only if B'. And yetit is almost universally rejected by writers of elementary logic texts as evenan acceptable reading.

And, of course, another consideration against the Correspondence Thesisis produced by notorious examples like Austin's

6. There are biscuits on the sideboard if you want some,

which sounds very odd indeed when phrased as an implication. Indeed, (6)poses perplexities of one kind or another for any theory of the conditional,and so should perhaps best be ignored as posing any special threat tot heAnderson and Belnap account of conditionals. Perhaps it was Austin-typeexamples that led Anderson and Belnap [1975, pp. 491{492] to say \wethink every use of ìmplies' or èntails' as a connective can be replaced by asuitable ìf-then'; however, the converse may not be true". They go on to say\But with reference to the uses in which we are primarily interested, we feelfree to move back and forth between ìf-then' and èntails' in a free-wheelingmanner".

Associated with the Correspondence Thesis is the idea that just as therecan be contingent conditionals (e.g. (1)), so then the corresponding implica-tions (e.g. (3)) must also be contingent. This goes against certain Quinean

RELEVANCE LOGIC 5

tendencies to `regiment' the English word ìmplies' so that it stands only forlogical implication. Although there is no objection to thus giving a technicalusage to an ordinary English word (even requiring in this technical usagethat ìmplication' be a metalinguistic relation between sentences), the pointis that relevance logicians by and large believe we are using ìmplies' in theordinary non-technical sense, in which a sentence like (3) might be truewithout there being any logical (or even necessary) implication from `Todayis Tuesday' to `This is Belgium'.

Relevance logicians are not themselves free of similar regimenting ten-dencies. Thus we tend to di�erentiate èntails' from ìmplies' on preciselythe ground that èntails', unlike ìmplies', stands only for necessary impli-cation [Meyer, 1966]. Some writings of Anderson and Belnap even suggesta more restricted usage for just logical implication, but we do not take thisseriously. There does not seem to be any more linguistic evidence for thusrestricting èntails' than there would be for ìmplies', though there may beat least more excuse given the apparently more technical history of èntails'(in its logical sense|cf. The oed).

This has been an explanation of, if not an apology for, the ways in whichrelevance logicians often express themselves. but it should be stressed thatthe reader need not accept all, or any, of this background in order to makesense of the basic aims of the relevance logic enterprise. Thus, e.g. thereader may feel that, despite protestations to the contrary, Anderson, Bel-nap and Co. are hopelessly confused about the relationships among èntails',ìmplies', and ìf-then', but still think that their system R provides a goodformalisation of the properties of ìf-then' (or at least ìf-then relevantly'),and that they system E does the same for some strict variant produced bythe modi�er `necessarily'.

One of the reasons the recent logical tradition has been motivated toinsist on the �erce distinction between implications and conditionals hasto do with the awkwardness of reading the so-called `material conditional'A! B as corresponding to any kind of implication (cf. [Quine, 1953]).

The material conditional A ! B can of course be de�ned as :A _ B,and it certainly does seem odd, modifying an example that comes by oraltradition from Anderson, to say that:

7. Picking a guinea pig up by its tail implies that its eyes will fall out.

just on the grounds that its antecedent is false (since guinea pigs have notails). But then it seems equally false to say that:

8. If one picks up a guinea pig by its tail, then its eyes will fall out.

And also both of the following appear to be equally false:

9. Scaring a pregnant guinea pig implies that all of her babies will beborn tailless.


10. If one scares a pregnant guinea pig, then all of her babies will be borntailless.

It should be noted that there are other ways to react to the oddity of sen-tences like the ones above other than calling them simply false. Thus thereis the reaction stemming from the work of Grice [1975] that says that at leastthe conditional sentences (8) and (10) above are true though nonethelesspragmatically odd in that they violate some rule based on conversationalco-operation to the e�ect that one should normally say the strongest thingrelevant, i.e. in the cases above, that guinea pigs have no tails (cf. [Fogelin,1978, p. 136 �.] for a textbook presentation of this strategy).

Also it should be noted that the theory of the `counterfactual' conditionaldue to Stalnaker{Thomason, D. K. Lewis and others (cf. Chapter [[??]] ofthis Handbook), while it agrees with relevance logic in �nding sentences like(8) (not (10) false, disagrees with relevance logic in the formal account itgives of the conditional.

It would help matters if there were an extended discussion of thesecompeting theories (Anderson{Belnap, Grice, Stalnaker-Thomason-Lewis),which seem to pass like ships in the night (can three ships do this withoutstrain to the image?) but there is not the space here. Such a discussionmight include an attempt to construct a theory of a relevant counterfactualconditional (if A were to be the case, then as a result B would be the case).The rough idea would be to use say The Routley{Meyer semantics for rel-evance logic (cf. Section 3.7) in place of the Kripke semantics for modallogic, which plays a key role in the Stalnaker{Thomason{Lewis semanticalaccount of the conditional (put the 3-placed alternativeness relation in therole of the usual 2-placed one). Work in this area is just starting. Seethe works of [Mares and Fuhrmann, 1995] and [Akama, 1997] which bothattempt to give semantics for relevant counterfactuals.

Also any discussion relating to Grice's work would surely make much ofthe fact that the theory of Grice makes much use of a basically unanalysednotion of relevance. One of Grice's chief conversational rules is `be relevant',but he does not say much about just what this means. One could look atrelevance logic as trying to say something about this, at least in the case ofthe conditional.

Incidentally, as Meyer has been at great pains to emphasise, relevancelogic gives, on its face anyway, no separate account of relevance. It is notas if there is a unary relevance operator (`relevantly').

One last point, and then we shall turn to more substantive issues. Or-thodox relevance logic di�ers from classical logic not just in having an ad-ditional logical connective (!) for the conditional. If that was the onlydi�erence relevance logic would just be an `extension' of classical logic, us-ing the notion of Haack [1974], in much the same way as say modal logicis an extension of classical logic by the addition of a logical connective �

RELEVANCE LOGIC 7

for necessity. The fact is (cf. Section 1.6) that although relevance logiccontains all the same theorems as classical logic in the classical vocabularysay, ^;_;: (and the quanti�ers), it nonetheless does not validate the sameinferences. Thus, most notoriously, the disjunctive syllogism (cf. Section 2)is counted as invalid. Thus, as Wolf [1978] discusses, relevance logic doesnot �t neatly into the classi�cation system of [Haack, 1974], and might bestbe called `quasi-extension' of classical logic, and hence `quasi-deviant'. In-cidentally, all of this applies only to `orthodox' relevance logic, and not tothe `classical relevance logics' of Meyer and Routley (cf. Section 3.11).

1.3 Hilbert-style Formulations

We shall discuss �rst the pure implicational fragments, since it is primarilyin the choice of these axioms that the relevance logics di�er one from theother. We shall follow the conventions of Anderson and Belnap [Andersonand Belnap, 1975], denoting by `R!' what might be called the `putativeimplicational fragment of R'. Thus R! will have as axioms all the axiomsof R that only involve the implication connective. That R! is in fact theimplicational fragment of R is much less than obvious since the possibilityexists that the proof of a pure implicational formula could detour in anessential way through formulas involving connectives other than implication.In fact Meyer has shown that this does not happen (cf. his Section 28.3.2 of[Anderson and Belnap, 1975]), and indeed Meyer has settled in almost everyinteresting case that the putative fragments of the well-known relevancelogics (at least R and E) are the same as the real fragments. (Meyer alsoshowed that this does not happen in one interesting case, RM, which weshall discuss below.)

For R! we take the rule modus ponens (A;A! B ` B) and the followingaxiom schemes.

A! A Self-Implication (1)

(A! B)! [(C ! A)! (C ! B)] Pre�xing (2)

[A! (A! B)]! (A! B) Contraction (3)

[A! (B ! C)]! [B ! (A! C)] Permutation: (4)

A few comments are in order. This formulation is due to Church [1951b]

who called it `The weak implication calculus'. He remarks that the axiomsare the same as those of Hilbert's for the positive implicational calculus (theimplicational fragment of the intuitionistic propositional calculus H) exceptthat (1) is replaced with

A! (B ! A) Positive Paradox: (10)

(Recent historical investigation by Do�sen [1992] has shown that Orlov con-structed an axiomatisation of the implication and negation fragment of R


in the mid 1920s, predating other known work in the area. Church andMoh, however, provided a Deduction Theorem (see Section 1.4) which isabsent from Orlov's treatment.)

The choice of the implicational axioms can be varied in a number of in-formative ways. Thus putting things quickly, (2) Pre�xing may be replacedby

(A! B)! [(B ! C)! (A! C)] SuÆxing. (20)

(3) Contraction may be replaced by

[A! (B ! C)]! [(A! B)! (A! C)] Self- Distribution, (30)

and (4) Permutation may be replaced by

A! [(A! B)! B] Assertion: (40)

These choices of implicational axioms are `isolated' in the sense that onechoice does not a�ect another. Thus

THEOREM 1. R! may be axiomatised with modus ponens, (1) Self-Implicationand any selection of one from each pair f(2); (20)g; f(3); (30)g, and f(4); (40)g.Proof. By consulting [Anderson and Belnap, 1975, pp. 79{80], and �ddling.

�

There is at least one additional variant of R! that merits discussion. Itturns out that it suÆces to have SuÆxing, Contraction, and the pair ofaxiom schemes

[(A! A)! B]! B Specialised Assertion, (4a)

A![(A! A)! A] Demodaliser. (4b)

Thus (4b) is just an instance of Assertion, and (4a) follows from Assertionby substitution A ! A for A and using Self-Implication to detach. That(4a) and (4b) together with SuÆxing and Contraction yield Assertion (and,less interestingly, Self-Implication) can be shown using the fact proven in[Anderson and Belnap, 1975, Section 8.3.3], that these yield (letting ~A ab-breviate A1 ! A2)

~A! [( ~A! B)! B] Restricted-Assertion. (400)

The point is that (4a) and (4b) in conjunction say that A is equivalent to

(A ! A) ! A, and so every formula A has an equivalent form ~A and so`Restricted Assertion' reduces to ordinary Assertion.2

2There are some subtleties here. Detailed analysis shows that both SuÆxing andPre�xing are needed to replace ~A with A (cf. Section 1.3). Pre�xing can be derived fromthe above set of axioms (cf. [Anderson and Belnap, 1975, pp. 77{78 and p. 26].

RELEVANCE LOGIC 9

Incidentally, no claim is made that this last variant of R! has the sameisolation in its axioms as did the previous axiomatisations. Thus, e.g.that SuÆxing (and not Pre�xing) is an axiom is important (a matrix ofJ. R. Chidgey's (cf. [Anderson and Belnap, 1975, Section 8.6]) can be usedto show this.

The system E of entailment di�ers primarily from the system R in thatit is a system of relevant strict implication. Thus E is both a relevance logicand a modal logic. Indeed, de�ning �A =df (A ! A) ! A one �nds Ehas something like the modality structure of S4 (cf. [Anderson and Belnap,1975, Sections 4.3 and 10]).

This suggests that E! can be axiomatised by dropping Demodaliser fromthe axiomatisation of R!, and indeed this is right (cf. [Anderson and Bel-nap, 1975, Section 8.3.3], for this and all other claims about axiomatisationsof E!).3

The axiomatisation above is a `�xed menu' in that Pre�xing cannot bereplaced with SuÆxing. There are other `�a la carte' axiomatisations in thestyle of Theorem 1.

THEOREM 2. E! may be axiomatised with modus ponens, Self-Implicationand any selection from each of the pairs fPre�xing, SuÆxingg, fContraction,Self-Distributiong and fRestricted-Permutation, Restricted-Assertiong (onefrom each pair).

Another implicational system of less central interest is that of `ticket en-tailment' T!. It is motivated by Anderson and Belnap [1975, Section 6] asderiving from some ideas of Ryle's about `inference tickets'. It was moti-vated in [Anderson, 1960] as `entailment shorn of modality'. The thoughtbehind this last is that there are two ways to remove the modal sting fromthe characteristic axiom of alethic modal logic, �A! A. One way is to addDemodaliser A ! �A so as to destroy all modal distinctions. The other isto drop the axiom�A! A. Thus the essential way one gets T! from E! isto drop Specialised Assertion (or alternatively to drop Restricted Assertionor Restricted Permutation, depending on which axiomatisation of E! onehas). But before doing so one must also add whichever one of Pre�xing andSuÆxing was lacking, since it will no longer be a theorem otherwise (this iseasiest to visualise if one thinks of dropping Restricted permutation, sincethis is the key to getting Pre�xing from SuÆxing and vice versa). Also(and this is a strange technicality) one must replace Self-Distribution withits permuted form:

(A! B)! [[A! (B ! C)]! (A! C)] Permuted Self-Distribution.

(300)

This is summarised in3The actual history is backwards to this, in that the system R was �rst axiomatised

by [Belnap, 1967a] by adding Demodaliser to E.


THEOREM 3 (Anderson and Belnap [Section 8.3.2, 1975]). T! is axioma-tised using Self-Implication, Pre�xing, SuÆxing, and either of fContraction,Permuted Self-Distributiong, with modus ponens.

There is a subsystem of E! called TW! (and P�W, and T�W inearlier nomenclature) axiomatised by dropping Contraction (which corre-sponds to the combinator W) from T!. This has obtained some interestbecause of an early conjecture of Belnap's (cf. [Anderson and Belnap, 1975,Section 8.11]) that A ! B and B ! A are both theorems of TW! onlywhen A is the same formula as B. That Belnap's Conjecture is now Bel-nap's Theorem is due to the highly ingenious (and complicated) work of E.P. Martin and R. K. Meyer [1982] (based on the earlier work of L. Powersand R. Dwyer). Martin and Meyer's work also highlights a system S! (forSyllogism) in which Self-Implication is dropped from TW!.

Moving on now to adding the positive extensional connectives ^ and _,in order to obtain R!;^;_ (denoted more simply as R+) one adds to R!

the axiom schemes

A ^ B ! A; A ^ B ! B Conjunction Elimination (5)

[(A! B) ^ (A! C)]! (A! B ^ C) Conjunction Introduction (6)

A! A _ B; B ! A _ B Disjunction Introduction (7)

[(A! C) ^ (B ! C)]! (A _ B ! C) Disjunction Elimination (8)

A ^ (B _ C)! (A ^ B) _ C Distribution (9)

plus the rule of adjunction (A;B ` A ^ B). One can similarly get thepositive intuitionistic logic by adding these all to H!.

Axioms (5){(8) can readily be seen to be encoding the usual eliminationand introduction rules for conjunction and disjunction into axioms, giving^ and _ what might be called `the lattice properties' (cf. Section 3.3). Itmight be thought that A ! (B ! A ^ B) might be a better encodingof conjunction introduction than (6), having the virtue that it allows forthe dropping of adjunction. This is a familiar axiom for intuitionistic (andclassical) logic, but as was seen by Church [1951b], it is only a hair's breadthaway from Positive Paradox (A! (B ! A)), and indeed yields it given (5)and Pre�xing. For some mysterious reason, this observation seemed toprevent Church from adding extensional conjunction/disjunction to whatwe now call R! (and yet the need for adjunction in the Lewis formulationsof modal logic where the axioms are al strict implications was well-known).

Perhaps more surprising than the need for adjunction is the need for ax-iom (9). It would follow from the other axioms if only we had Positive Para-dox among them. The place of Distribution in R is continually problematic.It causes inelegancies in the natural deduction systems (cf. Section 1.5) andis an obstacle to �nding decision procedures (cf. Section 4.8). Incidentally,all of the usual distributive laws follow from the somewhat `clipped' version

RELEVANCE LOGIC 11

(9).The rough idea of axiomatising E+ and T+ is to add axiom schemes

(5){(9) to E! and T!. This is in fact precisely right for T+, but for E+

one needs also the axiom scheme (remember �A =df (A! A)! A):

�A ^�B ! �(A ^ B) (10)

This is frankly an inelegance (and one that strangely enough disappears inthe natural deduction context of Section 1.5). It is needed for the inductiveproof that necessitation (` C ) ` �C) holds, handling he case where Cjust came by adjunction (cf. [Anderson and Belnap, 1975, Sections 21.2.2and 23.4]). There are several ways of trying to conceal this inelegance, butthey are all a little ad hoc. Thus, e.g. one could just postulate the rule ofnecessitation as primitive, or one could strengthen the axiom of RestrictedPermutation (or Restricted Assertion) to allow that ~A be a conjunction(A1 ! A1) ^ (A2 ! A2).

As Anderson and Belnap [1975, Section 21.2.2] remark, if propositionalquanti�cation is available, �A could be given the equivalent de�nition8p(p! p)! A, and then the o�ending (10) becomes just a special case ofConjunction Introduction and becomes redundant.

It is a good time to advertise that the usual zero-order and �rst-order rele-vance logics can be out�tted with a couple of optional convenience featuresthat come with the higher-priced versions with propositional quanti�ers.Thus, e.g. the propositional constant t can be added to E+ to play the roleof 8p(p! p), governed by the axioms.

(t! A)! A (11)

t! (A! A); (12)

and again (10) becomes redundant (since one can easily show (t ! A) $[(A! A)! A]).

Further, this addition of t is conservative in the sense that it leads tono new t-free theorems (since in any given proof t can always be replacedby (p1 ! p1) ^ � � � ^ (pn ! pn) where p1; : : : ; pn are all the propositionalvariables appearing in the proof | cf. [Anderson and Belnap, 1975]).

Axiom scheme (11) is too strong for T+ and must be weakened to

t: (11T)

In the context of R+, (11) and (11T) are interchangeable. and in R+, (12)may of course be permuted, letting us characterise t in a single axiom as`the conjunction of all truths':

A$ (t! A) (13)

(in E, t may be thought of as `the conjunction of all necessary truths').


`Little t' is distinguished from `big T ', which can be conservatively addedwith the axiom scheme

A! T (14)

(in intuitionistic or classical logic t and T are equivalent).Additionally useful is a binary connective Æ, labelled variously `inten-

sional conjunction', `fusion', `consistency' and `cotenability'. these lasttwo labels are appropriate only in the context of R, where one can de�neA ÆB =df :(A! :B). One can add Æ to R+ with the axiom scheme:

[(A ÆB)! C]$ [A! (B ! C)] Residuation (axiom): (15)

This axiom scheme is too strong for other standard relevance logics, butMeyer and Routley [1972] discovered that one can always add conservativelythe two way rule

(A ÆB)! C a ` A! (B ! C) Residutation (rule) (16)

(in R+ (16) yields (15)). Before adding negation, we mention the positivefragment B+ of a kind of minimal (Basic) relevance logic due to Routleyand Meyer (cf. Section 3.9). B+ is just like TW+ except for �nding theaxioms of Pre�xing and SuÆxing too strong and replacing them by rules:

A! B ` (C ! A)! (C ! B) Pre�xing (rule) (17)

A! B ` (B ! C)! (A! C) SuÆxing (rule) (18)

As for negation, the full systems R, E, etc. may be formed adding to theaxiom schemes for R+, E+, etc. the following 4

(A! :A)! :A Reductio (19)

(A! :B)! (B ! :A) Contraposition (20)

::A! A Double Negation. (21)

Axiom schemes (19) and (20) are intuitionistically acceptable negation prin-ciples, but using (21) one can derive forms of reductio and contrapositionthat are intuitionistically rejectable. Note that (19){(21) if added to H+

would give the full intuitionistic propositional calculus H.In R, negation can alternatively be de�ned in the style of Johansson, with

:A =df (A! f), where f is a false propositional constant, cf. [Meyer, 1966].Informally, f is the disjunction of all false propositions (the `negation' oft). De�ning negation thus, axiom schemes (19) and (20) become theorems

4Reversing what is customary in the literature, we use : for the standard negation ofrelevance logic, reserving � for the `Boolean negation' discussed in Section 3.11. We dothis so as to follow the notational policies of the Handbook.

RELEVANCE LOGIC 13

(being instances of Contraction and Permutation, respectively). But scheme(21) must still be taken as an axiom.

Before going on to discuss quanti�cation, we brie y mention a couple ofother systems of interest in the literature.

Given that E has a theory of necessity riding piggyback on it in thede�nition �A =df (A ! A) ! A, the idea occurred to Meyer of adding toR a primitive symbol for necessity � governed by the S4 axioms.

�A! A (�1)

�(A! B)! (�A! �B) (�2)

�A ^�B ! �(A ^ B) (�3)

�A! ��A; (�4)

and the rule of Necessitation (` A) ` �A).His thought was that E could be exactly translated into this system R�

with entailment de�ned as strict implication. That this is subtly not thecase was shown by Maksimova [1973] and Meyer [1979b] has shown how tomodify R� so as to allow for an exact translation.

Yet one more system of interest is RM (cf. Section 3.10) obtained byadding to R the axiom scheme

A! (A! A) Mingle: (22)

Meyer has shown somewhat surprisingly that the pure implicational systemobtained by adding Mingle to R is not the implicational fragment of RM,and he and Parks have shown how to axiomatise this fragment using aquite unintelligible formula (cf. [Anderson and Belnap, 1975, Section 8.18]).Mingle may be replaced equivalently with the converse of Contraction:

(A! B)! (A! (A! B)) Expansion: (23)

Of course one can consider `mingled' versions of E, and indeed it was inthis context that McCall �rst introduced mingle, albeit in the strict form(remember ~A = A1 ! A2),

~A! ( ~A! ~A) ~Mingle (24)

(cf. [Dunn, 1976c]).We �nish our discussion of axiomatics with a brief discussion of �rst-

order relevance logics, which we shall denote by RQ, EQ, etc. We shallpresuppose a standard de�nition of �rst-order formula (with connectives:;^;_;! and quanti�ers 8; 9). For convenience we shall suppose that wehave two denumerable stocks of variables: the bound variables x; y, etc.


and the free variables (sometimes called parameters) a; b, etc. The boundvariables are never allowed to have unbound occurrences.

The quanti�er laws were set down by Anderson and Belnap in accord withthe analogy of the universal quanti�er with a conjunction (or its instances),and the existential quanti�er as a disjunction. In view of the validity ofquanti�er interchange principles, we shall for brevity take only the universalquanti�er 8 as primitive, de�ning 9xA =df :8x:A. We thus need

8xA! A(a=x) 8-elimination (25)

8x(A! B)! (A! 8xB) 8- introduction (26)

8x(A _ B)! A _ 8xB Con�nement: (27)

If there are function letters or other term forming operators, then (25)should be generalised to 8xA ! A(t=x), where t is any term (subject toour conventions that the `bound variables' x; y, etc. do not occur (`free')in it). Note well that because of our convention that `bound variables' donot occur free, the usual proviso that x does not occur free in A in (26)and (27) is automatically satis�ed. (27) is the obvious `in�nite' analogyof Distribution, and as such it causes as many technical problems for RQas does Distribution for R (cf. Section 4.8). Finally, as an additional rulecorresponding to adjunction, we need:

A(a=x)

8xAGeneralisation: (28)

There are various more or less standard ways of varying this formulation.Thus, e.g. (cf. Meyer, Dunn and Leblanc [1974]) one can take all universalgeneralisations of axioms, thus avoiding the need for the rule of Generalisa-tion. Also (26) can be `split' into two parts:

8x(A! B)! (8xA! 8xB) (26a)

A! 8xA Vacuous Quanti�cation (26b)

(again note that if we allowed x to occur free we would have to require thatx not be free in A).

The most economical formulation is due to Meyer [1970]. It uses onlythe axiom scheme of 8-elimination and the rule.

A! B _ C(a=x)

A! B _ 8xC(a cannot occur in A or B) (29)

which combines (26){(28).

1.4 Deduction Theorems in Relevance Logic

Let X be a formal system, with certain formulas of X picked out as axiomsand certain (�nitary) relations among the formulas of X picked out as rules.

RELEVANCE LOGIC 15

(For the sake of concreteness, X can be thought of as any of the Hilbert-style systems of the previous section.) Where � is a list of formulas of X(thought of as hypotheses) it is customary to de�ne a deduction from � tobe a sequence B1; : : : ; Bn, where for each Bi(1 � i � n), either (1) Bi is in�, or (2) B is an axiom of X, or (3) Bi `follows from' earlier members ofthe sequence, i.e. R(Bj1 ; : : : ; Bjk ; Bi) holds for some (k+ 1)|any rule R ofX and Bj1 ; : : : ; Bjk all precede Bi in the sequence B1; : : : ; Bn. A formulaA is then said to be deducible from � just in case there is some deductionfrom � terminating in A. We symbolise this as � `X A (often suppressingthe subscript).

A proof is of course a deduction from the empty set, and a theorem isjust the last item in a proof. There is the well-known

Deduction Theorem (Herbrand). If A1; : : : ; An; A `H! B, then wehave also A1; : : : ; An `H! A! B.

This theorem is proven in standard textbooks for classical logic, but thestandard inductive proof shows that in fact the Deduction Theorem holdsfor any formal system X having modus ponens as its sole rule and H! �X (i.e. each instance of an axiom scheme of H! is a theorem of X). IndeedH! can be motivated as the minimal pure implicational calculus havingmodus ponens as its sole rule and satisfying the Deduction Theorem. Thisis because the axioms of H! can all be derived as theorems in any formalsystem X using merely modus ponens and the supposition that X satis�esthe Deduction Theorem. Thus consider as an example:

(1) A;B ` A De�nition of `(2) A ` B ! A (1), Deduction Theorem(3) ` A! (B ! A) (2), Deduction Theorem:

Thus the most problematic axiom of H! has a simple `a priori deduc-tion', indeed one using only the Deduction Theorem, not even modus ponens(which is though needed for more sane axioms like Self-Distribution).

It might be thought that the above considerations provide a very powerfulargument for motivating intuitionistic logic (or at least some logic having hesame implicational fragment) as The One True Logic. For what else shouldan implication do but satisfy modus ponens and the Deduction Theorem?

But it turns out that there is another sensible notion of deduction. This iswhat is sometimes called a relevant deduction.(Anderson and Belnap [1975,Section 22.2.1] claim that this is the only sensible notion of deduction, butwe need not follow them in that). If there is anything that sticks out in thea priori deduction of Positive Paradox above it is that in (1), B was notused in the deduction of A.

A number of researchers have been independently bothered by this pointand have been motivated to study a relevant implication that goes hand in


hand with a notion of relevant deduction. This, in this manner Moh [1950]

and Church [1951b] came up with what is in e�ect R!. And Andersonand Belnap [1975, p. 261] say \In fact, the search for a suitable deductiontheorem for Ackermann's systems : : : provided the impetus leading us tothe research reported in this book." This research program begun in the late1950s took its starting point in the system(s) of Ackermann [1956], and thebold stroke separating the Anderson{Belnap system E from Ackermann'ssystem �0 was basically the dropping of Ackermann's rule so as to havean appropriate deduction theorem (cf. Section 2.1).

Let us accordingly de�ne a deduction of B from A1; : : : ; An to be relevantwith respect to a given hypothesis Ai just in case Ai is actually used inthe given deduction of B in the sense (paraphrasing [Church, 1951b]) thatthere is a chain of inferences connecting Ai with the �nal formula B. Thislast can be made formally precise in any number of ways, but perhaps themost convenient is to ag Ai with say a ] and to pass the ag along inthe deduction each time modus ponens is applied to two items at least oneof which is agged. It is then simply required that the last step of thededuction (B) be agged. Such devices are familiar from various textbookpresentations of classical predicate calculus when one wants to keep trackwhether some hypothesis Ai(x) was used in the deduction of some formulaB(x) to which one wants to apply Universal Generalisation.

We shall de�ne a deduction of B from A1; : : : ; An to be relevant sim-pliciter just in case it is relevant with respect to each hypothesis Ai. Apractical way to test for this is to ag each Ai with a di�erent ag (say thesubscript i) and then demand that all of the ags show up on the last stepB.

We can now state a version of the

Relevant Deduction Theorem (Moh, Church). If there is a deductionin R! of B from A1; : : : ; An; A that is relevant with respect to A, then thereis a deduction in R! of A ! B from A1; : : : ; An. Furthermore the newdeduction will be `as relevant' as the old one, i.e. any Ai that was used inthe given deduction will be used in the new deduction.

Proof. Let the given deduction be B1; : : : ; Bk, and let it be given with aparticular analysis as to how each step is justi�ed. By induction we showfor each Bi that if A was used in obtaining Bi (Bi is agged), then there isa deduction of A! Bi from A1; : : : ; An, and otherwise there is a deductionof Bi from those same hypotheses. The tedious business of checking thatthe new deduction is as relevant as the old one is left to the reader. Wedivide up cases depending on how the step Bi is justi�ed.

Case 1. Bi was justi�ed as a hypothesis. Then neither Bi is A or itis some Aj . But A ! A is an axiom of R! (and hence deducible fromA1; : : : ; An), which takes care of the �rst alternative. And clearly on thesecond alternative Bi is deducible from A1; : : : ; An (being one of them).

RELEVANCE LOGIC 17

Case 2. Bi was justi�ed as an axiom. Then A was not used in obtainingBi, and of course Bi is deducible (being an axiom).

Case 3. Bi was justi�ed as coming from preceding steps Bj ! Bi andBj by modus ponens. There are four subcases depending on whether A wasused in obtaining the premises.

Subcase 3.1. A was used in obtaining both Bj ! Bi and Bj . Then by in-ductive hypothesis A1; : : : ; An `R! A ! (Bj ! Bi) and A1; : : : ; An `R!A! Bj . So A! B may be obtained using the axiom of Self-Distribution.

Subcase 3.2. A was used in obtaining Bj ! Bi but not Bj . Use theaxiom of Permutation to obtain A! Bi from A! (Bj ! Bi) and Bj .

Subcase 3.3. A was not used in obtaining Bj ! Bi but was used for Bj .Use the axiom of Pre�xing to obtain A! Bi from Bj ! Bi and A! Bj .

Subcase 3.4. A was not used in obtaining either Bj ! Bi nor Bj . ThenBi follows form these using just modus ponens.

Incidentally, R! can easily be veri�ed to be the minimal pure implica-tional calculus having modus ponens as sole rule and satisfying the RelevantDeduction Theorem, since each of the axioms invoked in the proof of thistheorem can be easily seen to be theorems in any such system (cf. the nextsection for an illustration of sorts).

There thus seem to be at least two natural competing pure implicationallogics R! and H!, di�ering only in whether one wants one's deductionsto be relevant or not.5 �

Where does the Anderson{Belnap's [1975] preferred system E! �t intoall of this? The key is that the implication of E! is both a strict anda relevant implication (cf. Section 1.3 for some subtleties related to thisclaim). As such, and since Anderson and Belnap have seen �t to give itthe modal structure of the Lewis system S4, it is appropriate to recall theappropriate deduction theorem for S4.

Modal Deduction Theorem [Barcan Marcus, 1946] If A1 ! B1; : : : ; An !Bn; A `S4 B (! here denotes strict implication), then A1 ! B1; : : : ; An !Bn `S4 A! B.

The idea here is that in general in order to derive the strict (necessary)implication A ! B one must not only be able to deduce B from A andsome other hypotheses but furthermore those other hypotheses must besupposed to be necessary. And in S4 since Ai ! Bj is equivalent to �(Ai !Bj), requiring those additional hypotheses to be strict implications at leastsuÆces for this.

Thus we could only hope that E! would satisfy the

5This seems to di�er from the good-humoured polemical stand of Anderson and Belnap[1975, Section 22.2.1], which says that the �rst kind of `deduction', which they call(pejoratively) `OÆcial deduction', is no kind of deduction at all.


Modal Relevant Deduction Theorem [Anderson and Belnap, 1975]

If there is a deduction in E! of B from A1 ! B1; : : : ; An ! Bn; A that isrelevant with respect to A, then there is a deduction in E! of A! B fromA1 ! B1; : : : ; An ! Bn that is as relevant as the original.

The proof of this theorem is somewhat more complicated than its un-modalised counterpart which we just proved (cf. [Anderson and Belnap,1975, Section 4.21] for a proof).

We now examine a subtle distinction (stressed by Meyer|see, for ex-ample, [Anderson and Belnap, 1975, pp. 394{395]), postponed until nowfor pedagogical reasons. We must ask, how many hypotheses can danceon the head of a formula? The question is: given the list of hypothesesA, A, do we have one hypothesis or two? When the notion of a deduc-tion was �rst introduced in this section and a `list' of hypotheses � wasmentioned, the reader would naturally think that this was just informallanguage for a set. And of course the set fA;Ag is identical to the setfAg. Clearly A is relevantly deducible from A. The question is whetherit is so deducible from A;A. We have then two di�erent criteria of use,depending on whether we interpret hypotheses as grouped together intolists that distinguish multiplicity of occurrences (sequences)6 or sets. Thisissue has been taken up elsewhere of late, with other accounts of deduc-tion appealing to `resource consciousness' [Girard, 1987; Troelstra, 1992;Schroeder-Heister and Do�sen, 1993] as motivating some non-classical log-ics. Substructural logics in general appeal to the notion that the number oftimes a premise is used, or even more radically, the order in which premisesare used, matter.

At issue in R and its neighbours is whether A! (A! A) is a correct rel-evant implication (coming by two applications of `The Deduction Theorem'from A;A ` A). This is in fact not a theorem of R, but it is the character-istic axiom of RM (cf. Section 1.3). So it is important that in the RelevantDeduction Theorem proved for R! that the hypotheses A1; : : : ; An be un-derstood as a sequence in which the same formula may occur more thanonce. One can prove a version of the Relevant Deduction Theorem withhypotheses understood as collected into a set for the system RMO!, ob-tained by adding A! (A! A) to R! (but the reader should be told thatMeyer has shown that RMO!, is not the implicational fragment of RM,cf. [Anderson and Belnap, 1975, Section 8.15]).7

6Sequences are not quite the best mathematical structures to represent this groupingsince it is clear that the order of hypotheses makes no di�erence (at least in the case ofR). Meyer and McRobbie [1979] have investigated `�resets' (�nitely repeatable sets) asthe most appropriate abstraction.

7Arnon Avron has defended this system, RMO!, as a natural way to charac-terise relevant implication [Avron, 1986; Avron, 1990a; Avron, 1990b; Avron, 1990c;Avron, 1992]. In Avron's system, conjunction and disjunction are intensional connec-tives, de�ned in terms of the implication and negation of RMO!. As a result, they do

RELEVANCE LOGIC 19

Another consideration pointing to the naturalness of R! is its connectionto the �I-calculus. A formula is a theorem of R! if and only if it is thetype of a closed term of the �I-calculus as de�ned by Church. A �I term isa � term in which every lambda abstraction binds at least one free variable.So, �x:�y:xy has type A ! �

(A ! B) ! B�, and so, is a theorem of R!,

while �x:�y:x, has type A ! (B ! A), which is an intuitionistic theorem,but not an R! theorem. This is re ected in the � term, in which the �ydoes not bind a free variable.

We now brie y discuss what happens to deduction theorems when thepure implication systems R! and E! are extended to include other con-nectives, especially ^. R will be the paradigm, its situation extendingstraight-forwardly to E. The problem is that the full system R seems notto be formulable with modus ponens as the sole rule; there is also need foradjunction (A;B ` A ^B) (cf. Section 1.3).

Thus when we think of proving a version of the Relevant Deduction Theo-rem for the full system R, it would seem that we are forced to think throughonce more the issue of when a hypothesis is used, this time with relation toadjunction. It might be thought that the thing to do would be to pass the ag ] along over an application of adjunction so that A^B ends up aggedif either of the premises A or B was agged, in obvious analogy with thedecision concerning modus ponens.

Unfortunately, that decision leads to disaster. For then the deductionA;B ` A ^ B would be a relevant one (both A and B would be ùsed'),and two applications of `The Deduction Theorem' would lead to the thesisA! (B ! A ^B), the undesirability of which has already been remarked.

A more appropriate decision is to count hypotheses as used in obtainingA^B just when they were used to obtain both premises. This corresponds tothe axiom of Conjunction Introduction (C ! A)^(C ! B)! (C ! A^B),which thus handles the case in the inductive proof of the deduction theoremwhen the adjunction is applied. This decision may seem ad hoc (perhapsùse' simpliciter is not quite the right concept), but it is the only decision tobe made unless one wants to say that the hypothesis A can (in the presenceof the hypothesis B) be ùsed' to obtain A^B and hence B (passing on the ag from A this way is something like laundering dirty money).

This is the decision that was made by Anderson and Belnap in the con-text of natural deduction systems (see next section), and it was applied byKron [1973; 1976] in proving appropriate deduction theorems for R, E (andT). It should be said that the appropriate Deduction Theorem requires si-multaneous agging of the hypothesis (distinct ags being applied to eachformula occurrence, say using subscripts in the manner of the `practicalsuggestion' after our de�nition of relevant deduction for R!), with the re-quirement that all of the subscripts are passed on to the conclusion. So the

not have all of the distributive lattice properties of traditional relevance logics.


Deduction Theorem applies only to fully relevant deductions, where everypremise is used (note that no such restriction was placed on the RelevantDeduction Theorem for R!).

An alternative stated in Meyer and McRobbie [1979] would be to adjustthe de�nition of deduction, modifying clause (2) so as to allow as a step ina deduction any theorem (not just axiom) of R, and to restrict clause (3) sothat the only rule allowed in moving to later steps is modus ponens.8 Thisis in e�ect to restrict adjunction to theorems, and reminds one of similarrestrictions in the context of deduction theorems of similarly restricting therules of necessitation and universal generalisation. It has the virtue thatthe Relevant Deduction Theorem and its proof are the same as for R!.(Incidentally, Meyer's and Kron's sense of deduction coincide when all ofA1; : : : ; An are used in deducing B; this is obvious in one direction, and lessthan obvious in the other.)

There are yet two other versions of the deduction theorem that meritdiscussion in the context of relevance logic (relevance logic, as Meyer oftenpoints out, allows for many distinctions).

First in Belnap [1960b] and Anderson and Belnap [1975], there is atheorem (stated for E, but we will state it for our paradigm R) calledThe Entailment Theorem, which says that A1; : : : ; An `entails' B i� `R(A1 ^ : : : ^ An) ! B. A formula B is de�ned in e�ect to be entailed by

hypothesis A1; : : : ; An just in case there is a deduction of B using their con-junction A1 ^ : : :^An. Adjunction is allowed, but subject to the restrictionthat the conjunctive hypothesis was used in obtaining both premises. TheEntailment Theorem is clearly implied by Kron's version of the DeductionTheorem.

The last deduction theorem for R we wish to discuss is the

Enthymematic Deduction Theorem (Meyer, Dunn and Leblanc [1974]).If A1; : : : ; An; A `R B, then A1; : : : ; An `R A ^ t! B.

Here ordinary deducibility is all that is at issue (no insistence on thehypotheses being used). It can either be proved by induction, or crankedout of one of the more relevant versions of the deduction theorem. Thus itfalls out of the Entailment Theorem that

`R X ^ A ^ T ! B;

where X is the conjunction of A1; : : : ; An, and T is the conjunction of allthe axioms of R used in the deduction of B. But since `R t! T , we have`R X ^ A ^ t! B.

8Of course this requires we give an independent characterisation of proof (and theo-rem), since we can no longer de�ne a proof as a deduction from zero premisses. We thusde�ne a proof as a sequence of formulas, each of which is either an axiom or follows frompreceding items by either modus ponens or adjunction (!).

RELEVANCE LOGIC 21

However, the following R theorem holds:

`R (X ^ A ^ t! B)! (X ^ t! (A ^ t! B)):

So `R X ^ t! (A ^ t! B), which leads (using `R t) to X `R A ^ t! B,which dissolving the conjunction gives the desired

A1; : : : ; An `R A ^ t! B:

In view of the importance of the notion, let us symbolise A ^ t ! Bas A !t B. This functions as a kind of `enthymematic implication' (Aand some truth really implies B) and there will be more about Anderson,Belnap and Meyer's investigations of this concept in Section 1.7. Let ussimply note now that in the context of deduction theorems, it functions likeintuitionistic implication, and allows us in R! to have two di�erent kindsof implication, each well motivated in its relation to the two di�erent kindsof deducibility (ordinary and relevant).9 For a more extensive discussionof deduction theorems in relevance logics and related systems, more recentpapers by Avron [1991] and Brady [1994] should be consulted.

1.5 Natural Deduction Formulations

We shall be very brief about these since natural deduction methods areamply discussed by Anderson and Belnap [1975], where such methods in factare used s a major motivation for relevance logic. Here we shall concentrateon a natural deduction system NR for R.

The main idea of natural deduction (cf. Chapters [[were I.1 and I.2]] of theHandbook) of course is to allow the making of temporary hypotheses, withsome device usually being provided to facilitate the book-keeping concerningthe use of hypotheses (and when their use is `discharged'). Several textbooks(for example, [Suppes, 1957] and [Lemmon, 1965])10 have used the deviceof in e�ect subscripting each hypothesis made with a distinct numeral, andthen passing this numeral along with each application of a rule, thus keepingtrack of which hypothesis are used. When a hypothesis is discharged, thesubscript is dropped. A line obtained with no subscripts is a `theorem' sinceit depends on no hypotheses.

Let us then let �; �, etc. range over classes of numerals. The rules for !are then naturally:

A! B�

A�

B�[�

[!E]

Afkg...B�

A! B��fkg (provided k 2 �)

[!I ]

9In E enthymematic implication is like S4 strict implication. See [Meyer, 1970a].10The idea actually originates with [Feys and Ladri�ere, 1955].


Two fussy, really incidental remarks must be made. First, in the rule !Eit is to be understood that the premises need not occur in the order listed,nor need they be adjacent to each other or to the conclusion. Otherwise wewould need a rule of `Repetition', which allows the repeating of a formulawith its subscripts as a later line. (Repetition is trivially derivable givenour `non-adjacent' understanding of!E|in order to repeat A�, just proveA ! A and apply !E.) Second, it is understood that we have what onemight call a rule of `Hypothesis Introduction': anytime one likes one canwrite a formula as a line with a new subscript (perhaps most conveniently,the line number).

Now a non-fussy remark must be made, which is really the heart of thewhole matter. In the rule for !I , a proviso has been attached which hasthe e�ect of requiring that the hypothesis A was actually used in obtainingB. This is precisely what makes the implication relevant (one gets theintuitionistic implication system H! if one drops this requirement). Thereader should �nd it instructive to attempt a proof of Positive Paradox(A ! (B ! A)) and see how it breaks down for NR! (but succeeds inNH!. The reader should also construct proofs in NR! of all the axiomsin one of the Hilbert-style formulations of R! from Section 1.3.

Then the equivalence of R! in its Hilbert-style and natural deductionformulations is more or less self-evident given the Relevant Deduction The-orem (which shows that the rule! I can be `simulated' in the Hilbert-stylesystem, the only point at issue).

Indeed it is interesting to note that Lemmon [1965], who seems to havethe same proviso on !I that we have for NR! (his actual language isa bit informal), does not prove Positive Paradox until his second chapteradding conjunction (and disjunction) to the implication-negation systemhe developed in his �rst chapter. His proof of Positive Paradox depends�nally upon an ìrrelevant' Î rule. The following is perhaps the moststraightforward proof in his system (di�ering from the proof he actuallygives):

(1) A1 Hyp(2) B2 Hyp(3) A ^ B1;2 1; 2;Î?(4) A1;2 3;Ê(5) B ! A1 2; 4;! I(6) A! (B ! A) 1; 5;! I .

We think that the manoeuvre used in getting B's 2 to show up attached toA in line (4) should be compared to laundering dirty money by running itthrough an apparently legitimate business. The correct `relevant' versions

RELEVANCE LOGIC 23

of the conjunction rules are instead

A�

B�

A ^ B�

[Î ]A ^ B�

A�

A ^ B�

B�[Ê]

What about disjunction? In R (also E, etc.) one has de Morgan's Lawsand Double Negation, so one can simply de�ne A _B = :(:A ^ :B). Onemight think that settling down in separate int-elim rules for _ would thenonly be a matter of convenience. Indeed, one can �nd in [Anderson andBelnap, 1975] in e�ect the following rules:

A�

A _ B�

B�

A _B�[_I ]

A _ B�

...Ak

...C�[fkgBh

...C�[fhgC�[�

[_E]

But (as Anderson and Belnap point out) these rules are insuÆcient. >Fromthem one cannot derive the following

A ^ (B _ C)�Distribution:

(A ^ B) _ C�And so it must be taken as an additional rule (even if disjunction is de�nedfrom conjunction and negation).

This is clearly an unsatisfying, if not unsatisfactory, state of a�airs. Thecustomary motivation behind int-elim rules is that they show how a connec-tive may be introduced into and eliminated from argumentative discourse(in which it has no essential occurrence), and thereby give the connective'srole or meaning. In this context the Distribution rule looks very much tobe regretted.

One remedy is to modify the natural deduction system by allowing hy-potheses to be introduced in two di�erent ways, `relevantly' and ìrrele-vantly'. The �rst way is already familiar to us and is what requires asubscript to keep track of the relevance of the hypothesis. It requires thatthe hypotheses introduced this way will all be used to get the conclusion.The second way involves only the weaker promise that at least some of thehypotheses so introduced will be used. This suggestion can be formalised by


allowing several hypotheses to be listed on a line, but with a single relevancenumeral attached to them as a bunch. Thus, schematically, an argument ofthe form

(1) A;B1

(2) C;D2

...(k) E1;2

should be interpreted as establishing

A ^ B ! (C ^D ! E):

Now the natural deduction rules must be stated in a more general formallowing for the fact that more than one formula can occur on a line. Keyamong these would be the new rule:

�; A _ B�

...�; Ak

...��[fkg

�; Bl

...��[flg

��[�

[_E0]

It is fairly obvious that this rule has Distribution built into it. Of course,other rules must be suitably modi�ed. It is easiest to interpret the formulason a line as grouped into a set so as not to have to worry about `structuralrules' corresponding to the commutation and idempotence of conjunction.

The rules !I;!E;_I;_E;Î , and Ê can all be left as they were (orexcept for !I and !E, trivially generalised so as to allow for the fact thatthe premises might be occurring on a line with several other ìrrelevant'premises), but we do need one new structural rule:

��

�;��

[Comma I ]

Once we have this it is natural to take the conjunction rules in `Ketonenform':

�; A;B�[Î 0]

�; A ^B�

RELEVANCE LOGIC 25

�; A ^B�[^E0]

�; A;B�

with the rule

�;��[Comma E]

��

It is merely a tedious exercise for the reader to show that this new systemN 0R is equivalent to NR. Incidentally, N 0R was suggested by re ectionupon the Gentzen System LR+ of Section 4.9.

Before leaving the question of natural deduction for R, we would like tomention one or two technical aspects. First, the system of Prawitz [1965]

di�ers from R in that it lacks the rule of Distribution. This is perhapscompensated for by the fact that Prawitz can prove a normal form theoremfor proofs in his system. A di�erent system yet is that of [Pottinger, 1979],based on the idea that the correct ^I rule is

A�

B�

A ^ B�[�

He too gets a normal form theorem. We conjecture that some appropriatenormal form theorem is provable for the system N 0R+ on the well-knownanalogy between cut-elimination and normalisation and the fact that cut-elimination has been proven for LR+ (cf. Section 4.9). Negation thoughwould seem to bring extra problems, as it does when one is trying to add itto LR+.

One last set of remarks, and we close the discussion of natural deduction.The system NR above di�ers from the natural deduction system for Rof Anderson and Belnap [1975]. Their system is a so-called `Fitch-style'formalism, and so named FR. The reader is presumed to know that inthis formalism when a hypothesis is introduced it is thought of as startinga subproof, and a line is drawn along the left of the subproof (or a box isdrawn around the subproof, or some such thing) to demarcate the scopeof the hypothesis. If one is doing a natural deduction system for classicalor intuitionistic logic, subproofs or dependency numerals can either one beused to do essentially the same job of keeping track the use of hypotheses(though dependency numerals keep more careful track, and that is why theyare so useful for relevant implication).

Mathematically, a Fitch-style proof is a nested structure, representingthe fact that subproofs can contain further subproofs, etc. But once onehas dependency numerals, this extra structure, at least for R, seems otiose,and so we have dispensed with it. The story for E is more complex, since


on the Anderson and Belnap approach E di�ers from R only in what isallowed to be `reiterable' into subproof. Since implication in E is necessaryas well as relevant, the story is that in deducing B from A in order to showA ! B, one should only be allowed to use items that have been assumedto be necessarily true, and that these can be taken to be formulas of theform C ! D. So only formulas of this form can be reiterated for use in thesubproof from A to B. Working out how best to articulate this idea usingonly dependency numerals (no lines, boxes, etc.) is a little messy. Thisconcern to keep track of how premises are used in a proof by way of labelshas been taken up in a general way by recent work on Labelled DeductiveSystems [D'Agostino and Gabbay, 1994; Gabbay, 1997].

We would be remiss not to mention other formulations of natural deduc-tion systems for relevance logics and their cousins. A di�erent generalisa-tion of Hunter's natural deduction systems (which follows more closely theGentzen systems for positive logics | see Section 4.9) is in [Read, 1988;Slaney, 1990].11

1.6 Basic Formal Properties of Relevance Logic

This section contains a few relatively simple properties of relevance logics,proofs for which can be found in [Anderson and Belnap, 1975]. With oneexception (the `Ackermann Properties'|see below), these properties all holdfor both the system R and E, and indeed for most of the relevance logicsde�ned in Section 1.3. For simplicity, we shall state these properties forsentential logics, but appropriate versions hold as well for their �rst-ordercounterparts.

First we examine the Replacement Theorem For both R and E,

` (A$ B) ^ t! (�(A)$ �(B)):

Here �(A) is any formula with perhaps some occurrences of A and �(B) isthe result of perhaps replacing one or more of those occurrences by B. Theproof is by a straightforward induction on the complexity of �(A), and oneclear role of the conjoined t is to imply � ! � when �(= �(A)) containsno occurrences of A, or does but none of them is replaced by B. It mightbe thought that if these degenerate cases are ruled out by requiring thatsome actual occurrence of A be replaced by B, then the need for t wouldvanish. This is indeed true for the implication-negation (and of course thepure implication) fragments of R and E, but not for the whole systems invirtue of the non-theoremhood of what V. Routley has dubbed `Factor':

11The reader should be informed that still other natural deduction formalisms for Rof various virtues can be found in [Meyer, 1979b] and [Meyer and McRobbie, 1979].

RELEVANCE LOGIC 27

1. (A! B)! (A ^ �! B ^ �).

Here the closest one can come is to

2. (A! B) ^ t! (A ^ �! B ^ �),

the conjoined g giving the force of having � ! � in the antecedent, andthe theorem (A ! B) ^ (� ! �) ! (A ^ � ! B ^ �) getting us home.(2) of course is just a special case of the Replacement Theorem. Of more`relevant' interest is the

Variable Sharing Property. If A! B is a theorem of R (or E), thenthere exists some sentential variable p that occurs in both A and B. Thisis understood by Anderson and Belnap as requiring some commonality ofmeaning between antecedent and consequent of logically true relevant impli-cations. The proof uses an ingenious logical matrix, having eight values, forwhich see [Anderson and Belnap, 1975, Section 22.1.3]. There are discussedboth the original proof of Belnap and an independent proof of Don�cenko,and strengthening by Maksimova. Of modal interest is the

Ackermann Property. No formula of the form A ! (B ! C) (A con-taining no !) is a theorem of E. The proof again uses an ingenious matrix(due to Ackermann) and has been strengthened by Maksimova (see [Ander-son and Belnap, 1975, Section 22.1.1 and Section 22.1.2]) (contributed byJ. A. Co�a) on `fallacies of modality'.

1.7 First-degree Entailments

A zero degree formula contains only the connectives ^;_, and :, and canbe regarded as either a formula of relevance logic or of classical logic, as onepleases. A �rst degree implication is a formula of the form A ! B, whereboth A and B are zero-degree formulas: Thus �rst degree implications canbe regarded as either a restricted fragment of some relevance logic (say Ror E) or else as expressing some metalinguistic logical relation between twoclassical formulas A and B. This last is worth mention, since then even aclassical logician of Quinean tendencies (who remains unconverted by theconsiderations of Section 1.2 in favour of nested implications) can still take�rst degree logical relevant implications to be legitimate.

A natural question is what is the relationship between the provable �rst-degree implications of R and those of E. It is well-known that the corre-sponding relationship between classical logic and some normal modal logic,say S4 (with the ! being the material conditional and strict implication,respectively), is that they are identical in their �rst degree fragments. Thesame holds of R and E (cf. [Anderson and Belnap, 1975, Section 2.42]).

This fragment, which we shall call Rfde (Anderson and Belnap [1975] callit Efde) is stable (cf. [Anderson and Belnap, 1975, Section 7.1]) in the sense


that it can be described from a variety of perspectives. For some semanticalperspectives see Sections 3.3 and 3.4. We now consider some syntacticalperspectives of more than mere òrthographic' signi�cance.

The perhaps least interesting of these perspectives is a `Hilbert-style'presentation of Rfde (cf. [Anderson and Belnap, 1975, Section 15.2]). Ithas the following axioms:

3. A ^ B ! A;A ^ B ! B Conjunction Elimination

4. A! A _B;B ! A _ B Disjunction Introduction

5. A ^ (B _ C)! (A ^ B) _ C Distribution

6. A! ::A;::A ! A Double Negation

It also has gobs of rules:

7. A! B;B ! C ` A! C Transitivity

8. A! B;A! C ` A! B ^ C Conjunction Introduction

9. A! C;B ! C ` A _B ! C Disjunction Introduction

10. A! B ` :B ! :A Contraposition.

More interesting is the characterisation of Anderson and Belnap [1962b;1975] of Rfde as `tautological entailments'. The root idea is to consider �rstthe `primitive entailments'.

11. A1 ^ : : : Âm ! B1 _ : : : _Bn,

where each Ai and Bj is either a sentential variable or its negate (an àtom')and make it a necessary and suÆcient criterion for such a primitive entail-ment to hold that same Ai actually be identically the same formula as someBj (that the entailment be `tautological' in the sense that Ai is repeated).This rules out both

12. p ^ :p! q,

13. p! q _ :q,where there is no variable sharing, but also such things as

14. p ^ :p ^ q ! :q,where there is (of course all of (12){(14) are valid classically, where a prim-itive entailment may hold because of atom sharing or because either theantecedent is contradictory or else the consequent is a logical truth).

Now the question remains as to which non-primitive entailments to countas valid. Both relevance logic and classical logic agree on the standard

RELEVANCE LOGIC 29

count as valid. Both relevance logic and classical logic agree on the stan-dard `normal form equivalences': commutation, association, idempotence,distribution, double negation, and de Morgan's laws. So the idea is, givena candidate entailment A! B, by way of these equivalences, A can be putinto disjunctive normal form and B may be put into conjunctive normalform, reducing the problem to the question of whether the following is avalid entailment:

15. A1 _ � � � _ Ak ! B1 ^ � � � ^ Bh.

But simple considerations (on which both classical and relevance logicagree) having to do with conjunction and disjunction introduction and elim-ination show that (15) holds if for each disjunct Ai and conjunct Bj , theprimitive entailment Ai ! Bj is valid. For relevance logic this means thatthere must be atom sharing between the conjunction Ai and the disjunctionBj .

This criterion obviously counts the Disjunctive Syllogism

16. :p ^ (p _ q)! q,

as an invalid entailment, for using distribution to put its antecedent intodisjunctive normal form, (16) is reduced to

160 (:p ^ p) _ (:p ^ q)! q.

But by the criterion of tautological entailments,

17. :p ^ p! q,

which is required for the validity of (160), is rejected.Another pleasant characterisation of Rfde is contained in [Dunn, 1976a]

using a simpli�cation of Je�rey's `coupled trees' method for testing clas-sically valid entailments. The idea is that to test A ! B one works outa truth-tree for A and a truth tree for B. One then requires that everybranch in the tree for A `covers' some branch in the tree for B in the sensethat every atom in the covered branch occurs in the covering branch. Thishas the intuitive sense that every way in which A might be true is also away in which B would be true, whether these ways are logically possibleor not, since `closed' branches (those containing contradictions) are not ex-empt as they are in Je�rey's method for classical logic. This coupled-treesapproach is ultimately related to the Anderson{Belnap tautological entail-ment method, as is also the method of [Dunn, 1980b] which explicates anearlier attempt of Levy to characterise entailment (cf. also [Clark, 1980]).

1.8 Relations to Familiar Logics

There is a sense in which relevance logic contains classical logic.


ZDF Theorem (Anderson and Belnap [1959a]). The zero-degree formulas(those containing only the connectives ^;_;:) provable in R (or E) areprecisely the theorems of classical logic.

The proof went by considering a `cut-free' formulation of classical logicwhose axioms are essentially just excluded middles (which are theorems ofR / E) and whose rules are all provable �rst-degree relevant entailments(cf. Section 2.7). This result extends to a �rst-order version [Anderson andBelnap Jr., 1959b]. (The admissibility of (cf. Section 2) provides anotherroute to the proof to the ZDF Theorem.)

There is however another sense in which relevance logic does not containclassical logic:

Fact (Anderson and Belnap [1975, Section 25.1]). R (and E) lack as aderivable rule Disjunctive Syllogism:

:A;A _ B ` B:

This is to say there is no deduction (in the standard sense of Section 1.4)of B from :A and A _ B as premises. This is of course the most notori-ous feature of relevance logic, and the whole of Section 2 is devoted to itsdiscussion.

Looking now in another direction, Anderson and Belnap [1961] beganthe investigation of how to translate intuitionistic and strict implicationinto R and E, respectively, as `enthymematic' implication. Anderson andBelnap's work presupposed the addition of propositional quanti�es to, let ussay R, with the subsequent de�nition of `A intuitionistically implies B' (insymbols A � B) as 9p(p^ (A^p ! B)). This has the sense that A togetherwith some truth relevantly implies B, and does seem to be at least in theneighbourhood of capturing Heyting's idea that A � B should hold if thereexists some `construction' (the p) which adjoined to A `yields' (relevantimplication) B. Meyer in a series of papers [1970a; 1973] has extendedand simpli�ed these ideas, using the propositional constant t in place ofpropositional quanti�cation, de�ning A � B as A^t! B. If a propositionalconstant F for the intuitionistic absurdity is introduced, then intuitionisticnegation can be de�ned in the style of Johansson as :A =df A � F . AsMeyer has discovered one must be careful what axiom one chooses to governF . F ! A or even F � A is too strong. In intuitionistic logic, the absurdproposition intuitionistically implies only the intuitionistic formulas, so thecorrect axiom is F � A�, where A� is a translation into R of an intuitionisticformula. Similar translations carry S4 into E and classical logic into R.

RELEVANCE LOGIC 31

2 THE ADMISSIBILITY OF

2.1 Ackermann's Rule

The �rst mentioned problem for relevance logics in Anderson's [1963] sem-inal `open problems' paper is the question of `the admissibility of '. Todemystify things a bit it should be said that is simply modus ponens forthe material conditions (:A _ B):

1.A

:A _ B:

B

It was the third listed rule of Ackermann's [1956] system of strenge Imp-likation (�; �; ; 1st, 2nd, 3rd). This was the system Anderson and Belnap`tinkered with' to produce E (Ackermann also had a rule Æ which theyreplaced with an axiom).

The major part of Anderson and Belnap's `tinkering' was the extremelybold step of simply deleting as a primitive rule, on the well- motivatedground that the corresponding object language formula

2. A ^ (:A _ B)! B

is not a theorem of E.

It is easy to see that (2) could not be a theorem of either E or R, sinceit is easy to prove in those systems

3. A ^ :A! A ^ (:A _ B)

(largely because :A ! :A _ B is an instance of an axiom), and of course(3) and (2) yield by transitivity the `irrelevancy'

4. A ^ :A! B.

The inference (1) is obviously related to the Stoic principle of the dis-junctive syllogism:

5.:AA _ B

:B

Indeed, given the law of double negation (and replacement) they are equiv-alent, and double negation is never at issue in the orthodox logics. Thus Eand R reject

6. :A ^ (A _ B)! B


as well as (2).This rejection is typically the hardest thing to swallow concerning rele-

vance logics. One starts o� with some pleasant motivations about relevantimplication and using subscripts to keep track of whether a hypothesis hasactually been used (as in Section 1.5), and then one comes to the pointwhere one says ànd of course we have to give up the disjunctive syllogism'and one loses one's audience. Please do not stop reading! We shall try tomake this rejection of disjunctive syllogism as palatable as we can.

(See [Belnap and Dunn, 1981; Restall, 1999] for related discussions, andalso discussion of [Anderson and Belnap, 1975, Section 16.1]); see Burgess[1981] for an opposing point of view.

2.2 The Lewis `Proof'

One reason that disjunctive syllogism has �gured so prominently int hecontroversy surrounding relevance logic is because of the use it was put to byC. I. Lewis [Lewis and Langford, 1932] in his so-called ìndependent proof':that a contradiction entails any sentence whatsoever (taken by Andersonand Belnap as a clear breakdown of relevance). Lewis's proof (with ournotations of justi�cation) goes as follows:

(1) p ^ :p(2) p 2, ^-Elimination(3) :p 1, ^-Elimination(4) p _ q 2, _-Introduction(5) q 3, 4 disjunctive syllogism

Indeed one can usefully classify alternative approaches to relevant implica-tion according to how they reject the Lewis proof. Thus, e.g. Nelson rejects^-Elimination and _-Introduction, as does McCall's connexive logic. Parry,on the other hand, rejects only _-Introduction. Geach, and more recently,Tennant [1994],accept each step, but says that èntailment' (relevant impli-cation) is not transitive. It is the genius of the Anderson{Belnap approachto see disjunctive syllogism as the culprit and the sole culprit.12

Lewis concludes his proof by saying, \If by (3), p is false; and, by (4), atleast one of the two, p and q is true, then q must be true". As is told in[Dunn, 1976a], Dunn was saying such a thing to an elementary logic classone time (with no propaganda about relevance logic) when a student yelledout, \But p was the true one|look again at your assumption".

12Although this point is complicated, especially in some of their earlier writings (see,e.g. [Anderson and Belnap Jr., 1962a]) by the claim that there is a kind of fallacy ofambiguity in the Lewis proof. the idea is that if _ is read in the ìntensional' way (as:A ! B), then the move from (3) and (4) to (5) is ok (it's just modus ponens forthe relevant conditional), but the move from (2) to (4) is not (now being a paradox ofimplication rather than ordinary disjunction introduction).

RELEVANCE LOGIC 33

That student had a point. Disjunctive syllogism is not obviously ap-propriate to a situation of inconsistent information|where p is assumed(given, believed, etc.) to be both true and false. This point has been ar-gued strenuously in, e.g. [Routley and Routley, 1972; Dunn, 1976a] andBelnap [1977b; 1977a]. The �rst two of these develop a semantical analysisthat lets both p and :p receive the value `true' (as is appropriate to modelthe situation where p ^ :p has been assumed true), and there will be moreabout these ideas in Section 3.4. The last is particularly interesting sinceit extends the ideas of Dunn [1976a] so as to provide a model of how acomputer might be programmed as to make inferences from its (possibly in-consistent) database. One would not want trivially inconsistent informationabout the colour of your car that somehow got fed into the fbi's computer(perhaps by pooled databases) to lead to the conclusion that you are PublicEnemy Number One.

We would like to add yet one more criticism of disjunctive syllogism,which is sympathetic to many of the earlier criticisms.

We need as background to this criticism the natural deduction frameworkof [Gentzen, 1934] as interpreted by [Prawitz, 1965] and others. the idea (asin Section 1.5) is that each connective should come with rules that introduceit into discourse(as principal connective of a conclusion) and rules that elim-inate it from discourse (as principal connective of a premise). further the`normalisation ideas of Prawitz, though of great technical interest and com-plication, boil down philosophically to the observation that an eliminationrule should not be able to get out of a connective more than an introductionrule can put into the connective. This is just the old conservation Principle,`You can't get something for nothing', applied to logic.

The paradigm here is the introduction and elimination rules for conjunc-tion. The introduction rule, from A;B to infer A ^ B packs into A ^ Bprecisely what the elimination rule, from A ^ B to infer either A or B(separately), then unpacks.

Now the standard introduction rule for disjunction is this: from either Aor B separately, infer A_B. We have no quarrel with an introduction rule.an introduction rule gives meaning to a connective and the only thing towatch out for is that the elimination rule does not take more meaning froma connective than the introduction rule gives to it (of course, one can alsoworry about the usefulness and/or naturalness of the introduction rules fora given connective, but that (pace [Parry, 1933]) seems not an issue in thecase of disjunction.

In the Lewis `proof' above, it is then clear that the disjunctive syllogismis the only conceivably problematic rule of inference. Some logicians (asindicated above) have queried the inferences from (1) to (2) and (4), andfrom (2) to (3), but from the point of view that we are now urging, this issimply wrongheaded. Like Humpty Dumpty, we use words to mean whatwe say. So there is nothing wrong with introducing connectives ^ and _


via the standard introduction rules. Other people may want connectives forwhich they provide di�erent introduction (and matching elimination) rules,but that is their business. We want the standard (`extensional') senses of^ and _.

Now the d.s. is a very odd rule when viewed as an elimination rule for _parasitical upon the standard introduction rules (whereas the constructivedilemma, the usual _-Elimination rule is not at all odd). Remember thatthe introduction rules provide the actual inferences that are to be stored inthe connective's battery as potential inferences, perhaps later to be releasedagain as actual inferences by elimination rules. The problem with the dis-junctive syllogism is that it can release inferences from _ that it just doesnot contain. (In another context, [Belnap, 1962] observed that Gentzen-style rules for a given connective should be `conservative', i.e. they shouldnot create new inferences not involving the given connective.)

Thus the problem with the disjunctive syllogism is just that p _ q mighthave been introduced into discourse (as it is in the Lewis `proof') by _-Introduction from p. So then to go on to infer q from p _ q and :p by thedisjunctive syllogism would be legitimate only if the inference from p;:p toq were legitimate. But this is precisely the point at issue. At the very leastthe Lewis argument is circular (and not independent).13

2.3 The Admissibility of

Certain rules of inference are sometimes `admissible' in formal logics inthe sense that whenever the premises are theorems, so is the conclusion atheorem, although these rules are nonetheless invalid in the sense that thepremises may be true while the conclusion is not. Familiar examples are therule of substitution in propositional logic, generalisation in predicate logic,and necessitation in modal logic. Using this last as paradigm, although theinference from A to �A (necessarily A) is clearly invalid and would indeedvitiate the entire point of modal logic, still for the (`normal') modal logics,whenever A is a theorem so is �A (and indeed their motivation would besomehow askew if this did not hold).

Anderson [1963] speculated that something similar was afoot with respectto the rule and relevance logic. Anderson hoped for a `sort of luckyaccident', but the admissibility of seems more crucial to the motivationof E and R than that. Kripke [1965] gives a list of four conditions that apropositional calculus must meet in order to have a normal characteristicmatrix, one of which is the admissibility of .14 `Normal' is meant in the

13This is a new argument on the side of Anderson and Belnap [1962b, pp. 19, 21].14The other conditions are that it be consistent, that it contain all classical tautologies,

and that it be `complete in the sense of Halld�en'. R and E can be rather easily seen tohave the �rst two properties (see Section 1.8 for the bit about classical tautologies), butthe last is rather more diÆcult (see Section 3.11).

RELEVANCE LOGIC 35

sense of Church, and boils down to being able to divide up its elements intothe `true' and the `false' with the operations of conjunction, disjunction,and negation treating truth and falsity in the style of the truth tables (aconjunction is true if both components are true, etc.). If one thinks ofE (as Anderson surely did) as the logic of propositions with the logicaloperations, and surely this should divide itself up into the true and the falsepropositions.15

2.4 Proof(s) of the Admissibility of

There are by now at least four variant proofs of the admissibility of forE and R. The �rst three proofs (in chronological order: [Meyer and Dunn,1969], [Routley and Meyer, 1973] and [Meyer, 1976a]) are all basically due toMeyer (with some help from Dunn on the �rst, and some help from Routleyon the second), and all depend on the same �rst lemma. The last proofwas obtained by Kripke in 1978 and is unpublished (see [Dunn and Meyer,1989]).

All of the Meyer proofs are what Smullyan [1968] would call `synthetic'in style, and are inspired by Henkin-style methods. The Kripke proof is`analytic' in style, and is inspired by Kanger{Beth{Hintikka tableau-stylemethods. In actual detail, Kripke's argument is modelled on completenessproofs for tableau systems, wherein a partial valuation for some open branchis extended to a total valuation. As Kripke has stressed, this avoids theapparatus of inconsistent theories that has hitherto been distinctive of thevarious proofs of 's admissibility.

We shall sketch the third of Meyer's proofs, leaving a brief descriptionof the �rst and second for Section 3.11. Since they depend on semanticalnotions introduced there.

The strategy of all the Meyer proofs can be divided into two segments:The Way Up and The Way Down. Of course we start with the hypothesesthat ` A and ` :A _ B, yet assume not ` B for the sake of reduction. Weshall be more precise in a moment, but The Way Up involves constructingin a Henkin-like manner a maximal theory T (containing all the logical the-orems) with B 62 T . The problem though is that T may be inconsistent inthe sense of having both C;:C 2 T for some formula C. (Of course thiscould not happen in classical logic, for by virtue of the paradox of implica-tion C ^ :C ! B;B would be a member of T contrary to construction.)The Way Down �xes this by �nding in e�ect some subtheory T 0 � T thatis both complete and consistent, and indeed is a `truth set' in the sense of[Smullyan, 1968] (Meyer has labelled it the Converse Lindenbaum Lemma).Thus for all formulas X and Y , :X 2 T 0 i� X 62 T 0, and X _ Y 2 T 0 i� at

15This would be less obvious to Routley and Meyer [1976], and Priest [1987; 1995] whoraise the `consistency of the world' as a real problem.


least one of X and Y is in T 0. So since :A _ B 2 T 0, at least one of :Aand B is in T 0. But since A 2 T 0, then :A is not in T 0. So B must be inT 0.16 But T 0 is a subset of T , which was constructed to keep B out. So Bcannot be in T 0, and so by reductio we obtain B as desired.

Enough of strategy! We now collect together a few notions needed for amore precise statement of The Way Up Lemma. Incidentally, we shall fromthis point on in our discussion of consider only the case of R. Results forE (and a variety of neighbours) hold analogously.

By an `R-theory' we mean a set of formulas T of R closed under adjunc-tion And logical relevant implication, i.e. such that

1. if A;B 2 T , then A ^ B 2 T ;

2. if `R A! B and A 2 T , then B 2 T .

Note that an arbitrary R-theory may lack some or all of the theorems ofR (in classical logic and most familiar logics this would be impossible be-cause of the paradox of strict implication which says that a logical theoremis implied by everything). We thus need a special name for those R{theoriesthat contain all of the R-theorems|those are called regular.17 In this sec-tion, since we have no use of irregular theories and shall be talking onlyof R, by a theory we shall always mean a regular R theory (irregular R-theories however play a great role in the completeness theorems of Section 3below and there we shall have to be more careful about our distinctions).

A theory T is called prime if whenever A _ B 2 T , then A 2 T orB 2 T . The converse of this holds for any theory T in virtue of the R-axioms A! A _B and B ! A _B and property (2). A theory T is calledcomplete if for every formula A;A 2 T or :A 2 T , and called consistent iffor no formula A do we have both A;:A 2 T . In virtue of the R-theoremA _ :A, we have that all prime theories are complete. A consistent primetheory is called normal, and it should by now be apparent that a normaltheory is a truth set in the sense of Smullyan given above.

Where � is a set of formulas, we write � `R A to mean that A is deduciblefrom � in the `oÆcial sense' of there being a �nite sequence B1; : : : ; Bn,

16The proof as given here would appear to use disjunctive syllogism in the meta-language at just this point, but it can be restructured (indeed we so restructured theoriginal proofs [Meyer and Dunn, 1969]) so as to avoid at least such an explicit useof disjunctive syllogism. The idea is to obtain by distribution (A 2 T 0 and A 62 T 0) or(B 2 T 0 and B 62 T 0) from the hypothesis B 62 T 0. The whole question of a `relevant' ver-sion of the admissibility of is a complicated one, and admits of various interpretations.See [Belnap and Dunn, 1981; Meyer, 1978].17It is interesting to note for regular theories, condition (2) may be replaced with the

condition

(20) if A 2 T and (A! B) 2 T , then B 2 T , in virtue of the R-theorem A^ (A! B) !B.

RELEVANCE LOGIC 37

with Bn = A and each Bi being either a member of �, or an axiom ofR, or a consequence of earlier terms by modus ponens or adjunction (incontext we shall often omit the subscript R). We write � `� A to meanthat � [ � `R A, and quite standardly we write things like �; A `R Bin place of the more formal � [ fAg `R B. Note that for any theory T ,writing `T A in place of � `T A boils down to saying that A is a theorem ofT (A 2 T ). Where � is a set of formulas not necessarily a theory, `� A canbe thought of as saying that A is deducible from the `axioms' �. The setfA :`� Ag is pretty intuitively the smallest theory containing the axioms�, and we shall label it as Th(�).

We can now state and sketch a proof of the

Way Up Lemma. Suppose not `R A. Then there exists a prime theory Tsuch that not `T A.

Proof. Enumerate the formulas of R : X1; X2; : : : . De�ne a sequence ofsets of formulas by induction as follows.

T0 = set of theorems of R.

Ti+1 = Th(Ti [ fXi+1g) if it is not the case that Ti; Xi+1 ` A;Ti, otherwise.

Let T be the union of all these Tn's. It is easy to see as is standard that Tis a theory not containing A. Also we can show that T is prime.

Thus suppose `T X _ Y and yet X;Y 62 T . Then it is easy to se thatsince neither X nor Y could be added to the construction when their turncame up without yielding A, we have both

1. X `T A,

2. Y `T A.

But by reasonably standard moves (R has distribution), we get

3. X _ Y `T A,

and so `T A contrary to the construction. �

The Way Down Lemma. Let T 0 be a prime theory. Then there exists anormal theory T � T 0.

The concept we need is that of a `metavaluation' (more precisely as weuse it here a `quasi-metavaluation', but we shall not bother the reader withsuch detail). The concept and its use re may be found in [Meyer, 1976a].(See also Meyer [1971; 1976b] for other applications.) For simplicity weassume for a while that the only primitive connectives are :;_ and ! (^can be de�ned via de Morgan). A metavaluation v is a function from theset of formulas into the truth values f0; 1g, such that


1. for a propositional variable p; v(p) = 1 i� p 2 T ;

2. v(:A) = 1 i� both (a) v(A) = 0 and (b) :A 2 T ;

3. v(A _B) = 1 i� either v(A) = 1 or v(B) = 1.

4. v(A! B) = 1 i� both (a) v(A) = 0 or v(B) = 1, and (b) A! B 2 T .

One surprising aspect of these conditions is the double condition in (2)that must be met for :A to be assigned the value 1. Not only must (a) Abe assigned 0 (the usual `extensional condition'), but also (b) :A must bea theorem of T (the `intensional condition'). and of course there are similarremarks about (4). The condition in (1) also relies upon G (actually to alesser extent than it might seem|when both p;:p 2 T , it would not hurtto let v(p) = 0).

We now set T 0 = fA : v(A) = 1g. The following lemma is useful, and hasan easy proof by induction on complexity of formulas (the case when A isa negation evaluated as 0 uses the completeness of T ).

Completeness Lemma. If v(A) = 1, then A 2 T . If v(A) = 0, then:A 2 T .

It is reasonably easy to see that T 0 is in fact a truth set. That it behavesok with respect to disjunction can be read right o� of clause (3) in thede�nition of v, so we need only look at negation where the issue is whetherT 0 is both consistent and complete. It is clear from clause (2) that T 0

is consistent, but T 0 is also complete. Thus, suppose A 62 T 0, then bythe Completeness Lemma :A 2 T . This is the intensional condition forv(:A) = 1, but our supposition that A 62 T 0 is just the extensional conditionthat v(A) = 0. Hence v(:A) = 1, i.e. :A 2 T 0 as desired.

It is also reasonably easy to check that T 0 is an R-theory. It is left tothe reader to do the easy calculation that T 0 is closed under adjunction andR-implication, i.e. that these preserve assignments by v of the value 1. Herewe will illustrate the more interesting veri�cation that the R-axioms all getassigned the value 1. We shall not actually check all of them, but ratherconsider several typical ones.

First we check suÆxing: (A ! B) ! [(B ! C) ! (A ! C)]. Supposev assigns it 0. Since it is a theorem of R and a fortiori of T , then itsatis�es the intensional condition and so must fail to satisfy the extensionalcondition. So v(A ! B) = 1 and v((B ! C) ! (A ! C)) = 0. By theCompleteness Lemma, then (A! B) 2 T , and so by modus ponens from thevery axiom in question (SuÆxing) we have that (B ! C)! (A! C) 2 T .So v((B ! C)! (A! C)) satis�es the intensional condition, and so mustfail to satisfy the extensional condition since it is 0. So v(B ! C) = 1and v(A ! C) = 0. By reasoning analogous to that above (one moremodus ponens) we derive that v(A ! C) must �nally fail to satisfy the

RELEVANCE LOGIC 39

extensional condition, i.e. v(A) = 1 and v(C) = 0. But clearly since all ofv(A! B) = 1, v(B ! C) = 1, v(A) = 1, then by the extensional condition,v(C) = 1, and we have a contradiction.

The reader might �nd it instructive in seeing how negation is handledto verify �rst the intuitionistically acceptable form of the Reductio axioms(A ! :A) ! :A, and then to verify its classical variant (used in someaxiomatisations of R), (:A ! A) ! A. The �rst is easier. Also ClassicalDouble Negation, ::A! A is fun.

This completes the sketch of Meyer's latest proof of the admissibility of for R.

2.5 for First-order Relevance Logics

The �rst proof of the admissibility of for �rst-order R, E, etc. (which weshall denote as RQ, etc.) was in Meyer, Dunn and Leblanc [1974], and usesalgebraic methods analogous to those used for the propositional relevancelogic in [Meyer and Dunn, 1969]. The proof we shall describe here thoughwill again be Meyer's metavaluation-style proof.

The basic trick needed to handle �rst-order quanti�ers is to produce thistime a �rst-order truth set. Assuming that only the universal quanti�er 8is primitive (the existential can be de�ned: 9x =df :8x:), this means weneed

(8) 8xA 2 T i� A(a=x) 2 T for all parameters (free variables) a:

This is easily accommodated by adding a clause to the de�nition of themetavaluation v so that

5. v(8xA) = 1 i� v(A(a=x)) = 1 for all parameters a.

This does not entirely �x things, for in proving the Completeness Lemmawe have now in the induction to consider the case when A is of the form8xB. If v(8xB) = 1, then (by (5)), va(B(a=x)) = 1 for all parameters a.By inductive hypothesis, for all a;B(a=x) 2 T . But, and here's the rub,this does not guarantee that 8xB(a=x) 2 T . We need to have constructedon The Way Up a theory T that is `!-complete' in just the sense that thisguarantee is provided. ([Meyer et al., 1974] call such a theory `rich'.) Ofcourse it is understood by `theory' we now mean a `regular RQ-theory',i.e. one containing all of the axioms of RQ and closed under its rules (seeSection 1.3). Actually things can be arranged as in [Meyer et al., 1974] sothat generalisation is in e�ect built into the axioms so that the only rulescan continue to be adjunction and modus ponens.

Thus we need the following

Way Up Lemma for RQ. Suppose A is not a theorem of �rst-order RQ.Then there exists a prime rich theory T so that A 62 T .


This lemma is Theorem 3 of [Meyer et al., 1974], and its proof is of basi-cally a Henkin style with one novelty. In usual Henkin proofs one can assure!-completeness by building into the construction of T that whenever :8xBis put in, then so is :B(a=x) for some new parameter a. This guarantees!-completeness since if B(a) 2 T for all a, but 8xB 62 T , then by complete-ness :8xB 2 T and so by the usual construction :B(a) 2 T for some a, andso by consistency (??) B(a) 62 T for some a, contradicting the hypothesisfor !-completeness. But we of course have for relevance logics no guaranteethat T is consistent, as has been remarked above.

The novelty then was to modify the construction so as to keep thingsout as well as put things in, though this last still was emphasised. Fullsymmetry with respect to `good guys' and `bad guys' was �nally obtainedby Belnap, 18 in what is called the Belnap Extension Lemma, which shallbe stated after a bit of necessary terminology.

We shall call an ordered pair (�;�) of sets of formulas of RQ and `RQ-pair'. We shall say that one RQ pair (�1;�1) extends another (�0;�0) if�0 � �1 and �0 � �1. An RQ pair is de�ned to be exclusive if for noA1; : : : ; Am 2 �; B1; : : : ; Bn 2 � do we have ` A1^� � �^Am ! B1_� � �_Bn.It is called exhaustive if for every formula A, either A 2 � or A 2 �.19

It is now easiest to assume that ^ and 9 are back as primitive. We call aset of formulas � _-prime (^-prime) if whenever A _ B 2 �(A ^ B 2 �),at least one of A or B 2 � (clearly _-primeness is the same as primeness).Analogously, we call � 9-prime (8- prime) if whenever 9xA 2 �(8xA 2 �),then A(a=x) 2 � for some a. Given an RQ pair (�;�) we shall call �(�)completely prime if � is both _- and 9-prime (� is both ^- and 8-prime).the pair (�;�) is called completely prime if both � and � are completelyprime. We can now state the

Belnap Extension Lemma. Let (�;�) be an exclusive RQ pair. Then(�;�) can be extended to an exclusive, exhaustive, completely prime RQpair (T; F ) in a language just like the language of RQ except for havingdenumerably many new parameters.

We shall not prove this lemma here, but simply remark that it is asurprisingly straightforward application of Henkin methods to construct amaximal RQ-pair and show it has the desired properties (indeed it simplysymmetrises the usual Henkin construction of �rst-order classical logic).

18Belnap's result is unpublished, although he communicated it to Dunn in 1973. Dunncirculated a write-up of it about 1975. It is cited in some detail in [Dunn, 1976d]. Gabbay[1976] contains an independent but precise analogue for the �rst-order intuitionistic logicwith constant domain.19We choose our terminology carefully, not calling (�;�) a `theory', not using `consis-

tency' for exclusiveness, and not using `completeness' for exhaustiveness. We do this soas to avoid con ict with our earlier (and more customary) usage of these terms and inthis we di�er on at least one term from usages on other occasions by Gabbay, Belnap, orDunn.

RELEVANCE LOGIC 41

In order to derive the RQ Way Up Lemma we simply set � = RQand � = fAg and extend it to the pair (T; F ) using the Belnap ExtensionLemma. It is easy to see that T is a (regular) RQ-theory, and clearlyG is prime. but also T is !-complete. Thus suppose B(a=x) 2 T forall a, but 8xB 62 T . Then by exhaustiveness 8xB 2 FR. Then by 8-primeness, B(a=x) 2 FR for some a. But since `RQ B(a=x) ! B(a=x),this contradicts the exclusiveness of the pair (T; F ).

2.6 for Higher-order Relevance Logics and Relevant Arith-metic

The whole point about being merely an admissible rule is that it mightnot hold for various extensions of F (cf. [Dunn, 1970] for actual counterexamples). Thus, as we just saw, it was an achievement to show that continues to e admissible in R when it is extended to include �rst-orderquanti�cation. The question of the admissibility of naturally has greatinterest when R is further extended to include theories in the foundationsof mathematics such as type theory (set theory) and arithmetic.

Meyer [1976a] contains investigations of the admissibility of for relevanttype theory (R!). We shall report nothing in the way of detail here exceptto observe that Meyer's result is invariant among various restrictions of theformulas A in the Comprehension Axiom scheme:

9Xx+18yn(Xn+1(yn)$ A):

As for relevantly formulated arithmetic, most work has gone on in study-ing Meyer's systems R] , R]] and their relatives, based on Peano arithmetic,though Dunn has also considered a relevantly formulated version of Robin-son Arithmetic [Anderson et al., 1992]. Here we will recount the resultsfor R] and R]] for they are rather surprising. In a nutshell, is admis-sible in relevant arithmetics with the in�nitary !-rule (from A(0), A(1),A(2), : : : to infer 8xA(x)), but not without it [Friedman and Meyer, 1992;Meyer, 1998].

The system R] is given by rewriting the traditional axioms of Peanoarithmetic with relevant implication instead of material implication in thenatural places. You get the following list of axioms

Identity y = z ! (x = y ! x = z)Successor x0 = y0 ! x = y

x = y ! x0 = y0

0 6= x0

Addition x+ 0 = xx+ y0 = (x + y)0

Multiplication x0 = 0xy0 = xy + x

Induction A(0) ^ 8x�A(x)! A(x0)�! 8xA(x)


which you add to those of RQ in order to obtain an arithmetic theory.The question about the admissibility of was open for many years, untilFriedman teamed up with Meyer to show that it is not [Friedman and Meyer,1992]. The proof does not provide a direct counterexample to . Instead, ittakes a more circuitous route. First, we need Meyer's classical containmentresult for R] . When we map formulae in the extensional vocabulary ofarithmetic to the language of R] by setting �(x = y) to (x = y) _ (0 6= 0)and leaving the rest of the map to respect truth functions (so �(A ^ B) =�(A) ^ �(B), �(:A) = :�(A) and �(8xA) = 8x�(A)) then we have thefollowing theorem:

�(A) is a theorem of R] i� A is a theorem of classical Peanoarithmetic.

This is a subtle result. The proof goes through by showing, by induction,that �(A) is equivalent either to (A^ (0 = 0))_ (0 6= 0) or to (A_ (0 6= 0))^(0 = 0), and then that and the classical form of induction (with materialimplication in place of relevant implication) is valid for formulae of this formin R] . Then, if we had the admissibility of for R] , we could infer Afrom �(A). (If �(A) is equivalent to (A ^ (0 = 0)) _ (0 6= 0), then we canuse 0 = 0 and to derive A^ (0 = 0), and hence A. Similarly for the othercase).

The next signi�cant result is that not all theorems of classical Peanoarithmetic are theorems of R] . Friedman provided a counterexample, whichis simple enough to explain here. First, we need some simple preparatoryresults.

� R] is a conservative extension of the theory R]+ axiomatised by thenegation free axioms of R] [Meyer and Urbas, 1986].

� If classical Peano theorem is to be provable in R] and if it containsno negations, then it must be provable in R]+.

� Any theorem provable in R]+ must be provable in the classical positivesystem PA+ which is based on classical logic, instead of R.

The proofs of these results are relatively straightforward. The next resultis due to Friedman, and it is much more surprising.

� The ring of complex numbers is a model of PA+.

The only diÆcult thing to show is that it satis�es the induction axiom.For any formula A(x) in the vocabulary of arithmetic, the set of complexnumbers � such that A(�) is true is either �nite or co�nite. If A(x) isatomic, then it is equivalent to a polynomial of the form f(x) = 0, and fmust either have �nitely many roots or be 0 everywhere. But the set ofeither �nite or co�nite sets is closed under boolean operations, so no A(x)

RELEVANCE LOGIC 43

we can construct will have an extension which is neither �nite or co�nite.)As a result, the induction axiom must be satis�ed. For if A(0) holds andif A(x) � A(x0) holds then there are in�nitely many complex numbers �such that A(�). So the extension of A is at least co�nite. But if there is apoint � such that A(�) fails, then so would A(� � 1), A(� � 2) and so onby the induction step A(x) � A(x0), and this contradicts the con�nitude ofthe extension of A. As a result, A(�) holds for every �.

We can then use this surprising model of positive Peano arithmetic toconstruct a Peano theorem which is not a theorem of R] . It is known thatfor any odd prime p, there is a positive integer y which is not a quadraticresidue mod p. That is, 9y8z:(y � z2 mod p) is provable in Peano arith-metic. This formula can be rewritten in the language or arithmetic witha little work. However, the corresponding formula is false in the complexnumbers, so it is not a theorem of PA+. Therefore it isn't a theorem of R]

+, and by the conservative extension result, it is not a theorem of R] . Asa consequence, R] is not closed under .

Where is the counterexample to ? Meyer's containment result providesa proof of �(B), where B is the quadratic residue formula. The rule wouldallow us to derive B from �(B), and it is here that must fail.

If we replace the induction axiom by the in�nitary rule !, we can provethe admissibility of using a modi�cation of the Belnap Extension Lemmafor the Way Up and using the standard metavaluation technique for theWay Down. The modi�cation of the Belnap Extension Lemma is due toMeyer [1998].

Belnap Extension Lemma, with Witness Protection:

Let (�;�) be an exclusive R]] pair in the language of arithmetic(that is, with 0 as the only constant). Then (�;�) can beextended to an exclusive, exhaustive, completely prime R]] pair(T; F ) in the same language.

This lemma requires the !-rule for its proof. Consider the induction stagein which you wish to place 8xA(x) in �i. The witness condition dictatesthat there be some term t such that A(t) also appear in �i. The !-ruleensures that we can do this without the need for a new term, for if no term000��0 could be consistently added to �i, then each A(000��0) is a consequenceof �i, and by the !-rule, so is 8xA(x), contradicting the fact that we canadd 8xA(x) to �i. So, we know that some 000��0 will do, and as a result, weneed add no new constants to form the complete theory T . The rest of theway up lemma and the whole of the way down lemma can then be provedwith little modi�cation. (for details, see [Meyer, 1998]). Consequently, isadmissible in R]] .

These have been surprising results, and important ones, for relevant arith-metic is an important `test case' for accounts of relevance. It is a theory inwhich we can have some fairly clear idea of what it is for one formulae to


properly follow from another. In R] and R]] , we have 0 = 2! 0 = 4 be-cause there is an `arithmetically appropriate' way to derive 0 = 4 from0 = 2 | by multiplying both sides by 2. However, we cannot derive0 = 2 ! 0 = 3, and, correspondingly, there is no way to derive 0 = 3from 0 = 2 using the resources of arithmetic. The only way to do it withinthe vocabulary is to appeal to the falsity of 0 = 3, and this is not a rele-vantly acceptable move. 0 6= 3 ! (0 = 3 ! 0 = 2) does not have much torecommend as pattern of reasoning which respects the canons of relevance.

We are left with important questions. Are there axiomatisable extensionsof R] which are closed under ? Can theories like R] and R]] be ex-tended to deal with more interesting mathematical structures, while keepingaccount of some useful notion of relevance? Early work on this area, from aslightly di�erent motivation (paraconsistency, not relevance) indicates thatthere are some interesting results at hand, but the area is not without itsdiÆculties [Mortensen, 1995].

The admissibility of would also seem to be of interest for relevanttype theory (even relevant second-order logic) with an axiom of in�nity (see[Dunn, 1979b]).

One of the chief points of philosophical interest in showing the admissibil-ity of for some relevantly formulated version of a classical theory relates tothe question of the consistency of the classical theory (this was �rst pointedout in Meyer, Dunn and Leblanc [1974]). As we know from G�odel's work,interesting classical theories cannot be relied upon to prove their own con-sistency. To exaggerate perhaps only a little, the consistency of systems likePeano (even Robinson) arithmetic must be taken in faith.

But using relevance logic in place of classical logic in formulating suchtheories gives us a new strategy of faith. It is conceivable that since rele-vance logic is weaker than classical logic, the consistency of the resultanttheory might be easier to demonstrate. This has proved true at least in thesense of absolute consistency (some sentence is unprovable) as shown by[Meyer, 1976c] for Peano arithmetic using elementary methods. Classicallyof course there is no di�erence between absolute consistency and ordinary(negation) consistency (for no sentence are both A and :A provable), andif is admissible for the theory, then this holds for relevance logic, too. Theinteresting thing then would be to produce a proof of the admissibility of , which we know from G�odel would itself have to be non-elementary.

One could then imagine arguing with a classical mathematician in thefollowing Pascal's Wager sort of way [Dunn, 1980a].

Look. You have equally good reason to believe in the negationconsistency of the classical system and the (relative) complete-ness of the relevant system. In both cases you have a non-elementary proof which secures your belief, but which might bemistaken. Consider the consequences in each case if it is mis-

RELEVANCE LOGIC 45

taken. If you are using the classical system, disaster! Since evenone contradiction classically implies everything, for each theo-rem you have proven, you might just as well have proven itsnegation. But if you are using the relevant system, things arenot so bad. For at least large classes of sentences, it can beshown by elementary methods (Meyer's work) that not both thesentences and their negations are theorems.

2.7 Ackermann's and Gentzen's Cut: Gentzen Systems asRelevance Logic

In [Meyer et al., 1974] an analogy was noted between the role that theadmissibility of plays in relevance logic and the role that cut eliminationplays in Gentzen calculi (even those for classical systems). For the readerunfamiliar with Gentzen calculi, this subsection will make more sense aftershe has read Sections 4.6 and 4.7. The Gentzen system for the classicalpropositional calculus LK with the material conditional and negation asprimitive (as is well-known, all of the other truth-functional connectivescan be de�ned from these) may be obtained by adding to the rules of LR:

!

of Section 4.7. the rule of Thinning (see Section 4.6) on both the left andright. Gentzen also had as a primitive rule:

� ` A;B ;A ` Æ;

�; ` B; Æ (Cut)

which has as a special case

` A ` B:` B

(1)

Since A ` B is derivable just when ` A ! B is derivable, and since inclassical logic A! B is equivalent to :A _ B, (1) above is in e�ect

` A ` :A _ B;` B

(10)

which is just .All of the Gentzen rules except Cut have the Subformula Property: Every

formula that occurs in the premises also occurs in the conclusion, thoughperhaps there as a subformula. Gentzen showed via his Hauptsatz thatCut was redundant|it could be eliminated without loss (hence this is oftencalled the Elimination Theorem). Later writers have tended to think ofGentzen systems as lacking the Cut Rule, and so the Elimination Theoremis stated as showing that Cut is admissible in the sense that whenever thepremises are derivable so is the conclusion. There is thus even a parallel


historical development with Ackermann's rule in relevance logic, sincewriters on relevance logic have tended to follow Anderson and Belnap'sdecision to drop as a primitive rule.

Note that the Subformula Property can be thought of as a kind of rela-tion of relevance between premises and conclusion. Thus Cut as primitivedestroys a certain kind of relevance property of Gentzen systems, just as as primitive destroys the relevance of premises to conclusion in relevancelogic. The analogies become even clearer if we reformulate Gentzen's systemaccording to the following ideas of [Sch�utte, 1956].

The basic objects of Gentzen's calculus LK were the sequents A1; : : : ; Am `B1; : : : ; Bn, where the Ai's and Bj 's are formulas (any or all of which mightbe missing). Such a sequent may be interpreted as a statement to the ef-fect that either one of the Ai's is false or one of the Bj 's is true. To everysuch sequent there corresponds what we might as well call its `right-handedcounterpart':

` :A1; : : : ;:Am; B1; : : : ; Bn

It is possible to develop a calculus parallel to Gentzen's using only `right-handed' sequents, i.e. those with empty left side. This is in e�ect whatSch�utte did, but with one further trick. Instead of working with a right-handed sequent ` A1; : : : ; Am, which can be thought of as a sequence offormulas, he in e�ect replaced it with the single formula A1 _ � � � _Am.20

With these explanations in mind, the reader should have no trouble inperceiving Sch�utte's calculus K1 as `merely' a notational variant of Gentzen'soriginal calculus LK (albeit, a highly ingenious one). Also Sch�utte's sys-tem had the existential quanti�er which we have omitted here purely forsimplicity. Dunn and Meyer [1989] treats it as well.

The axioms of K1 are all formulas of the form A _ :A. The inferencerules divide themselves into two types:

Structural rules:

M_ A _B _ N[Interchange]M_ B _ A _ N

N _ A _A[Contraction]N _ A

Operational rules:

N[Thinning]N _B

N _ :A N _ :B[de Morgan]N _ :(A _ B)

N _ A[Double Negation]N _ ::A

It is understood in every case but that of Thinning that either both ofM and N may be missing. Also there is an understanding in multipledisjunctions that parentheses are to be associated to the right.

20It ought be noted that similar \single sided" Gentzen systems �nd extensive use inthe proof theory for Linear Logic [Girard, 1987; Troelstra, 1992].

RELEVANCE LOGIC 47

In [Meyer et al., 1974] it was said that the rule Cut is just `in peculiarnotation'. In the context of Sch�utte's formalism the notation is not even sodi�erent. Thus:

M_A :A _ N[Cut]M_N

A :A _B[ ]:

B

Since either M or N may be missing, obviously is just a special case ofCut.

It is pretty easy to check that each of the rules above corresponds toa provable �rst-degree relevant implication. Indeed [Anderson and BelnapJr., 1959a] with their `Simple Treatment' formulation of classical logic (ex-tended to quanti�ers in [Anderson and Belnap Jr., 1959b]) independentlyarrived at a Cut-free system for classical logic much like Sch�utte's (but withsome improvements, i.e. they have more general axioms and avoid the needfor structural rules). They used this to show that E contains all the clas-sical tautologies as theorems, the point being that the Simple Treatmentrules are all provable entailments in E (unlike the usual rule for axiomaticformulations of classical logic, modus ponens for the material conditional,i.e. ). Thus the later proven admissibility of was not needed for this pur-pose, although it surely can be so used. Sch�utte's system can also clearlybe adapted to the purpose of showing that classical logic is contained in rel-evance logic, and indeed [Belnap, 1960a] used K1 (with its quanti�cationalrules) to show that EQ contains all the theorems of classical �rst-orderlogic.

It turns out that one can give a proof of the admissibility of Cut fora classical Gentzen-style system, say Sch�utte's K1, along the lines of aMeyer-style proof of the admissibility of (see [Dunn and Meyer, 1989],�rst reported in 1974).21 We will not give many details here, but the keyidea is to treat the rules of K1 as rules of deducibility and not merely astheorem generating devices. Thus we de�ne a deduction of A from a set offormulas � as a �nite tree of formulas with A as its origin, members of � oraxioms of K1 at its tips, and such that each point that is not a tip followsfrom the points just above it by one of the rules of K1 (this de�nition has tobe slightly more complicated if quanti�ers are present due to usual problemscaused by generalisation). We can then inductively build a prime complete

21We hasten to acknowledge the nonconstructive character of this prof. In this our proofcompares with that of Sch�utte [1956] (also proofs for related formalisms due to Andersonand Belnap, Beth, Hintikka, Kanger) in its uses of semantical (model-theoretic) notions,and di�ers from Gentzen's. Like the proofs of Sch�utte et al. this proof really provides acompleteness theorem. We may brie y label the di�erence between this proof and thoseof Sch�utte and the others by using (loosely) the jargon of Smullyan [1968]. Calling bothHilbert-style formalisms and their typical Henkin-style completeness proofs `synthetic',and calling both Gentzen-style formalisms and their typical Sch�utte-style completenessproof `analytic', it looks as if we can be said to have given an synthetic completenessproof for an analytic formalism.


theory (closed under deducibility) on The Way up, which will clearly beinconsistent since because of the `Subformula Property' clearly, e.g. q is notdeducible from p;:p. but this can be �xed on The Way Down by usingmetavaluation techniques so as to �nd a complete consistent subtheory.

In 1976 E. P. Martin, Meyer and Dunn extended and analogised theresult of Meyer concerning the admissibility of for relevant type theorydescribed in the last subsection, in much the same way as the argumentfor the �rst-order logic has been analogised here, so as to obtain a new proofof Takeuti's Theorem (Cut-elimination for simple type theory). This un-published proof dualises the proof of Takahashi and Prawitz (cf. [Prawitz,1965]) in the same way that the proof here dualises the usual semanticalproofs of Cut-elimination for classical �rst-order logic. This dualisation isvividly described by saying that in place of `Sch�utte's Lemma' that everysemi- (partial-) valuation may be extended to a (total) valuation, there is in-stead the `Converse Sch�utte Lemma' that every àmbi-valuation' (sometimesassigns a sentence both the values 0, 1) may be restricted to a (consistent)valuation.

3 SEMANTICS

3.1 Introduction

In Anderson's [1963] òpen problems' paper, the last major question listed,almost as if an afterthought, was the question of the semantics of E andEQ. Despite this appearance Anderson said (p. 16) `the writer does notregard this question as \minor"; it is rather the principle large questionremaining open'. Anderson cited approvingly some earlier work of Belnap's(and his) on providing an algebraic semantics for �rst-degree entailments,and said (p. 16), `But the general problem of �nding a semantics for thewhole of E, with an appropriate completeness theorem, remains unsolved'.

It is interesting to note that Anderson's paper appeared in the sameActa Filosphica Fennica volume as the now classic paper of Kripke [1963]

which provided what is now simply called `Kripke-style' semantics for avariety of modal logics (Kripke [1959a] of course provided a semantics forS5, but it lacked the accessibility relationR which is so versatile in providingvariations).

When Anderson was writing his òpen problems' paper, the paradigm of asemantical analysis of a non-classical logic was probably still something likethe work of McKinsey and Tarski [1948], which provided interpretationsfor modal logic and intuitionistic logic by way of certain algebraic struc-tures analogous to the Boolean algebras that are the appropriate structuresfor classical logic. But since then the Kripke-style semantics (sometimes re-ferred to as `possible-worlds semantics', or `set-theoretical semantics') seems

RELEVANCE LOGIC 49

to have become the paradigm. Fortunately, E and R now have both an al-gebraic semantics and a Kripke-style semantics. We shall �rst distinguish ina kind of general way the di�erences between these two main approaches tosemantics, before going on to explain the particular details of the semanticsfor relevant logics (again R will be our paradigm).

3.2 Algebraic vs. Set-theoretical Semantics

It is convenient to think of a logical system as having two distinct aspectssyntax (well-formed strings of symbols, e.g. sentences) and semantics (what,e.g. these sentences mean, i.e. propositions). These double aspects competewith one another as can be seen in the competing usages `sentential calculus'and `propositional calculus', but we should keep �rmly in mind both aspects.

Since sentences can be combined by way of connectives, say he conjunc-tion sign ^, to form further sentences, typically there is for each logicalsystem at least one natural algebra arising at the level of syntax, the alge-bra of sentences (if one has a natural logical equivalence relation there isyet another that one obtains by identifying logically equivalent sentences to-gether into equivalence classes|the so-called `Lindenbaum algebra'). Andsince propositions can be combined by the corresponding logical operations,say conjunction, to form propositions, here is an analogous algebra of propo-sitions.

Now undoubtedly some readers, who were taught to `Quine' propositionsfrom an early age, will have troubles with the above story. The same readerwould most likely not �nd compelling any particular metaphysical accountwe might give of numbers. We ask that reader then to at least suspenddisbelief in propositions so that we can get on with the mathematics.

There is an alternative approach to semantics which can be describedby saying that rather than taking propositions as primitive, it `constructs'them out of certain other semantical primitives. Thus there is as a paradigmof this approach the so-called `UCLA proposition' as a set of `possibleworlds'.22 We here want to stress the general structural idea, not placingmuch emphasis upon the particular choice of `possible worlds ' as the se-mantical primitive. Various authors have chosen `reference points', `cases',`situations', `set-ups', etc.|as the name for the semantical primitive varyingfor sundry subtle reasons from author to author. We have both in relevancelogic contexts have preferred `situations', but in a show of solidarity we shallhere join forces with the Routley's [1972] in their use of `set-ups'.

Such `set-theoretical' semantical accounts do not always explicitly verifysuch a construction of propositions. Indeed perhaps the more commonapproach is to provide an interpretation that says whether a formula A is

22Actually the germ of this idea was already in Boole (cf. [Dipert, 1978]), althoughapparently he thought of it as an analogy rather than as a reduction.


true and false at a given set-up S writing �(a; S) = T or S � A or somesuch thing. Think of Kripke's [1963] presentation of his semantics for modallogic. But (unless one has severe ontological scruples about sets) one mightjust as well interpret A by assigning it a class of set-ups, writing �(A) orjAj or some such thing. One can go from one framework to the other byway of equivalence

S 2 jAj i� S � A:

3.3 Algebra of First-degree Relevant Implications

Given two propositions a and b, it is natural to consider the implicationrelation among them, which we write as a � b (`a implies b'). It might bethought to be natural to write this the other way around as a � b on someintuition that a is the stronger or `bigger' one if it implies b. Also it suggestsa � b (`b is contained in a'), which is a natural enough way to think of impli-cation. There are good reasons though behind our by now almost universalchoice (of course at one level it is just notation, and it doesn't matter whatyour convention is). Following the idea that a proposition might be identi-�ed with the set of cases in which it is true, a implies b corresponds to a � b,which has the same direction as a � b. Then conjunction ^ corresponds tointersection \, and they have roughly the same symbol (and similarly for _and [).

It is also natural to assume, as the notation suggests, that implication isa partial order, i.e.

(p.o.1) a � a (Re exivity),(p.o.2) a � b and b � a) a = b (Antisymmetry),(p.o.3) a � b and b � c) a � c (Transitivity).

It is natural also to assume that there are operations of conjunction ^ anddisjunction _ that satisfy

(^lb) a ^ b � a, a ^ b � b,(^glb) x � a and x � b) x � a ^ b,(_ub) a � a _ b, b � a _ b,(_lub) a � x and b � x) a _ b � x.

Note that (^lb) says that a ^ b is a lower bound both of a and of b, and(^glb) says it is the greatest such lower bound. Similarly a _ b is the leastupper bound of a and b.

A structure (L;�;^;_) satisfying all the properties above is a well-knownstructure called a lattice. Almost any logic would be compatible with the as-sumption that propositions form a lattice (but there are exceptions, witnessParry's [1933] Analytic Implication which would reject (_ub)).

RELEVANCE LOGIC 51

Lattices can be de�ned entirely operationally as structures (L;^;_) withthe relation a � b de�ned as a ^ b = a. Postulates characterising theoperations are:

Idempotence: a ^ a = a, a _ a = aCommutativity: a ^ b = b ^ a, a _ b = b _ aAssociativity: a ^ (b ^ c) = (a ^ b) ^ c, a _ (b _ c) = (a _ b) _ cAbsorption: a ^ (a _ b) = a, a _ (a ^ b) = a.

An (upper) semi-lattice is a structure (S;_), with _ satisfying Idempotence,Commutativity, and Associativity.

Given two lattices (L;^;_) and (L0;^0;_0), a function h from L into 0

is called a (lattice) homomorphism if both h(a ^ b) = h(a) ^0 h(b) andh(a _ b) = h(a) _0 h(b). If h is one{one, h is called an isomorphism.

Many logics (certainly orthodox relevance logic) would insist as well thatpropositions form a distributive lattice, i.e. that

a ^ (b _ c) � (a ^ b) _ c:This implies the usual distributive laws a ^ (b _ c) = (a ^ b) ^ (a ^ c) anda _ (b ^ c) = (a _ b) ^ (a _ c). (Again there are exceptions, important onesbeing quantum logic with its weaker orthomodular law, and linear logic withits rejection of even the orthomodular law.)

The paradigm example of a distributive lattice is a collection of sets closedunder intersection and union (a so-called `ring' of sets). Stone [1936] indeedshowed that abstractly all distributive lattices can be represented in thisway. Although we will not argue this here, it is natural to think that ifpropositions correspond to classes of cases, then conjunction should carryover to intersection and disjunction to union, and so productions shouldform a distributive lattice.

Certain subsets of lattices are especially important. A �lter is a non-empty subset F such that

1. a; b 2 F ) a ^ b 2 F ,

2. a 2 F and a � b) b 2 F .

Filters are like theories. Note by easy moves that a �lter satis�es

10. a; b 2 F , a ^ b 2 F ,

20. a 2 F or b 2 F ) a _ b 2 F .

When a �lter also satis�es the converse of (20) it is called prime, and is likea prime theory. A �lter that is not the whole lattice is called proper. Stone[1936] showed (using an equivalent of the Axiom of Choice) the

Prime Filter Separation Property. In a distributive lattice, if a 6� b,then there exists a prime �lter P with a 2 P and b 62 P .


This is related to the Belnap Extension Lemma of Section 2.5.So far we have omitted discussion of negation. This is because there is

less agreement among logics as to what properties it should have.23 Thereis, however, widespread agreement that it should at least have these:

1. (Contraposition) a � b) :b � :a,

2. (Weak Double Negation) a � ::a.

These can both be neatly packaged in one law:

3. (Intuitionistic Contraposition) a � :b) b � :a.

We shall call any unary function : satisfying (3) (or equivalently (1) and(2)) a minimal complement. The intuitionists of course do not accept

4. (Classical Contraposition) :a � :b) b � a, or

5. (Classical Double Negation) ::a � a.

If one adds either of (4) or (5) to the requirements for a minimal complementone gets what is called a de Morgan complement (or quasi-complement),because, as can be easily veri�ed, it satis�es all of de Morgan's laws

(deM1) :(a ^ b) = :a _ :b,(deM2) :(a _ b) = :a ^ :b.

Speaking in an algebraic tone of voice, de Morgan complement is just a(one{one) order-inverting mapping (a dual automorphism) of period two.

De Morgan complement captures many of the features of classical nega-tion, but it misses

(Irrelevance 1) a ^ :a � b,(Irrelevance 2) a � b _ :b.

If (either of) these are added to a de Morgan complement it becomes aBoolean complement. If Irrelevance 1 is added to a minimal complement, itbecomes a Heyting complement (or pseudo-complement).

A structure (L;^;_;:), where (L;^;_) is a distributive lattice and :is a de Morgan (Boolean) complement is called a de Morgan (Boolean)lattice. Note that we did not try to extend this terminological frameworkto `Heyting lattices', because in the literature a Heyting lattice requires anoperation called `relative pseudo-complementation' in addition to Heytingcomplementation (plain pseudo-complementation).

As an example of de Morgan lattices consider the following (here we useordinary Hasse diagrams to display the order; a � b is displayed by puttinga in a connected path below b):

23Cf. Dunn [1994; 1996] wherein the various properties below are related to variousways of treating incompatibility between states of information.

RELEVANCE LOGIC 53

1s

s

0

2:

1s

s

s

0

3: p = :p

1s

��s@@@s��

s@@@

0

p :p4:

1s

��s@@@s�

��

s@@@

0

p = :p q = :q4:

The backwards numeral labelling the third lattice over is not a misprint.It signi�es that not only has the de Morgan complement been obtained byinverting he order of the diagram, as in the order three (of course :I = �and vice versa), but also by rotating it from right to left at the same time.2 and 4are Boolean lattices.

A homomorphism (isomorphism) h between de Morgan Lattice with deMorgan complements : and :0 respectively is a lattice homomorphism (iso-morphism) such that h(:a) = :0h(a).

A valuation in a lattice out�tted with one or the other of these `com-plementations' is a map v from the zero-degree formulas into its elementssatisfying

v(:A) = :v(A);

v(A ^ B) = v(A) ^ v(B);

v(A _ B) = v(A) _ v(B):

Note that the occurrence of `:' on the left-hand side of the equation denotesthe negation connective, whereas the occurrence on the right-hand side de-notes the complementation operation in the lattice (similarly for ^ and _).Such ambiguities resolve themselves contextually.

A valuation v can be regarded as in interpretation of the formulas aspropositions.

De Morgan lattices have become central to the study of relevance oflogic, but they were antecedently studied, especially in the late 1950s byMoisil and Monteiro, by Bia lynicki-Birula and Rasiowa (as `quasi-Booleanalgebras'), and by Kalman (as `distributive i-lattices') (see Anderson andBelnap [1975] or Rasiowa [1974] for references and information).


Belnap seems to have �rst recognised their signi�cance for relevance logic,though his research favoured a special variety which he called an intension-ally complemented distributive lattice with truth �lter (`icdlw/tf'), shortenedin Section 18 of [Anderson and Belnap, 1975] to just intensional lattice. Anintensional lattice is a structure (L;^;_;:; T ), where (L;^;_;:) is a deMorgan lattice and T is a truth-�lter, i.e. T is a �lter which is complete inthe sense T contains at least one of a and :a for each a 2 L, and consistentin the sense that T contains no more than one of a and :a.

Belnap and Spencer [1966] showed that a necessary and suÆcient condi-tion for a de Morgan lattice to have a truth �lter is that negation have no�xed point, i.e. for no element a; a = :a (such a lattice was called an icdl).For Boolean algebras this is a non-degeneracy condition, assuring that thealgebra has more than one element, the one element Boolean algebra beingbest ignored for many purposes. But the experience in relevance logic hasbeen that de Morgan lattices where some elements are �xed points are ex-tremely important (not all elements can be �xed points or else we do havethe one element lattice).

The viewpoint of [Dunn, 1966] was to take general de Morgan lattices asbasic to the study of relevance logics (though still results were analogisedwherever possible to the more special icdl's). Dunn [1966] showed that uponde�ning a �rst-degree implication A ! B to be (de Morgan) valid i� forevery valuation v in a de Morgan lattice, v(A) � v(B); A ! B is valid i�A ! B is a theorem of Rfde (or Efde). The analogous result for icdl's (ine�ect due to Belnap) holds as well.

Soundness (`RfdeA ! B ) A ! B is valid) is a more or less triv-

ial induction on the length of proofs in Rfde fragment|cf. [Anderson andBelnap, 1975, Section 18].

Completeness (A ! B valid ) `RfdeA ! B) is established by proving

the contrapositive. We suppose not `RfdeA ! B. We then form the

`Lindenbaum algebra', which has as an element for each zero degree formula(zdf) X; [X ] =df fY : Y is a zdf and `Rfde

X $ Y g. Operations are de�nedso that :[X ] = [:X ], [X ] ^ [Y ] = [X ^ Y ], and [X ] _ [Y ] = [X _ Y ], andwe set [X ] � [Y ] whenever `Rfde

X ! Y . It is more or less transparent,given Rfde formulated as it is, that the result is a de Morgan lattice. Itis then easy to see that A ! B is invalidated by the canonical valuationvc(X) = [X ], since clearly [A] 6� [B].

The above kind of soundness and completeness result is really quite triv-ial (though not unimportant), once at least the logic has had its axiomschopped so that they look like the algebraic postulates merely written in adi�erent notation. The next result is not so trivial.

Characterisation Theorem of Rfde with Respect to 4. `RfdeA!

B i� A! B is valid in 4, i.e. for every valuation v in 4, v(A) � v(B).

Proof. Soundness follows from the trivial fact recorded above that Rfde is

RELEVANCE LOGIC 55

sound with respect to de Morgan lattices in general. For completeness weneed the following:

4-Valued Homomorphism Separation Property. Let D be a de Morganlattice with a 6� b. Then there exists a homomorphism h of D into 4 so thath(a) 6� h(b).

Completeness will follow almost immediately from this result, for uponsupposing that not `Rfde

A! B, we have v(A) = h[A] 6� h[B] = v(B) (thecomposition of a homomorphism with a valuation is transparently a valua-tion). So we go on to establish the Homomorphism Separation Property.

Assume that a 6� b. By the Prime Filter Separation Property, we knowthere is a prime �lter P with a 2 P and b 62 P . for a given element x,we de�ne h(x) according to the following four possible `complementationpatterns' with respect to P .

1. x 2 P;:x 62 P : set h(x) = 1;

2. :x 2 P; x 62 P : set h(x) = 0;

3. x 2 P;:x 2 P : set h(x) = p;

4. x 62 P;:x 62 P : set h(x) = q.

It is worth remarking that if D is a Boolean lattice, (3) (inconsistency)and (4) (incompleteness) can never arise, which explains the well-knownsigni�cance of 2 for Boolean homomorphism theory. Clearly these speci�-cations assure that h(a) 2 fp; Ig and h(b) 2 fq; 0g, and so by inspectionh(a) 6� h(b). It is left to the reader to verify that h in fact is a homomor-phism. (Hint to avoid more calculation: set [p) = fp; Ig and [q) = fq; Ig(the principal �lters determined by p and q). Observe that the de�nitionof h above is equivalent to requiring of h that h(x) 2 [p) i� x 2 P , andh(x) 2 [q) i� :a 62 P . Observe that if whenever i = p; q; y 2 [i) i� z 2 [i),then y = z. Show for i = p; q; h(a^b) 2 [i) i� h(a)^H(b) 2 [i), h(a_b) 2 [i)i� h(a) _ h(b) 2 [i), and h(:a) 2 [i) i� :h(a) 2 [i). �

3.4 Set-theoretical Semantics for First-degree Relevant Impli-cation

Dunn [1966] (cf. also [Dunn, 1967]) considered a variety of (e�ectivelyequivalent) representations of de Morgan lattices as structures of sets. Weshall here discuss the two of these that have been the most in uential in thedevelopment of set-theoretical semantics for relevance logic.

The earliest one of these is due to Bia lynicki-Birula and Rasiowa [1957]

and goes as follows. Let U be a non-empty set and let g : U ! U be suchthat it is of period two, i.e.

1. g(g(x)) = x, for all x 2 U .


(We shall call the pair (U; g) and involuted set|g is the involution, and isclearly 1{1). Let Q(U) be a `ring' of subsets of U (closed under \ and [)closed as well under the operation of `quasi-complement'

2. :X = U � g[X ](X � U).

(Q(U);[;\;:) is called a quasi-�led of sets and is a de Morgan lattice.

Quasi-fields of Sets Theorem [Bia lynicki-Birula and Rasiowa, 1957].Every de Morgan lattice D is isomorphic to a quasi-�eld of sets.

Proof. Let U be the set of all prime �lters of D, and let P range over U .Let :P ! f:a : a 2 Pg, and de�ne g(P ) = D�:P . We leave to the readerto verify that U is closed under g. For each element a 2 D, set

f(a) = fP : a 2 Pg:Clearly f is one{one because of the Prime Filter Separation Property, so

we need only check that f preserves the operations.

ad^: P 2 f(a ^ b), a ^ b 2 P , ((10) of Section 3.3) a 2 P and b 2 P ,P 2 f(a) and P 2 f(b), P 2 f(a) \ f(b). So f(a ^ b) = f(a) \ f(b)as desired.

ad_: The argument that f(a _ b) = f(a) [ f(b) is exactly parallel using(20) (or alternately this can be skipped using the fact that a _ b =:(:a ^ :b).

ad:: P 2 f(:a) , :a 2 P , a 2 :P , a 62 g(P ) , g(P ) 62 f(a) , P 62g[f(a)], P 2 U � g[f(a)].

We shall now discuss a second representation. Let U be a non-empty setand let R be a ring of subsets of U (closed under intersection and union, butnot necessarily under complement, quasi-complement, etc.). by a polarityin R we mean an ordered pair X = (X1; X2) such that X1; X2 2 R. Wede�ne a relation and operations as follows, given polarities X = (X1; X2)and Y = (Y1; Y2):

X � Y , X1 � Y1 and Y2 � X2

X ^ Y = (X1 \ Y1; X2 [ Y2)

X _ Y = (X1 [ Y1; X2 \ Y2)

:X = (X2; X1):

By a �eld of polarities we mean a structure (P (R);�;^;_;:) where P (R)is the set of all polarities in some ring of sets R, and the other componentsare de�ned as above. We leave to the reader the easy veri�cation that every�eld of polarities is a de Morgan lattice. �

RELEVANCE LOGIC 57

We shall prove the following

Polarities Theorem [Dunn, 1966]. Every de Morgan lattice is isomorphicto a �eld of polarities.

Proof. Given he previous representation, it clearly suÆces to show thatevery quasi-�eld of sets is isomorphic to a �eld of polarities.

The idea is to set f(X) = (X;U �g[X ]). Clearly f is one{one. We checkthat it preserves operations.

ad^: f(X\Y ) = (X\Y; U�g[X\Y ]) = (X\Y; (U�g[X ])\(U�g[Y ])) =(X;U � g[X ]) ^ (Y; U � g[Y ]) = f(X) ^ f(Y ).

ad_: Similar.

ad:: f(:X) = (:X;U � g(:X)) = (U � g[X ]; U � g(U � g[X ])) = (U �g[X ]; X) = :f(X).

�

We now discuss informal interpretations of the representation theoremsthat relate to semantical treatments of relevant �rst-degree implicationsfamiliar in the literature.

Routley and Routley [1972] presented a semantics for Rfde, the main in-gredients of which were a set K of àtomic set-ups' (to be explained) onwhich was de�ned an involution �. An àtomic set-up' is just a set of propo-sitional variables, and it is used to determine inductively when complexformulas are also ìn' a given set-up. A set-up is explained informally as be-ing like a possible world except that it is not required to be either consistentor complete. The Routley's [1972] paper seems to conceive of set-ups verysyntactically as literally being sets of formulas, but the Routley and Meyer[1973] paper conceives of them more abstractly. We shall think of them thislatter way here so as to simplify exposition. The Routleys' models can thenbe considered a structure (K; �;�), where K is a non-empty set, � is aninvolution on K, and � is a relation from K to zero-degree formulas. Weread à � A' as the formula A holds at the set-up a:

1. (^ �) a � A ^ B , a � A and a � B

2. (_ �) a � A _ B , a � A or a � B

3. (: �) a � :A, not a� � A.

The connection of the Routleys' semantics with quasi-�elds of sets will be-come clear if we let (K; �) induce a quasi-�eld of sets Q with quasi- com-plement :, and let j j interpret sentences in Q subject to the followingconditions:


10. j ^ j jA ^Bj = jAj \ jBj20. j _ j jA _Bj = jAj [ jBj30. j:j j:Aj = :jAj.Clause (^ �) results from clause j ^ j by translating a 2 jX j as a � X (cf.

Section 3.2). Thus clause j ^ j says

a 2 jA ^ Bj , a 2 jAj and a 2 jBj;

i.e. it translates as clause (^ �). The case of disjunction is obviously thesame. The case of negation is clearly of special interest, so we write it out.

Thus clause j:j says

a 2 j:Aj , a 2 :jAj;, a 2 K � jAj�;, a 62 jAj�;, a� 62 jAj:

But the translation of this last is just clause (: �).Of course the translation works both ways, so that the Routleys' seman-

tics is just an interpretation in the quasi-�elds of sets of Bia lynicki-Birulaand Rasiowa written in di�erent notation. Incidentally soundness and com-pleteness of Rfde relative to the Routleys' semantics follows immediatelyvia the translation above from the corresponding theorem of the previoussection vis �a vis de Morgan lattices together with their representation asquasi-�elds of sets. Of course the Routleys' conceived their results and de-rived them independently from the representation of Bia lynicki-Birula andRasiowa.

We will not say very much here about what intuitive sense (if any) can beattached to the Routleys' use of the �-operator in their valuational clause fornegation. Indeed this question has had little extended discussion in the lit-erature (though see [Meyer, 1979a; Copeland, 1979]). The Routleys' [1972]

paper more or less just springs it on the reader, which led Dunn in [Dunn,1976a] to describe the switching of a with a� as à feat of prestidigitation'.Routley and Meyer [1973] contains a memorable story about how a� `weaklyasserts', i.e. fails to deny, precisely what a asserts, but one somehow feelsthat this makes the negation clause vaguely circular. Still, semantics oftengives one this feeling and maybe it is just a question of degree. One wayof thinking of a and a� is to regard them as `mirror images' of one anotherreversing ìn' and òut'. Where one is inconsistent (containing both A and:A), the other is incomplete (lacking both A and :A), and vice versa (whena = a�, a is both consistent and complete and we have a situation appro-priate to classical logic). Viewed this way the Routleys' negation clause

RELEVANCE LOGIC 59

makes sense, but it does require some anterior intuitions about inconsistentand incomplete set-ups. More about the interpretation of this clause willbe discussed in Section 5.1.

Let us now discuss the philosophical interpretation(s) to be placed onthe representation of de Morgan lattices as �elds of polarities. In Dunn[1966; 1971] the favoured interpretation of a polarity (X1; X2) was as a`proposition surrogate', X1 consisting of the `topics' the proposition givesde�nite positive information about and, X2 of the topics the propositiongives de�nite negative information about. A valuation of a zero degreeformula in a de Morgan lattice can be viewed after a representation of theelements of the lattice as polarities as an assignment of positive and negativecontent to the formula. The `mistake' in the `classical' Carnap/Bar-Hillelapproach to content is to take the content of :A to be the set-theoreticalcomplement of the content of A (relative to a given universe of discourse).In general there is no easy relation between the content of A and that of:A. They may overlap, they may not be exhaustive. Hence the need forthe double-entry bookkeeping done by proposition surrogates (polarities).If A is interpreted as (X1; X2), :A gets interpreted as the interchanged(X2; X1).

Another semantical interpretation of the same mathematics is to be foundin Dunn [1969; 1976a]. There given a polarity X = (X1; X2); X1 is thoughtof as the set of situations in which X is true and X2 as the set of situationsin which X is false. These situations are conceived of as maybe inconsistentand/or incomplete, and so again X1 and X2 need not be set-theoretic com-plements. This leads in the case when the set of situations being assessedis a singleton fag to a rather simple idea. The �eld of polarities looks likethis

(fag; ;) T (= fTg)s

��s@@@s��

s@@@

(;; fag) F (= fFg)

(fag; fag) B (= fT; Fg) (;; ;) N (= ;)

We have taken the liberty of labelling the points so as to make clear theinformal meaning. (Thus the top is a polarity that is simply true in a andthe bottom is one that is simply false, but the left-hand one is both trueand false, and the right-hand one is neither.) Note that the de Morgancomplement takes �xed points on both B and N. This is of course our oldfriend 4, which we know to be characteristic for Rfde.

This leads to the idea of an `ambi-valuation' as an assignment to sentencesof one of the four values T, F, B, N, conceived either as primitive or realised


as sets of the usual two truth values as suggested by the labelling. On thislatter plan we have the valuation clauses (with double entry bookkeeping):

(^) T 2 v(A ^ B), T 2 v(A) and T 2 v(B);F 2 v(A ^ B), F 2 v(A) or F 2 v(B);

(_) T 2 v(A _ B), T 2 v(A) or T 2 v(B);F 2 v(A _ B), F 2 v(A) and F 2 v(B);

(:) T 2 v(:A), F 2 v(A);F 2 v(:A), T 2 v(A):

We stress here (as in [Dunn, 1976a]) that all this talk of something'sbeing both true and false or neither is to be understood epistemically andnot ontologically. One can have inconsistent and or incomplete assumptions,information, beliefs, etc. and this is what we are trying to model to see whatfollows from them in an interesting (relevant!) way. Belnap [1977b; 1977a]

calls the elements of the lattice `told values' to make just this point, and goeson to develop (making connections with Scott's continuous lattices) a theoryof `a useful four-valued logic' for `how a computer should think' withoutletting minor inconsistencies in its data lead to terrible consequences.

Before we leave the semantics of �rst-degree relevant implications, weshould mention the interesting semantics of van Fraassen [1969] (see also An-derson and Belnap [1975, Section 20.3.1] and van Fraassen [1973]), whichalso has a double-entry bookkeeping device. We will not mention detailshere, but we do think it is an interesting problem to try to give a represen-tation of de Morgan lattices using van Fraassen's facts so as to try to bringit under the same umbrella as the other semantics we have discussed here.

3.5 The Algebra of R

This section is going to be brief. Dunn has already exposited on this topic inSection 28.2 of [Anderson and Belnap, 1975] and the interested reader shouldconsult that and then Meyer and Routley [1972] for information about howto algebraise related weaker systems and how to give set-theoretical repre-sentations.

De Morgan monoids are a class of algebras that are appropriate to Rin the sense that (i) the Lindenbaum algebra of R is one of them and (ii)all R theorems are valid in them ((ii) gives soundness, and of course (i)delivers completeness by way of the canonical valuation). In thinking aboutde Morgan monoids it is essential that R be equipped with the sententialconstant t. Also it is nice to think of fusion (Æ) as a primitive connective,with even perhaps ! de�ned (A ! B =df :(A Æ :B)) but this is notessential since in R (but not the weaker relevance logics) fusion can bede�ned as A ÆB =df :(A! :B).

A de Morgan monoid is a structure D = (D;^;_;:; Æ; e) where

RELEVANCE LOGIC 61

(I) (D;^;_;:) is a de Morgan lattice,

(II) (D; Æ; e) is an Abelian monoid, i.e. Æ is a commutative, associativebinary operation on D with e its identity, i.e. e 2 D and e Æ a = a forall a 2 D,

(III) the monoid is ordered by the lattice, i.e. a Æ (b _ c) = (a Æ b) _ (a Æ c),(IV) Æ is upper semi-idempotent (`square increasing'), i.e. a � a Æ a,

(V) a Æ b � c i� a Æ :c � :b (Antilogism).

De Morgan monoids were �rst studied in [Dunn, 1966] (although [Meyer,1966] already had isolated some of the key structural features of fusionthat they abstract). They also were described in [Meyer et al., 1974] andused in showing admissible. Similar structures were investigated quiteindependently by Maksimova [1967; 1971].

The key trick in relating de Morgan monoids to R is that they are resid-uated, i.e. there is a `residual' operation ! so that

(VI) a Æ b � c i� a � b! c.

Indeed this operation turns out to be :(b Æ:c) (with the weaker systemsor with positive R it is important to postulate this law of the residual).Thus

(1) a Æ b � c, b Æ a � c Commutativity(2) a Æ b � c, b Æ :a 1, (V)(3) a Æ b � c, a � :(b Æ :c) 2, de Morgan lattice.

As an illustration of the power of (VI) we show how the algebraic analogueof the Pre�xing axiom follows from Associativity. First note that one canget from (III) the law of

(Monotony) a � b) c Æ a � c Æ b.Now getting down to Pre�xing:

1. a! b � a! b

2. (a! b) Æ a � b 1, (VI)

3. (c! a) Æ c � a 2, Substitution

4. (a! b) Æ ((c! a) Æ c) � b 2,3, Monotony

5. ((a! b) Æ (c! a)) Æ c � b 4, Associatively

6. (a! b) Æ (c! a) � c! b 5, (VI)

7. a! b � ((c! a)! (c! b)) 6, (VI).


Incidentally, something better be said at this point about how validityin de Morgan monoids is de�ned. Unlike the case with Rfde, there aretheorems which are of the form A! B, e.g. A_:A. We need some way ofde�ning validity which is broader than insisting that always v(A) � v(B).The identity e interprets the sentential constant t. By virtue of the Raxiom A$ (t! A) characterising t, it makes sense to count all de Morganmonoid elements a such that e � a as `designated', and to de�ne A asvalid i� v(A) � e for all valuations in all de Morgan monoids. We have thefollowing law

a � b, e � a! b;

which follows immediately from (VI) and the fact that e is the identityelement. This means that (7) just above can be transformed into

e � (a! b)! ((c! a)! (c! b))

validating pre�xing as promised.Other axioms of R can be validated by similar moves. Commutativity

validates Assertion, that e is the identity validates self-implication, square-increasingness validates Contraction, antilogism validates Contraposition,and the other axioms fall out of de Morgan lattice properties with latticeordering and the residual law pitching in.

We shall not here investigate the `converse' questions about how thefusion connective in R is associative, etc. (that the Lindenbaum algebra ofR is indeed a de Morgan monoid (cf. Dunn's Section 28.2.2 of [Andersonand Belnap, 1975])), but the proof is by `�ddling' with contraposition beingthe key move.

Not as much is known about the algebraic properties of de Morganmonoids as one would like. Getting technical for a moment and using unex-plained but standard terminology from universal algebra, it is known that deMorgan monoids are equationally de�nable (replace (V) with aÆ:(aÆ:b) �b, which can be replaced by the equation (a Æ :(a Æ :b)) _ b = b). Soby a theorem of Birkho� the class of de Morgan monoids is closed undersub-algebras, homomorphic images, and subdirect products. Further, givena de Morgan monoid D with a prime �lter P with e 2 P , the relationa � b , (a ! b) ^ (b ! a) 2 P is a congruence, and the quotient algebraD=� is subdirectly irreducible, and every de Morgan monoid is a subdirectproduct of such. It would be nice to have some independent interestingcharacterisation of the subdirectly irreducibles.

One signi�cant recent result about the algebra of R has been providedby John Slaney. He has shown that there are exactly 3088 elements in thefree De Morgan monoid generated by the identity e. Or equivalently, inthe language of R including the constant t, there are exactly 3088 non-equivalent formulae free of propositional variables. The proof technique is

RELEVANCE LOGIC 63

quite subtle, as generating a large algebra of 3088 elements is not feasible,even with computer assistance. Instead, Slaney attacked the problem usinga \divide and conquer" technique [Slaney, 1985]. Since R contains all for-mulae of the form A _ :A, for any A, whenever L is a logic extending R,L = (L+A)\ (L+:A), where L+A is the result of adding A as an axiomto L and closing under modus ponens and adjunction. Given this simpleresult, we can proceed as follows. R is (R+f ! t)\(R+:(f ! t)). Now itis not diÆcult to show that the algebra of R+ (f ! t) generated by t is thetwo element boolean algebra. Then you can restrict your attention to thealgebra generated by t in the logic R + :(f ! t). If this has some charac-teristic algebra, then you can be sure that the elements freely generated byt in R are bounded above by the number of elements in the direct productof the two algebras. To get the characteristic algebra of R + :(f ! t),Slaney goes on to divide and conquer again. He ends up considering sixmatrices, characterising six di�erent extensions of R. This would give himan upper bound on the number of constants (the matrices were size 2, 4,6, 10, 10 and 14, so the bound was their product, 67200, well above 3088).Then you have to consider how many of these elements are generated by theidentity in the direct product algebra. A reasonably direct argument showsthat there are exactly 3088 elements generated in this way, so the result isproved.

3.6 The Operational Semantics (Urquhart)

This set-theoretical semantics is based upon an idea that occurred indepen-dently to Urquhart and Routley in the very late 1960s and early 70s. Weshall discuss Routley's contribution (as perfected by Meyer) in the next sec-tion and also just mention some related independent work of [Fine, 1974].Here we concentrate upon the version of Urquhart [1972c] (cf. also Urquhart[1972b; 1972a; 1972d]).

Common to all the versions is the idea that one has some set K whoseelements are `pieces of information', and that there is a binary operation Æon K that combines pieces of information. Also there is an `empty pieceof information' 0 2 K. We shall write x � A to mean intuitively `A holdsaccording to the piece of information x'. The whole point of the semanticsis disclosed in the valuational clause

(!) x � A! B i� 8y 2 K (if y � A, then x Æ y � B):

The idea of the clause from left-to-right is immediately clear: if A ! Bis given by the information x, then if A is given by y, then the combinedpiece of information x Æ y ought to give B (by modus ponens). The ideaof the clause from right-to-left is to say that if this happens for all piecesof information y, this can only be because x gives us the information thatA! B.


Perhaps saying the whole point of the semantics is given in the clause (!)along is an exaggeration. There are at least two quick surprises. The �rstis that we do not require (or want) a certain condition analogised from acondition required by Kripke's (relational) semantics for intuitionistic logic:

(The Hereditary Condition) If x � A, then x Æ y � A:

This would yield that if x � A, then x � B ! A, i.e. if y � B, tenx Æ y � A. This would quickly involve us in irrelevance.

The other surprise is related to the failure of the Hereditary Condition:Validity cannot be de�ned as a formula's holding at all pieces of informationin all models, since even A ! A would not then turn out to be valid.Thus x � A ! A requires that if y � A then x Æ y � A. But this lastis just a commuted form of the rejected Hereditary Condition, and thereis no more reason to think it holds. We shall see in a moment that theappropriate de�nition of validity is to require that 0 � A for the emptypiece of information in all models.

Enough talk of what properties Æ does not have! What property does ithave? We have just been irting with one of them. Clearly 0 � A ! Arequires that if x � A then 0 Æ x � A, and how more naturally would thatbe obtained than requiring that 0 be a (left) identity?

0 Æ x = x: (Identity)

This then seems the minimal algebraic condition on a model. Urquhartin fact requires others, all naturally motivated by the idea that Æ is the`union' of pieces of information.

x Æ y = y Æ x (Commutativity)

x Æ (y Æ z) = (x Æ y) Æ z (Associativity)

x Æ x = x: (Idempotence)

These conditions combined may be expressed by saying that (K; Æ; 0)is a `(join) semi-lattice with least element 0', and accordingly Urquhart'ssemantics is often referred to as the `semi-lattice semantics'. It is well-knownthat every semi-lattice is isomorphic to a collection of sets with union as Æand the empty set as 0 (map x to fy : x Æ y = yg so that henceforward Æwill be denoted by [.

Each of the conditions above of course corresponds to an axiom of R!

when it is nicely axiomatised. Thus commutativity plays a natural role inverifying the validity of assertion. The following use of natural deduction

RELEVANCE LOGIC 65

in the metalanguage makes this point nicely (we write `A; x' rather thanx � A for a notational analogy):

1. A; x Hypothesis2. A! B; z Hypothesis3. B; x [ z 1, 2, (!)4. B; z [ x 3, Comm.5. (A! B)! B; x 2, 4(!)6. (A! B)! B; 0 [ x 5, Identity7. A! ((A! B)! B); 0.

The reader may �nd it amusing to write out an analogous pair of proofsfor Pre�xing, seeing how Associativity of [ enters in, and for Contractionwatching the Idempotence.24

The game has now been given away. There is some �ddling to be sure inproving a completeness theorem for R! re the semi-lattice semantics, butbasically the idea is that the semi-lattice semantics is just the system FR!

`written in the metalanguage'.There is not a problem in extending the semi-lattice semantics so as to

accommodate conjunction. The clause

x � A ^ B i� x � A and x � B (^)

does nicely. Somewhat strangely, the `dual' clause

x � A _ B i� x � A of x � B (_)

causes trouble. It is analogous to having the rule of _-Elimination NRread:

A _ B; xA; x Hyp.

...c; x [ yB; x Hyp.

...C; x [ yC; x [ y:

With this rule we can prove

(]) (A! B _ C) ^ (B ! C)! (A! C);

24Though unfortunately veri�cation of this last does not depend purely on Idempo-tence, but rather on (xy)y = xy, which of course is equivalent to Idempotence givenAssociativity and Identity. The veri�cation of the formula A ^ (A ! B) ! B `exactly'uses Idempotence, but of course this is hardly a formula of the implicational fragment.


which is not a theorem of R (see [Urquhart, 1972c]|the observation isMeyer's and Dunn's.)25 And of course one can analogously verify that it isvalid in the semi-lattice semantics.

Note that the condition (_) is not nearly as intuitive as the condition(^). The condition (^) is plausible for any piece of information x, at leastif the relation x � c does not require that C be explicitly contained in x.On the other hand the condition (_) is much less than natural. Does not ithappen all the time that a piece of information x determines A_B to hold,without saying which? Is not this one of the whole points of disjunctions?Pieces of information x that satisfy (_) might be called `prime' (in analogywith this epithet applied to theories of Section 2.4), and they have a kind ofcompleteness or e�eminateness that is rare in ordinary pieces of information.This by itself counts as no criticism of the semantics, since it is quite usualin semantical treatments to work with such idealised notions.

The condition (_) is not really as `dual' to the condition (^) as one mightthink. Thus the formula

(]d) (B ^ C ! A) ^ (C ! B)! (C ! A);

which is the dual of (]) is easily seen not to be valid in the semantics. Thisseems to be connected with another feature (problem?) of the semantics, towit, no one has ever �gured out how to add a natural semantical treatmentof classical negation to the semantics (although it is straightforward to add aspecies of constructive negation|see [Urquhart, 1972c]).26 The point of theconnection is that (]d) would follow from (]) given classical contrapositionprinciples, and yet the �rst is valid and the second one invalid in the positivesemantics. So something about the positive semantics would have to bechanged as well to accommodate negation.

The semi-lattice semantics has been extensively investigated in Charlewood[1978; 1981]. He �ts it out with (two) natural deduction systems one withsubscripts and one without. This last is in fact the (positive) system ofPrawitz [1965], which Prawitz wrongly conjectured to be the same as Ander-son and Belnap's. Charlewood proves normalisation theorems (somethingthat was anticipated by Prawitz for his system|incidentally the problemof normalisation for the Anderson{Belnap R seems still open). Inciden-tally, one advantage of these natural deduction systems is that, unlike theAnderson{Belnap one for their system R (cf. Section 1.5), they allow for aproof of distribution.

Charlewood also carries out in detail the engineering needed to implementK. Fine's axiomatisation of the semi-lattice semantics. What is needed is

25It would be with C ! C as an additional conjunct in the antecedent.26Charlewood and Daniels have investigated a combination of the semi-lattice semantics

for the positive connectives and a four-valued treatment of negation in the style of [Dunn,1976a]. they avoid the problem just described by in e�ect building into their de�nition ofa model that it must satisfy classical contraposition. This does not seem to be natural.

RELEVANCE LOGIC 67

to add to the Anderson{Belnap's R+ the following rule:

R1: From B0^((A1^q1; : : : ; qn^An)! X)! ((B1^q1; : : : ; Bn^qn)! E)for X = B;C, and n � 0 infer the same thing with B_C put in place ofthe displayed X , provided that he qi are distinct and occur only whereshown.

We forbear taking cheap shots at such an ungainly rule, the true eleganceof which is hidden in the details of the completeness proof that we shall notbe looking into. Obviously Anderson and Belnap's R is to be preferredwhen the issue is simplicity of Hilbert-style axiomatisations.27

3.7 The Relational Semantics (Routley and Meyer)

As was indicated in the last section, Routley too had the basic idea of theoperational semantics at about the same time as Urquhart. Priority wouldbe very hard to assess. At any rate Dunn �rst got details concerning boththeir work in early 1971, although J. Garson told him of Urquhart's workin December of 1970 and he has seen references made to a typescript ofRoutley's with a 1970 date on it (in [Charlewood, 1978]).

Meyer and Dunn were colleagues at the time, and Routley sent Meyera somewhat incomplete draft of his ideas in early 1971. This was a coura-geous and open communication in response to our keen interest in the topic(instead he might have sat on it until it was perfected). The draft favouredthe operational semantics, indeed the semi-lattice semantics, and was notclear that this was not the way to go to get Anderson and Belnap's R. Butthe draft started with a more general point of view suggesting the use of a3-placed accessibility relation Rxyz (of course a 2-placed operation like [ isa 3-placed relation, but not always conversely), with the following valuationclause for !:

(!) x � A! B i� 8y; z 2 K (if Rxyz and y � A, then z � B):

Forgetting negation for the moment, the clauses for ^ and _ are `truthfunctional', just as for the operational semantics.

Meyer, having observed with Dunn the lack of �t between the semi-lattice semantics and R, was all primed to make important contributions toRoutley's suggestion. In particular he saw that the more general 3-placedrelation approach could be made to work for all of R. In interpreting Rxyzperhaps the best reading is to say that the combination of the pieces ofinformation x and y (not necessarily the union) is a piece of informationin z (in bastard symbols, x Æ y � z). Routley himself called the x; y, etc.

27However, the semi-lattice semantics has been taken up and generalised in the �eld ofsubstructural logics in the work of [Do�sen, 1988; Do�sen, 1989] and [Wansing, 1993].


`set-ups', and conceived of them as being something like possible worldsexcept that they were allowed to be inconsistent and incomplete (but alwaysprime). On this reading Rxyz can be regarded as saying that x and y arecompatible according to z, or some such thing.

Before going on we want to advertise some work that we are not going todiscuss in any detail at all because of space limitations. The work of Fine[1974] independently covers some of the same ground as the Routley-Meyerpapers, with great virtuosity making clear how to vary the central ideasfor various purposes. The book of Gabbay [1976, see chapter 15] is alsodeserving of mention.

We now set out in more formal detail a version of the Routley{Meyersemantics for R+ (negation will be reserved for the next section). Thetechniques are novel and the completeness proof quite complicated, so weshall be reasonably explicit about details. The presentation here is verymuch indebted to work (some unpublished) of Routley, Meyer and Belnap.

By an (R+) frame (or model structure) is meant a structure (K;R; 0),where K is a non-empty set (the elements of which are called set-ups), Ris a 3-placed relation on K; 0 2 K, all subject to some conditions we shallstate after a few de�nitions. We de�ne for a; b 2 K; a � b (Routley andMeyer used >) i� R0ab, and R2abcd i� 9x (Rabx and Rxcd). We also writethis last as R2(ab)cd and distinguish it from R2a(bc)d =df 9x(Raxd^Rbcx).The variables a; b, etc. will be understood as ranging over the elements ofsome K �xed by the content of discussion.

Transcribing the conditions on the semi-lattice semantics as closely as wecan into this framework we get the requirements

1. (Identity) R0aa,

2. (Commutativity) Rabc) Rbac,

3. (Associativity) R2(ab)cd) R2a(bc)d,28

4. (Idempotence) Raaa.

It should be remarked that these conditions fail to pick up the wholestrength of the corresponding semi-lattice conditions. Thus, e.g. Identityhere only picks up 0�a � a and not conversely, and similarly for Idempotence(also of course Commutativity and Associativity do not require any identity,but this is a slightly di�erent point). We need for technical reasons one morecondition:

5. (Monotony) Rabc and a0 � a) Ra0bc.

28In the original equivalent conditions of Routley and Meyer [1973] this was instead`Pasch's Law': R2abcd) R2acbd. Also Monotony (condition (5) below) was misprintedthere.

RELEVANCE LOGIC 69

By a model we mean a structure M = (K;R; 0;�), where (K;R; 0) is aframe and � is a relation from K to sentences of R+ satisfying the followingconditions:

(1) (Atomic Hereditary Condition). For a propositional variable p, if a � pand a � b, then b � p.

(2) (Valuational Clauses). For formulas A;B

(!) a � A! B i� 8b; c 2 K (if Rabc and b � A, then c � B);

(^) a � A ^ B i� a � A and a � B;

(_) a � A _ B i� a � or a � B.

We shall say that A is veri�ed on M if 0 � A, and that A entails B onM if 8a 2 K (if a � A, then a � B). We say that A is valid if A is veri�edon all models.

It is easy to prove by an induction onA, the following (note how Monotonyenters in):

Hereditary Condition. For an arbitrary formula A, if a � A and a � b,then b � A.

Verification Lemma. If in a given model (K;R; 0;�)A entails B in thesense that for every a 2 K; a � A only if a � B, then A ! B is veri�ed inthe model, i.e. 0 � A! B.

Proof. suppose that R0ab and a � A. By the hypothesis of the Lemma,a � B, and by the Hereditary Condition, b � B, as is required for 0 � A!B. �

We are now in a position to prove the

Soundness Theorem. If `R A, then A is valid.

Proof. Most of this will be left to the reader. We �rst show that the axiomsof R+ are valid. Since they are all of the form A ! B we can simplifymatters a little by using the Veri�cation Lemma. As an illustration weverify Assertion (the reader may wish to compare this to the correspondingveri�cation vis �a vis the semi-lattice semantics of the last section).

To show A ! [(A ! B) ! B] is valid, it suÆces by the Veri�cationLemma to assume a � A and show a � A ! B ! B. For this last weassume Rabc and b � A ! B, and show c � B. By Commutativity, Rabc.By (!) since we have b � A! B and a � A, we get c � B as desired.

The veri�cation of the implicational axioms of Self-Implication and Pre-�xing are equally routine, falling right out of the Veri�cation Lemma and


Associativity for the relation R. Unfortunately the veri�cation of Contrac-tion is a bit contrived (cf. note 24 above), so we give it here.

To verify Contraction, we assume that (1) a � A ! :A ! B and showa � A! B. To show this last we assume that (2) Rabc and (3) b � A, andshow c � B. From (2) we get, by Commutativity, Rbac. But Rbbb holds byIdempotence. so we have R2(bb)ac. By Associativity we get R2b(ba)c, i.e.for some x, both (4) Rbxc and (5) Rbax. by Commutativity, from (5) weget Rabx. Using (!), we obtain from this, (1), and(3) hat (6) x � A! B.by Commutativity from (4) we get Rxbc, and from this, (6), and (3) we atlast get the desired c � B.

Veri�cation of the conjunction and disjunction axioms is routine and issafely left to the reader.

It only remains to be shown then that the rules modus ponens and ad-junction preserve validity. Actually something stronger holds. It is easy tose that for any a 2 K (not just 0), if a � A ! B and a � A, then a � B(by virtue of Raaa), and of course it follows immediately from (^) that ifa � A and a � B, then a � A ^ B. �

We next go about the business of establishing the

Completeness Theorem. If A is valid, then �R+ A.

The main idea of the proof is similar to that of the by now well-knownHenkin-style completeness proofs for modal logic. We suppose that no �R+

A and construct a so-called `canonical model', the set-us of which are certainprime theories (playing the role of the maximal theories of modal logic).The base set-up 0 is constructed as a regular theory (for the terminology`regular', `prime', etc. consult Section 2.4; of course everything is relativisedto R+). From this point on for simplicity we shall assume that we aredealing with R+ out�tted with the optional extra fusion connective Æ andthe propositional constant t (recall these can be conservatively added | cf.Section 1.3). We then de�ne Rabc to hold precisely when for all formulasA and B, whenever A 2 a and B 2 b, then A ÆB 2 c.29

Let us look now at the details. Pick 0 as some prime regular theory Twith A 62 T . We can derive that at least one such exists using the BelnapExtension Lemma (it was stated in Section 2.5 for RQ, but it clearly holdsfor R+ as well). thus set � = R+ and � = fAg.

De�ne K = set of prime theories,30 and de�ne the accessibility relationR canonically as above.29The use of Æ and t is a luxury to make things prettier at least at the level of descrip-

tion. Thus, e.g. as we shall see, the associativity of R follows from the associativity ofÆ, and other mnemonically pleasant things happen. We could avoid its use by de�ningRabc to hold whenever if A 2 a and A! B 2 b, then B 2 c. Incidentally, the valuationalclause for fusion is : x � A Æ B i� for some a; b such that Rabx; a � A and b � B. Thevaluational clause for t is x � t i� 0 � x.30One actually has a choice here. We have required of theories that they be closed

under implications provable in 0, i.e. require of T that whenever A 2 T and A! B 2 0,

RELEVANCE LOGIC 71

THEOREM 4. The canonically de�ned structure (K; 0; R) is an R+ frame.

LEMMA 5. The relation R de�ned canonically above satis�es Identity, Com-mutativity, Idempotence, and Associativity.

Proof.

ad Identity. We need to show that R0aa, i.e. if X 2 0 and A 2 a, thenX ÆA 2 a. By virtue of the R -theorem A! t ÆA, we have t ÆA 2 a. Butusing the R-theorem X ! :t ! x, we have t ! X 2 0. By Monotony wehave X ÆA 2 a as desired.

ad Commutativity. Suppose Rabc. We need show Rbac, i.e. if B 2 b andA 2 a, then B ÆA 2 c. From Rabc, it follows that A ÆB 2 c. But by virtueof the R-theorem A ÆB ! B ÆA (commutativity of Æ) we have B ÆA 2 C,as desired.

ad Idempotence. We need show Raaa, i.e. if A 2 a and B 2 a, thenA ÆB 2 a. This follows from the R-theorem A ^B ! A ÆB, which followsultimately from the square increasingness of Æ; (X ! X ÆX), as the proofsketch below makes clear.

1. A ^ B ! A Axiom

2. A ^ B ! B Axiom

3. (A ^ B) Æ (A ^ B)! A ÆB 1, 2, Monotony

4. A ^ B ! A ÆB 3, square increasingness

ad Associativity. This is by far the least trivial property. Let us thenassume that R2(ab)cd, i.e. 9x(Rabx and Rxcd). We need then show thatthere is a prime theory y such that Rayd and Rbcy, i.e. R2a(bc)d.

Set y0 = fY : 9B 2 b; C 2 c :`R BÆC ! Y g. (This is sometimes referredto as b Æ c). Clearly the de�nition of y0 assures that Rbcy0.

Observe that y0 is a theory.31 Thus it is clear that y0 is closed underprovable R-implication, since this is just transitivity. We show it is alsoclosed under adjunction. Thus suppose for some B;B0 2 b; C; C 0 2 c;�RB Æ C ! Y and `R B0 Æ C 0 ! Y 0. Then `R (B Æ C) ^ (B0 Æ C 0) !Y ^ Y 0 using easy properties of conjunction. But we have the R-theorem(B ^B0) Æ (C ^C 0)! (B ÆC) ^ (B0 ÆC 0) (which follows basically from theone-way distribution of Æ over ^; X Æ (Y ^ Z) ! (X Æ Y ) ^ (X Æ Z), which

then B 2 T . The latter is a stronger requirement and leads to the `smaller' reducedmodels of [Routley et al., 1982], which are useful for various purposes.31The presentation of Routley{Meyer [1973] is more elegant than ours, developing as

they do properties of what they call the calculus of `intensional R-theories', showing thatit is a partially ordered (under inclusion) commutative monoid (Æ as de�ned above) withidentity 0. Further Æ is monotonous with respect to �, i.e. if a � b then c Æ a � c Æ b, andÆ is square increasing, i.e. a � a Æ a. Then de�ning Rabc to mean a Æ b � c, the requisiteproperties of R fall right out.


follows basically from Monotony, Y1 ! Y2 ! X Æ Y1 ! X Æ Y2, which iseasy). So by transitivity we get `R (B ^ B0) Æ (C ^ C 0) ! Y ^ Y 0, fromwhich it follows that Y ^ Y 0 2 y0 as promised (B ^ B0 2 b; C ^ C 0 2 c ofcourse, since b; c are closed under adjunction).

We next verify that Ray0d. Suppose that A 2 a and Y 2 y0. Thenfor some B 2 b; C 2 c;`R B Æ C ! Y . Since Rabx;A Æ B 2 x. Andsince Rxcd(A Æ B) Æ C 2 d. By the associativity of Æ (since d is a theory),then A Æ (B Æ C) 2 d. but by Monotony, since `R B Æ C ! Y , we have`R A Æ (B Æ C)! A Æ Y . Hence A Æ Y 2 d, as needed.

The reader is excused if he has lost the thread a bit and thinks that weare now �nished verifying the associativity of R. We wanted some primetheory y which �lls in the blanks

1. Ra d and

2. Rbc ,

and we have just �nished verifying that y0 is a theory that does �ll in theblanks. The kicker is that y0 need not be prime. So we work next atpumping up y0 to make it prime while continuing to �ll in the blanks.

It clearly suÆces to prove

The Squeeze Lemma. Let a0 and y0 be theories that need not be prime,and let d be a prime theory. If Ra0y0d, then there exists a prime theory ysuch that (i) y0 � y and (ii) Ra0yd.

This can be accomplished by a Lindenbaum-style construction like thatof Section 2.3 (or alternatively Zorn's Lemma may be used as in Routleyand Meyer [1973]). The idea is to de�ne y as the union of a sequenceof sets of formulas yn, where (relative to some �xed enumeration of theformulas) yn+1 is de�ned inductively as yn [ fAn+1g if Ra(yn [ fAn+1g)d,and otherwise yn+1 is just yn.

But it is instructive to crank the existence of the given y out of the BelnapExtension Lemma for R.

Thus set � = y0 and � = fA : 9B(A ! B) 2 a and B 62 dg. We needcheck that (�; �) is exclusive.

We observe �rst that � is closed under disjunction. Thus supposeA1; A2 2�. Then for some B1; B)2; A1 ! B1; A2 ! B2 2 a, and yet B1; B2 62 d.Then (since d is prime) B1 _ B2 62 d. but since a is a theory, thenA1 _ A2 ! B1 _ B2 by an appropriate theorem of R in the proximityof the disjunction axioms. So A1 _ A2 2 � as desired. Since � is closeddually under adjunction (that was the point of observing above that y0 is atheory), this means that if the pair (�; �) fails to be exclusive, then for someX 2 �; A 2 �;`R X ! A. So for some B;A! B 2 a and B 62 d. But sincea is a theory, by transitivity we derive that X ! B 2 a. But since Raxd

RELEVANCE LOGIC 73

and X 2 x, we get (X ! B) ÆX 2 d. But since `R X Æ (X ! B)! B, wehave B 2 d, contrary to the choice of B.

Now that we know (�;�) is an exclusive pair we apply the Belnap Ex-tension Lemma to get a pair (y; y0) with y0 = � � y and y a prime theory,completing the proof of the Squeeze Lemma, which actually does completethe proof that the relation R is Associativity.

ad Monotony. (Yes, we still have something left to do.) Let us supposethat R0a0a and Rabc, and show Ra0bc. Note that it follows from R0a0athat a0 � a,32 from which it follows at once from Rabc and Ra0bc. Thusif X 2 a0 then since X ! X 2 0, then (X ! X) Æ X 2 a. But since`R+ (X ! X) ÆX ! X , then X 2 a.

Having now �nally veri�ed that the canonical (K; 0; R) has all the prop-erties of an R+-frame, we need now to de�ne an appropriate relation � onit. The natural de�nition is a � A i� A 2 a, but we need now to verify thatthis has the properties (1) and (2) required of � above.

Theorem 2. The canonically de�ned (K; 0; R;�) is indeed an R-model.

Proof. ad (1) (the Hereditary Condition). Suppose a � b, i.e. R0ab. Weshow that a � b, from which the Hereditary Condition immediately follows.Suppose then that A 2 a. Since t 2 0; t Æ A 2 b. But via the R-theoremt ÆA! A, we have A 2 b as desired.

ad (2) (the valuation of clauses). The clauses (^) and (_) are more orless immediate (primeness is of course needed for half of (_)). The clauseof interest is (!). Applying the canonical de�nition of �, this amounts to

(!c) A! B 2 a i� 8b; c(if Rabc and A 2 b; then B 2 c):

Left-to-right is argued as follows. Suppose A ! B 2 a;Rabc; A 2 b,and show B 2 c. Rabc of course means canonically that whenever X 2 aand Y 2 b, then X Æ Y 2 c. Setting X = A ! B and Y = A, we getA Æ (A! B)! C. Then using the R+- theorem

A Æ (A! B)! B; we obtain B 2 c:

Right-to-left is harder, and in fact involves the third (and last) applicationof the Belnap Extension Lemma in the proof of Completeness. Thus supposecontrapositively that A ! B 62 a. We need to construct prime theories band c, with A 2 b and B 62 c. We let �b = Th(fAg) and set �c = a Æ�b,i.e. fZ : 9X 2 a; 9Y 2 �b `R+ X ÆY ! Zg. This is the same as fZ : 9X 2a `R+ X ÆA! Zg. We set �c = fBg. Clearly (�c; �c) is an exclusive pair,for otherwise `R+ X ÆA! B, i.e. `R+ X ! (A! B) for some X 2 a, andso A ! B 2 a contrary to our supposition. We apply Belnap's ExtensionLemma to get an exclusive pair (c; c0) with �c � c and c prime theory. Note

32In the `reduced models (cf. note 46) one can show that R0a0a i� a0 � a.


that by de�nition of �b and �c; Ra�b�c, and so Ra�bc. We are now in aposition to apply the Squeeze Lemma getting a prime theory b � �b suchthat Rabc. Clearly A 2 b, but also B 62 c since B 2 �c � c0 (c and c0 areexclusive).

This at last completes the proof of the Completeness Theorem for R+.�

Remark. It is fashionable these days to always prove strong completeness.This could have been done. Thus de�ne A to be a logical consequence ofa set of formulas � i� for every R+-model M , if 0 � B for every B 2 �,then 0 � A. This is a kind of classical notion and should not be confusedwith some kind of relevant consequence. Thus, e.g. where B is a theoremof R+, since always 0 � B, B will be a logical consequence of any set �.De�ne B to be deducible from � (again in a neo-classical sense) to meanB 2 Th(� [R+). Appropriate modi�cations of the work above will showthat logical consequence is equivalent to deducibility.

3.8 Adding Negation to R+

We now discuss the Routley{Meyer semantics for the whole system R.The idea is simply to add the Routley's treatment of negation using the�-operator (discussed in Section 3.4). (This is not diÆcult and there is verylittle reason to segregate it o� into this separate section, except that wethought that the treatment of R+ was complicated enough.)

Thus an R-frame is a structure, (K;R; 0; �) where (K;R; 0) is an R+-frame and K is closed under the unary operation � satisfying:

(Period two) A�� = a,(Inversion) Rabc) Rac�b�

For an R-model the valuational clauses for the positive connectives are asfor an R-model, and we of course add

(:) a � :A i� a� 6� A:

The soundness and completeness results are relatively easy modi�cationsof those for R+. That � is of period two naturally is used n the veri�cationof Double Negation and Inversion is central to the veri�cation of Contra-position. For completeness, a� is de�ned canonically as fA : :A 62 ag (cf.the de�nition of the analogue g[P ] in the proof of Bia lynicki{Birula andRasiowa's representation of de Morgan lattices in Section 3.4), and one ofcourse has to show that a� is a prime theory when a is. One also has to showthat canonical � is of period two and satis�es (Inversion), and that canonical� satis�es (:) above, i.e. A 2 a , :A 62 a�, i.e. :A 62 fB : :B 62 ag, i.e.::A 2 a, which of course just uses Double Negation.

RELEVANCE LOGIC 75

It is worth remarking that since the canonical 0 is a prime regular theory,then since `R A _ :A, then 0 is complete (but not necessarily consistent|this is relevant to the development in Section 3.9). For your garden varietyRoutley{Meyer model (not necessarily canonical) notice also that 0 � A or0 � :A. This follows ultimately from 0� � 0, i.e. R00�0, proven below.

1. R0�0�0�

2. R0�00 1, (Inversion), (Period two)

3. R00�0 2, (Commutation).

Now 0� � 0 means by the Hereditary Condition that if 0 � A then 0� � A,i.e. 0 � :A as desired.

It should be said that although either the four-valued treatment or the�-operator treatment of negation work equally well for �rst-degree relevantimplications (at least from a technical point of view), the �-operator treat-ment seems to win hands down in the context of all of R. Meyer [1979a]

has succeeded in giving a four-valued treatment of all of R, but at the priceof great technical complexity (e.g. the accessibility relation has to be madefour-valued as well, and that is just for starters). Further, as Meyer pointsout, one's models still have to be closed under �, so it still can be said tosneak in the back door.

3.9 Routley-Meyer Semantics for E and other Neighbours of R

Once one sets down a set of conditions on an accessibility relation, they canbe played with n various ways so as to produce semantics for a wide varietyof systems as the experience with modal logic has taught us. Also otherfeatures of the frames can be generalised.

We can here only give the avour of a whole range of possible and actualresults. In all the results below � will satisfy the same conditions as forR+ (or R ) models (as appropriate). To begin with we follow Routley andMeyer [1973] with the description of a series of conditions on positive framesand corresponding axioms for propositional logic. They begin by requiringof a B+-frame (K;R; 0)

B1. a � a

B2. a � b and b � x) a � cB3. a0 � a and Rabc) Ra0bc.

B1{B2 of course say that � is a quasi-order, and B3 says something likethat it is monotone.


`B' appears to be for `Basic', for they regard the above postulates as anatural minimal set on their approach.33 Gabbay [1976] investigates evenweaker logics where no conditions at all are placed on the frame, but thesehave no theorems and are characterised only by rules of deducibility (un-less Boolean negation and/or the Boolean material conditional is present,options which he does explore).

The sense in which the above postulates are minimal goes something likethis. B3 is needed in proving the Hereditary Condition for implications,and the Hereditary Condition is needed in turn for verifying 0 � A !A (indeed anything) so we have at least some minimal theorems. TheHereditary Condition is used in showing the equivalence of the veri�cationof an implication in a model and entailment in that mode, i.e. 0 � A ! Bi� 8x 2 K(x � A ) x � B) (cf. Section 3.7 to see how these conditionswere used to establish these facts about R+-models). What about B2? Wethink it is just a `freebie'. It seems to play no role in verifying axioms orrules, but the completeness proof can be made to yield canonical (`reduced')models (cf. note 3.7) that satisfy it, so why not have it? This seems to bewhat Routley et. al. [1982] say. It appears that B1 is even more a freebie.

It may be shown that A is a theorem of the system B+ (formulated inSection 1.3) i� A is valid in all B+ models.

Routley and Meyer establish the following correspondence between con-ditions on the accessibility relation R and axioms:

(1) Raaa A ^ (A! B)! B(2) Rabc) R2a(ab)c (A! B) ^ (B ! C)! (A! C)(3) R2abcd) R2a(bc)d A! B ! ([B ! C]! [A! C])(4) R2abcd) R2b(ac)d A! B ! ([C ! A]! [C ! B])(5) Rabc) R2abbc (A! [A! B])! (A! B)(6) Ra0a ([A! A]! B)! B(7) Rabc) Rbac A! ([A! B]! B)(8) 0 � a A! (B ! B)(9) Rabc) b � c A! (B ! A).

Routley and Meyer connect these conditions on accessibility relations toaxioms extending the basic logic B. The correspondence is more perspicu-ous when you consider the structural rules corresponding to each axiom orcondition. We can express these as conditions on fusion:

33However, some notational confusion is possible, with Fine's use of `B' as anotherbasic relevance logic di�ering slightly from Routley and Meyer's usage [Fine, 1974]. ForFine, B includes the law of the excluded middle, and for Routley and Meyer, it does not.

RELEVANCE LOGIC 77

(1) Raaa A ` A ÆA(2) Rabc) R2a(ab)c A ÆB ` A Æ (A ÆB)(3) R2abcd) R2a(bc)d (A ÆB) Æ C ` A Æ (B Æ C)(4) R2abcd) R2b(ac)d (A ÆB) Æ C ` B Æ (A Æ C)(5) Rabc) R2abbc A ÆB ` (A ÆB) ÆB(6) Ra0a A Æ t ` A(7) Rabc) Rbac A ÆB ` B ÆA(8) 0 � a B ÆA ` B (or A ` t)(9) Rabc) b � c A ÆB ` B.

General recipes for translating between structural rules and conditionson accessibility relations are to be found in Restall [1998; 2000].

If one wants to add to B+ any of the axioms on the right to get a sententiallogic X, one merely adds the corresponding conditions to those for a B+

model to get the appropriate notion of an X-model, with a resultant soundand complete semantics.

Some logics of particular interest arising in this way are (nomenclature asin [Anderson and Belnap, 1975]) (note well that T has nothing to do withFeys' t of modal logic fame):

TW+ : B+ + (3; 4)T+ : TW+ + (5)E+ : T+ + (6)R+ : E+ + (7)H+ : R+ + (8)S4+ : E+ + (8):

These are far from the most elegant formulations from a postulationalpoint of view, being highly redundant (in particular the Pre�xing and Suf-�xing rules of B+ are supplanted already in TW+ by the correspondingaxioms. further the rule of Necessitation (A ` (A! A)! A) is also redun-dant already in TW+ (this is not so obvious|proof is by browsing through[Anderson and Belnap, 1975]).

What minimal conditions should be imposed on the �-operator when it isadded to a B+-frame so as to give a B-frame? Routley et. al. [1982] choose

B4. a�� = a, and

B5. a � b) b� � a�.The minimality of B5 can be defended in terms of its being needed for

showing that negations satisfy the Hereditary Condition. B4 would seem tohave little place in a minimal system except for the fact that the dominanttrend in relevance logic has been to keep classical double negation.34

34In fact, B5 is too strong for a purely minimal logic of negation. See Section 5.1 formore discussion on this.


One can get semantics for the full systems TW, T, etc. simply by addingthe appropriate postulates to the conditions on a B-model.

We could go on, but will instead refer the reader to Routley et al. [1982],Fine [1974] and Gabbay [1976] for a variety of variations producing systemsin the neighbourhood of R.

Some �nd the conditions on the \base point" 0 on frames rather puz-zling or unintuitive. Why should the basic conditions on frames includeconditions such as the fact that a � b de�ned as R0ab generate a partialorder? Some recent work by Priest and Sylvan and extended by Restall hasshown that these conditions can be done away with and the frames given aninterpretation rather reminiscent of that of non-normal modal logics [Priestand Sylvan, 1992; Restall, 1993]. The idea is as follows. We have two sortsof set-ups in a frame | normal ones and non-normal ones. Then we splitthe treatment of implication along this division. Normal points are givenan S5-like interpretation.

� x � A! B i� for every y if y � A then y � B

and non-normal points are given the condition which appeals to the ternaryrelation R

� x � A! B i� for every y and z where Rxyz if y � A then z � B

The other connectives are treated in just the same way as in the originalrelational semantics. To prove soundness and completeness for this seman-tics, it is simplest to go through the original semantics | for it is not toodiÆcult to show that this account is merely a notational variant, where wehave set Rxyz i� y = z when x is a normal set-up. This satis�es all ofthe conditions in the original semantics, for we have set a � b to be simplya = b.

We turn now to one such system RM deserving of special treatment.

3.10 Algebraic and Set-theoretical Semantics for RM

RM has been described by Meyer as `the laboratory of relevance logic'. Itplays a role somewhat like S5 among modal logics, being a place whereconjectures can be tested relatively easily (e.g. the admissibility of was�rst shown for RM). This then could be a very long section because RM isby far the best understood of the Anderson{Belnap style systems. We shalltry to keep it short by being dogmatic. The interested reader can verifythe results claimed by consulting Meyer's Section 29.3 and Section 29.4 of[Anderson and Belnap, 1975] (see also [Dunn, 1970; Tokarz, 1980]).

In the �rst place the appropriate algebras for RM are the idempotent deMorgan monoids (strengthening a � a Æ a to a = a Æ a). The subdirectlyirreducible ones are all chains with de Morgan complement where aÆb = a^b

RELEVANCE LOGIC 79

if a � :b, and a Æ b = a _ b otherwise. The designated elements are allelements a such that :a � a, and of course these must have a greatestlower bound to serve as the identity e. (This is just another descriptionwith Æ as primitive instead of ! of the `Sugihara matrices' described in thepublications cited above.) Meyer showed that if `RM A, then A is valid inall the �nite Sugihara matrices, establishing the �nite model property forRM.

Dunn showed that every extension of RM closed under substitution andthe rules of R has some �nite Sugihara matrix as a characteristic matrix(RM is `pretabular'). A similar result was shown by Scroggs to hold for themodal logic S5, and researchers (particularly Maksimova) have obtainedresults characterising all such pretabular extensions of S4 and of the in-tuitionistic logic. Curiously enough there are only �nitely many, and it isan interesting open problem to �nd some similar results for R. RM corre-sponds to the super-system of the intuitionistic propositional calculus LC(indeed LC can be translated into RM; see [Dunn and Meyer, 1971]). Muchstudy has been done of the `superintuitionistic' calculi (with an emphasison the decision problem), and it would be good to see some of the ideas ofthis carried over to the `super-relevant' calculi. A small start was begun in[Dunn, 1979a].

Routley and Meyer [1973] add the postulate

0 � a or 0 � a�

to the requirement on an R-frame to get an RM-frame. Dunn [1979a]

instead adds the requirement

Rac) a � c or b � c;

which neatly generalised to give a family of postulates yielding set-theoreticalsemantics for a denumerable family of weakenings of RM which are alge-braised by adding various weakenings of idempotence (an+1 = an). It is anopen problem whether R itself is the intersection of this family and whetherthey all have the �nite model property (if so, R is decidable). Since R isundecidable, one of these must be false. However, it is unknown at the timeof writing which one fails.

Proof Sketch


1. :X ! (:X ! :X) Mingle Axiom, Subst.2. X ! (:X ! X) 1, Permutation and

Contraposition3. (A _ :A) ^ (B _ :B)! 2, Subst.:((A _ :A) ^ (B _ :B))!((A _ :A) ^ (B _ :B))

4. :(A _ :A) _ :(B _ :B)! 3, MP, de M(A _ :A) ^ (B _ :B)

5. A ^ :A! B _ :B 4, _I;^E, de M.

Kalman [1958] especially investigated de Morgan lattices with the prop-erty a ^ :a � b _ :b. We will call these Kalman lattices. he showed thatevery Kalman lattice is isomorphic to a subdirect product of the de Morganlattice 3. This implies a three- valued Homomorphism Separation Propertyfor Kalman lattices (which also can be proven by modifying the proof of itsfour-valued analogue, noting that each `side' of 4 is just a copy of 3). Therepresentation in terms of polarities uses polarities X = (X1; X2) whereX1 [X2 = U , i.e. X1 and X2 are exhaustive.

This means informally that X always receives at least one of the valuestrue and false. This leads to a semantics using ambivaluations into theleft-hand side of 4:

F = ffg.

B = ft; fg

T = ftg

s

s

s

This idea leads to a simpler Kripke-style semantics for RM using an or-dinary binary accessibility relation instead of the Routley{Meyer ternaryone (actually this semantics antedates the Routley{Meyer one, the resultshaving been presented in [Dunn, 1969]|cf. [Dunn, 1976b] for a full presen-tation. No details will be supplied here. This semantics has been generalisedto �rst-order RM with a constant domain semantics [Dunn, 1976c]). Theanalogous question with Routley{Meyer semantics is has now been closed inthe negative in the work of [Fine, 1989], which we consider in Section 3.12.

Meyer [1980] has used this `binary semantics' to give a proof of an ap-propriate Interpolation Lemma for RM. (Unfortunately, interpolation failsfor E and R [Urquhart, 1993].)

3.11 Spin O�s from the Routley{Meyer Semantics

The Routley{Meyer semantical techniques can be used to prove a variety ofresults concerning the system R and related logics which were either more

RELEVANCE LOGIC 81

complicated using other methods (usually algebraic or Gentzen methods), oreven impossible. Thus (cf. [Routley and Meyer, 1973]), it is possible to givea variety of conservative extension results (being careful in constructing thecanonical model to use only connectives and sentential constant availablein the fragment being extended). Also it is possible to give a proof of theadmissibility of (see [Routley and Meyer, 1973]) that is easier than theoriginal algebraic proof (though not as easy as Meyer's latest proof usingmetavaluations|cf. Section 2.4). Admissibility of amounts to showingthat if A is refutable in a given R-model (K;R; 0;�) then A is refutable ina normal R-model (K 0; R0; 00;�0) (one where 00� = 00) gotten by adding anew `zero' and rede�ning R0 and �0 in a certain way from R and �.

Perhaps the most interesting new property to emerge this way is `Halld�encompleteness', i.e. if `R A_B and A and B share no propositional variablesin common, then `R A or `R B ([Routley and Meyer, 1973, Section 2.3]).

Another direction that the Routley{Meyer semantics has taken quicklyends up in heresy: classical (Boolean) negation � can be added to R withhorrible theorems resulting like A^ � A! B, and yet R does not collapseto classical logic. Indeed no new theorems emerge in the original vocabularyof R. The idea is to take a normal R-model (K;R; 0; �;�) and turn it in fora new R-model (K 0; R0; 00; �0;�0) , whose 00 is a new element K 0 = K[f00g,�0 is like � but with 00�0 = 00, and R0 is like R with the additional features:

1. R000ab i� R; a00b i� a = b,

2. R0ab00 i� a = b�.

Also �0 is just like � but with 00 � A if 0 � A.The whole point of this exercise is to provide refuting R-models for all

non-R-theorems that have the property

a � b (i.e. R00ab) ) a = b:

These are called `classical R-models' (�rst studied in Meyer and Routley[1973a; 1973b]) and upon them one can de�ne

a � �A, not a � A:

One could not do this on ordinary R-models without things coming apartat the seams, because in order to have the theorem �p ! �p valid, onewould need the Hereditary Condition to hold for � p, i.e. if a � b, then ifa � �p then b � �p, i.e. if a � p then b � p. But one has no reason tothink that this is the case, since all one has is the converse coming fromthe fact that the Hereditary condition holds for p. The inductive proofthe Hereditary condition breaks down in the presence of Boolean negation,but of course with classical R-models the Hereditary Condition becomesvacuous and there is no need for a proof.


This leads to certain technical simplicities, e.g. it is possible to giveG�odel{Lemmon style axiomatisations of relevance logics like the familiarones for modal logics, where one takes among one's axioms all classicaltautologies (using �)|cf. [Meyer, 1974].

But it also leads to certain philosophical perplexities. For example, whatwas all the fuss Anderson and Belnap made against contradictions implyingeverything and disjunctive syllogism? Boolean negation trivially satis�esthem, so what is the interest in de Morgan negation failing to satisfy them.Will the real negation please stand up?

A certain schism developed in relevance logic over just how Boolean nega-tion should be regarded. See [Belnap and Dunn, 1981; Restall, 1999] for the`con' side and [Meyer, 1978] for the `pro' side.

Belnap and Dunn [1981] point out that although Meyer's axiomatisationsof R with Boolean negation do not lead to any new theorems in the standardvocabulary of R, they do lead to new derivable rules, e.g. A ^ :A ` B and:A^ (A_B) ` B (note well that the negation here is de Morgan negation).This can be seen quite readily if one recognises that the semantic correlateof X ` Y is that 0 � X ) 0 � Y in all classical R-models, and that sinceall such are normal, : behaves at 0 in these just like classical negation. Weboth think this point counts against enriching R with Boolean negation,but Meyer [1978, note 21] thinks otherwise.

3.12 Semantics for RQ

The question of how to extend these techniques to handle quanti�ed rel-evance logics was open for a long time. The �rst signi�cant results wereby Routley, who showed that the obvious constant domain semantics weresuÆcient to capture BQ, the natural �rst-order extension of B [Routley,1980b]. However, extending the result to deal with systems involving tran-sitivity postulates in the semantics (such as Rabc^Rcde) R2abde) proveddiÆcult. To verify that the frame of prime theories on some constant domainactually satis�es this condition (given that the logic satis�es a correspond-ing condition, here the pre�xing axiom) requires constructing a new primetheory x such that Rabx and Rxde. And there seems to be no general wayto show that such a theory can be constructed using the domain sharedby the other theories. This is not a problem for logics like BQ, in whichthe frame conditions do not have conditions which, to be veri�ed in thecompleteness proof, require the construction of new theories.

Fine showed that this is not merely a problem with our proof techniques.Logics like RQ, EQ, TQ and even TWQ are incomplete with respect to theconstant domain semantics on the frames for the propositional logics [Fine,1989]. He has given a technical argument for this, constructing a formula inthe language of RQ which is true in all constant domain models, but whichis not provable. The argument is too detailed to give here. It consists of a

RELEVANCE LOGIC 83

simple part, which shows that the formula�(p! 9xEx) ^ 8x((p! Fx) _ (Gx! Hx))

�! �8x(Ex ^ Fx! q) ^ 8x((Ex! q) _Gx)! 9xHx _ (p! q)

�is valid in the constant domain semantics. This is merely a tedious veri�ca-tion that there is no counterexample. The subtle part of his argument is theconstruction of a countermodel. Clearly the countermodel cannot be a con-stant domain frame. Instead, he constructs a frame with variable domains,in which each of the axioms of RQ is valid (and in which the rules preservevalidity) but the o�ending formula fails. This is quite a tricky argument, forvariable domain semantics tend not to verify RQ's analogue to the Barcanformula

8x(p! Fx)! (p! 8xFx)

But Fine constructs his example in such a way that this formula is valid,despite the variable domains.

Despite this problem, Fine has found a semantics with respect to whichthe logic RQ is sound and complete. This semantics rests on a di�erent viewof the quanti�ers. For Fine's account, a statement of the form 8xA(x) is trueat a set-up not only when A(c) is true for each individual c in the domainof the set-up, but instead, when A(c) is true for an arbitrary individual c.In symbols,

a � 8xA(x) i� (9a")(9c 2 Da" �Da)(a" � A(c)).

That is, for every set-up a there are expansions of the form a" where we addnew elements to the domain, but these are totally arbitrary. The framesFine de�nes are rather complex, needing not only the " operator but alsoa corresponding # operator which cuts down the domain of a set-up, andan across operator which identi�es points in setups (! (a; fc; dg) is theminimal extension of the set-up a in which the individuals c and d areidenti�ed. Instead of discussing the details of Fine's semantics, we referthe reader to his paper which introduced them [Fine, 1988]. Fine's workhas received some attention, from Mares, who considers options for thesemantics of identity [Mares, 1992]. However, it must be said that whilethe semantic structure pins down the behaviour of RQ and related systemsexactly, it is not altogether clear whether the rich and complex structureof Fine's semantics is necessary to give a semantics for quanti�ed relevancelogics.

Whatever one's thoughts about the theoretical adequacy of Fine's se-mantics, they do raise some important issues for anyone who would give asemantic structure for quanti�ed relevance logics. There are a number ofissues to be faced and a number of options to be weighed up. One optionis to give complete primacy to the frames for the propositional logics, and


to use the constant domain semantics on these frames. The task then is toaxiomatise this extension. The task is also to give some interpretation ofwhat the points in these semantic structures might be. For if they are the-ories (or prime theories) then the evaluation clauses for the quanti�ers donot make a great deal of sense without further explanation. No-one thinksthat a claim of the form 9xA(x) can be a member of a theory only if there isan object in the language of the theory which satis�es A according to thattheory. Nor are we so readily inclined to think that all theories need sharethe same domain of quanti�cation.

If, on the other hand, we take the set-ups in frames to be quite like (someclass of) theories, then we must face the issue of the relationships betweenthese theories. No doubt, if 8xA(x) is in some theory, then A(c) will be inthat theory for any constant c in the language of the theory. However, theconverse need not be the case.

Anyway, it is clear that there is a lot of work to be done in the semanticsof relevance logics with quanti�ers. One area which hasn't been explored atany depth, but which looks like it could bring some light is the semantics ofpositive quanti�ed relevance logics. Without the distribution of the univer-sal quanti�er over disjunction, these systems are subsystems of intuitionisticlogic.

4 THE DECISION PROBLEM

4.1 Background

When the original of this Handbook article was published back in 1985,without a doubt the outstanding open problem in relevance logics was thequestion as to whether there exists a decision procedure for determiningwhether formulas are theorems of the system E or R. Anderson [1963] listedit second among his now historic open problems (the �rst was the admissi-bility of Ackermann's rule discussed in Section 2). Through the work ofUrquhart [1984], we now know that there is no such decision procedure.

Harrop [1965] lends interest to the decision problem with his remark that`all \philosophically interesting" propositional calculi for which the decisionproblem has been solved have been found to be decidable : : : '.35 We nowhave a very good counterexample to Harrop's claim.

In this section we shall examine Urquhart's proof, but before we get therewe shall also consider various fragments and subsystems of R for which thereare decision procedures. R will be our paradigm throughout this discussion,though we will make clear how things apply to related systems.

35He continues somewhat more technically ` : : : and none is known for which it hasbeen proved that it does not possess the �nite model property with recursive bound.'

RELEVANCE LOGIC 85

4.2 Zero-degree Formulas

These are formulas containing only ^;_, and :. As was explained in Sec-tion 1.7, the zero-degree theorems of R (or E) are precisely the same asthose of the classical propositional calculus, so of course the usual two val-ued truth tables yield a decision procedure.

4.3 First-degree Entailments

Two di�erent (though related) `syntactical' decision procedures were de-scribed for these in Section 1.7 (the method of `tautological entailments'and the method of `coupled trees'). A `semantical' decision procedure us-ing a certain four element matrix 4 is described in Section 3.3. The storythus told leaves out the historically (and otherwise) very important role ofa certain eight element matrix M0 (cf. [Anderson and Belnap, 1975, Section22.1.3]). This matrix is essential for the study of �rst-degree formulas andhigher (see Section 4.4 below), in so much as it is impossible to de�ne animplication operation on 4 and pick out a proper subset of designated ele-ments so as to satisfy the axioms of E (a fortiori R). Indeed M0 was used in[Anderson and Belnap Jr., 1962b] and [Belnap, 1960b] to isolate the �rst-degree entailments of R, and the formulation of Section 1.7 presupposesthis use.

4.4 First-degree Formulas

These are `truth functions' of �rst-degree entailments and/or formulas con-taining no ! at all (the `zero-degree formulas'). Belnap [1967a] gave adecision procedure using certain �nite `products' of M0. No one such prod-uct is characteristic for Dfdf, but every non-theorem of Efdf is refutable insome such products Mn

0 (where n may in fact be computed as the largestnumber of �rst-degree entailments occurring in a disjunction once the can-didate theorem has been put in conjunctive normal form). Hence Efdf hasthe �nite model property which suÆces of course for decidability (cf. [Har-rop, 1965]). This is frankly one of the most diÆcult proofs to follow in thewhole literature of relevance logics. A sketch may be found in [Andersonand Belnap, 1975, Section 19].

4.5 `Career Induction'

This is what Belnap has labelled the approach, exempli�ed in Sections 4.1{4.3 above of extending the positive solution to the decision problem `a degreeat a time'. The last published word on the Belnap approach is to be foundin his [1967b] where he examines entailments between conjunctions of �rst-degree entailments and �rst degree entailments.


Meyer [1979c], by an amazingly general and simple proof, shows that apositive answer to the decision problem for `second-degree formulas' (no !within the scope of an arrow within the scope of an !) is equivalent to�nding a decision procedure for all of R.

4.6 Implication Fragment

We now start another tack. Rather than looking at fragments of the wholesystem R delimited by complexity of formulas, we instead consider frag-ments delimited by the connectives which they contain. The earliest resultof this kind is due to [Kripke, 1959b], who gave a Gentzen system for theimplicational fragments of E and R, and showed them decidable. We shallhere examine the implicational fragment of R (R!) in some detail as akind of paradigm for this style of argument.36

The appropriate Gentzen calculus37 LR! is the same as that given byGentzen [1934] except for two trivial di�erences and one profound di�erence.The �rst trivial di�erence is the obvious one that we take only the opera-tional rules for implication, and the second trivial di�erence consequent onthis (with negation it would have to be otherwise) is that we can restrict oursequents to those with a single formula in the consequent. The profounddi�erence is that we drop the structural rule variously called `thinning' or`weakening'. This leaves:

Axioms.

A ` A:

Structural Rules.

Permutation�;A;B; �;` C�;B;A; �;` C Contraction

�; a;A ` B�;A ` B

Operational Rules.

(`!)�;A ` B� ` A! B

(!`)� ` A �;B ` C

:�; �;A! B ` C

It is easy to see why thinning would be a disaster for relevant implication.

36Actually this and various other results discussed below using Gentzen calculi presup-poses `separation theorems' due to Meyer, showing, e.g. as is relevant to this case, thatall of the theorems containing only ! are provable from the axioms containing only !.37We do not follow Anderson and Belnap [1975] in calling Gentzen systems `consecution

calculi', much as their usage has to recommend it.

RELEVANCE LOGIC 87

Thus:

A ` AThinning

A;B ` A(`!)

A ` B ! A(`!)` A! (B ! A)

It is desirable to prove `The Elimination Theorem', which says that thefollowing rule would be redundant (could be eliminated).

(Cut)� ` A �;A ` B

:�; � ` B

This is needed to show the equivalence of LR! to its usual Hilbert-style(axiomatic system `HR!' R! one of the formulations of Section 1.3). Wewill not pause on details here, but the principal question regarding theequivalence is whether modus ponens (The sole rule for HR!) is admissiblein the sense that whenever ` A and ` A ! B are both derivable in LR!,so is ` B (let � and � be empty).

The strategy of the proof of the Elimination Theorem can essentiallybe that of Gentzen with one important but essentially minor modi�cation.Thus, Gentzen actually proved something stronger than Cut elimination,namely,

(Mix)� ` A � ` B

;�; [� �A] ` B

where [� �A] is the result of deleting all occurrences of A from �. This isuseful in the induction, but sometimes it takes out too many occurrences ofA. In Gentzen's framework these could always be thinned back in, but ofcourse this is not available with LR!. We thus instead generalise Cut tothe rule

(Fusion)� ` A � ` B

;�; (� � A) ` B

where � contains some occurrences of A and (��A) is the result of deletingas many of those occurrences as one wishes (but at least one).

The main strategy of the decision procedure for LR! is to limit appli-cations of the contraction rule so as to prevent a proof search from runningon forever in the following manner: `Is p ` q derivable? Well it is if p; p ` qis derivable. Is p; p ` q derivable? Well it is if p; p; p ` q is, etc.'.

We need one simple notion before strategy can be achieved. We shallsay that the sequent of �0 ` A is a contraction of sequent � ` A just in


case �0 ` A can be derived from � ` A by (repeated) applications of therules Contraction and Permutation (with respect to this last it is helpfulnot even to distinguish two sequents that are mere permutations of oneanother). The idea that we now want to put in e�ect is to drop the ruleContraction, replacing it by building into the operational rules a limitedamount of contraction (in the generalised sense just explained).

More precisely, the idea is to allow a contraction of the conclusion of anoperational rule only in so far as the same result could not be obtained by�rst contracting the premises. A little thought shows that this means nochange for the rule (`!), and that the following will suÆce for

(!`0)� ` A �;B ` C[�; �;A! B] ` C

where [�; �;A! B] is any contraction of �; �;A! B such that :

1. A! B occurs only 0, 1, or 2 times fewer than in �; �;A! B;

2. Any formula other than A! B occurs only 0 or 1 time fewer.

It is clear that after modifying LR! by building some limited contractioninto (!`) in the manner just discussed, the following is provable by aninduction on length of derivations:

Curry's Lemma.38 If a sequent �0 is a contraction of a sequent � and �has a derivation of length n, then �0 has a derivation of length � n.

Clearly this lemma shows that the modi�cation of LR! leaves the samesequents derivable (since the lemma says the e�ect of contraction is re-tained). So henceforth we shall by LR! always mean the modi�ed version.

Besides the use just adverted to, Curry's Lemma clearly shows that everyderivable sequent has an irredundant derivation in the following sense: onecontaining no branch with a sequent �0 below a sequent � of which it is acontraction.

We are �nally ready to begin explicit talk about the decision procedure.Given a sequent �, one begins the test for derivability as follows (building

38This is named (following [Anderson and Belnap, 1975]) after an analogous lemma in[Curry, 1950] in relation to classical (and intuitionistic) Gentzen systems. There, withfree thinning available, Curry proves his lemma with (!`) (in its singular version) statedas:

�; A! B ` A �; A! B;B ` C:

�; A! B ` C

This in e�ect requires the maximum contraction permitted in our statement of (!`)above, but this is ok since items contracted `too much' can always be thinned back in.Incidentally, our statement of (!`) also di�ers somewhat from the statement of Andersonand Belnap [1975] or Belnap and Wallace [1961], in that we build in just the minimalamount of contraction needed to do the job.

RELEVANCE LOGIC 89

a `complete proof search tree'): one places above � all possible premisesor pairs of premises from which � follows by one of the rules. Note wellthat even with the little bit of contraction built into (!`) this will still beonly a �nite number of sequents. Incidentally, one draws lines from thosepremises to �. One continues in this way getting a tree. It is reasonablyclear that if a derivation exists at all, then it will be formed as a subtreeof this `complete proof search there', by the paragraph just above, the iscomplete proof search tree can be constructed to be irredundant. But theproblem is that the complete proof search tree may be in�nite, which wouldtend to louse up the decision procedure. There is a well-known lemma whichbegins to come to the rescue:

K�onig's Lemma. A tree is �nite i� both (1) there are only �nitely manypoints connected directly by lines to a given point (`�nite fork property') and(2) each branch is �nite (`�nite branch property').

By the `note well' in the paragraph above, we have (1). The questionremaining then is (2), and this is where an extremely ingenious lemma ofKripke's plays a role. To state it we �rst need a notion from Kleene. Twosequents � ` A and �0 ` A are cognate just when exactly the same formulas(not counting multiplicity) occur in � as in �0. Thus, e.g. all of the followingare cognate to each other:

(1) X;Y ` A(2) X;X; Y ` A(3) X;Y; Y ` A(4) X;X; Y; Y ` A(5) X;X;X; Y; Y ` A.

We call the class of all sequents cognate to a given sequent a cognation class.

Kripke's Lemma. Suppose a sequence of cognate sequents �0;�1; : : : ; isirredundant in the sense that for no �i;�j with i < j, is �i a contractionof �j . Then the sequence is �nite.

We postpone elaboration of Kripke's Lemma until we see what use it isto the decision procedure. First we remark an obvious property of LR!

that is typical of Gentzen systems (that lack Cut as a primitive rule):

Subformula Property. If � is a derivable sequent of LR!, then anyformula occurring in any sequent in the derivation is a subformula of someformula occurring in �.

This means that the number of cognation classes occurring in any deriva-tion (and hence in each branch) is �nite. But Kripke's Lemma further shows


that only a �nite number of members of each cognation class occur in abranch (this is because we have constructed the complete proof search treeto be irredundant). So every branch is �nite, and so both conditions ofK�onig's lemma hold. Hence the complete proof search tree is �nite and sothere is a decision procedure.

�0

1

2

3

4

5

6

7

2 3 4 5 6 7

Figure 1. Sequents in the Plane

Returning now to Kripke's Lemma, we shall not present a proof (forwhich see [Belnap Jr. and Wallace, 1961] or [Anderson and Belnap, 1975]).Instead we describe how it can be geometrically visualised. For simplicitywe consider sequents cognate to X;Y ` A ((1), (2), (3), etc. above). Eachsuch sequent can be represented as a point in the upper right-hand quadrantof the co-ordinate plane (where origin is labelled with 1 rather than 0 since(1) is the minimal sequent in the cognation class). See Figure 1. Thus, e.g.(5) gets represented as `3 X units' and `2 Y units'.

Now given any sequent, say

(�0) X;X;X; Y; Y ` A

as a starting point one might try to build an irredundant sequence by �rstbuilding up the number of Y 's tremendously (for purposes of keeping on thepage we let this be to six rather than say a million). But in so doing one

RELEVANCE LOGIC 91

has to reduce the number of X 's (say, to be strategic, by one). The graphnow looks like 2 for the �rst two members of the sequence �0;�1.

�0

�1

1

2

3

4

5

6

7

2 3 4 5 6 7

Figure 2. Descending Regions

The purpose of the intersecting lines at each point is to mark o� areas(shaded in the diagram) into which no further points of the sequence maybe placed. Thus if �2 were placed as indicated at the point (6, 5), it wouldreduce to �0. What this means is that each new point must march eitherone unit closer to the X axis or one unit closer to the Y axis. Clearly after a�nite number of points one or the other of the two axes must be `bumped',and then after a short while the other must be bumped as well. Whenthis happens there is no space left to play without the sequence becomingredundant.

The generalisation to the case of n formulas in the antecedent to Eu-clidean n-space is clear (this is with n �nite|with n in�nite no axis needever be bumped).

Incidentally, Kripke's Lemma (as Meyer discovered) is equivalent to atheorem of Dickson about prime numbers: Let M be a set of natural num-bers all of which are composed out of the �rst m primes. Then every n 2Mis of the form Pn1

1 �Pn22 � : : : Pnk

k , and hence (by unique decomposition) canbe regarded as a sequence of the Pi's in which each Pi is repeated ni times.Divisibility corresponds then to contraction (at least neglecting the case


ni = 0). Dickson's theorem says that if no member of M has a properdivisor in M , then M is �nite.

Before going on to consider how the addition of connectives changes thecomplexity, let us call the reader's attention to a major open problem: It isstill unknown whether the implication fragment of T is decidable.

4.7 Implication{Negation Fragment

The idea of LR:! is to accommodate the classical negation principles pre-

senting R in the same way that Gentzen [1934] accommodated them forclassical logic: provide multiple right-hand sides for the sequents. thismeans that a sequent is of the form � ` �, where � and � are (possibleempty) �nite sequences of formulas. One adds structural rules for Permuta-tion and Contraction on the right-hand side, reformulates (`!) and (!`)as follows

(`!)�;A ` B; �� ` A! B; �

(!`)� ` A; �;B ` Æ

;�; �; A! B ` ; Æ

and adds ` ip and op' rules for negation:

(` :)�;A ` �� ` :A; � (: `)

� ` A; �:

�;:A ` �LE:! is the same except that in the rule (`!)� must be empty and �

must consist only of formulas whose main connective is !. The decisionprocedure for LE:! was worked out by Belnap and Wallace [1961] alongbasically the lines of the argument of Kripke just reported in the last section,and is clearly reported in [Anderson and Belnap, 1975, Section 13]. themodi�cation to LR:

! is straightforward (indeed LR:! is easier because one

need not prove the theorem of p. 128 of [Anderson and Belnap, 1975], and soone can avoid all the apparatus there of `squeezes'). McRobbie and Belnap[1979] have provided a nice reformulation of LR:

! in an analytic tableaustyle, and Meyer has extended this to give analytic tableau for linear logicand other systems in the vicinity of R [Meyer et al., 1995].

4.8 Implication{Conjunction Fragment, and RWithout Distri-bution

This work is to be found in [Meyer, 1966]. The idea is to add to LR! theGentzen rules:

(^ `)�;A ` C

�;A ^B ` C�; � ` C

�;A ^ B ` C (` ^)� ` A � ` B

:� ` A ^ B

Again the argument for decidability is a simple modi�cation of Kripke's.

RELEVANCE LOGIC 93

Note that it is important that the rule (^ `) is stated in two parts, andnot as one `Ketonen form' rule:

(K^ `)�;A;B ` C

:�;A ^ B ` C

The reason is that without thinning it is impossible to derive the rule(s)(^ `) from (K^ `).

Early on it was recognised that the distribution axiom

A ^ (B _ C)! (A _ B) _ Cwas diÆcult to derive from Gentzen-style rules for E and R. Thus Anderson[1963] saw this as the sticking point for developing Gentzen formulations,and Belnap [1960b, page 72]) says with respect to LE! that `the standardrules for conjunction and disjunction could be added : : : the EliminationTheorem (suitably modi�ed) remaining provable. However, [since distribu-tion would not be derivable], the game does not seem worth the candle'.Meyer [1966] carried out such an addition to LR:

!, getting a system hecalled LR�, whose Hilbert-style version is precisely R without the distri-bution axiom. He showed using a Kripke-style argument that this systemis decidable. This system is now called LR, for \lattice R".

Meyer [1966] also showed how LR can be translated into R!;^ rathersimply. Given a formula A in the language of LR+, let V be the set ofvariables in A, and let two atomic propositions pt and pf not in V . Set :Afor the moment to be A! pf , to de�ne a translation A0 of A as follows.

p0 = p

t0 = pt

(A! B)0 = A0 ! B0

(A ^ B)0 = A0 ^ B0(A _ B)0 = :(:A0 ^ :B0)(A ÆB)0 = :(A0 ! :B0)

then setting t(A) =Vfpt ! (p ! p) : p 2 V [ fpt; pfgg and f(A) =Vf::p! p : p 2 V [ fpt; pfgg, we get the following theorem:

Translation Theorem (Meyer). If A is a formula in LR+ then A isprovable in LR+ if and only if (t(A)^f(A)^pt)! A0 is provable in R!;^.

The proof is given in detail in [Urquhart, 1997], and we will not present ithere.

Some recent work of Alasdair Urquhart has shown that although R!;^

is decidable, it is only just decidable [Urquhart, 1990; Urquhart, 1997].


More formally, Urquhart has shown that given any particular formula inthe language of R!;^, there is no primitive recursive bound on either thetime or the space taken by a computation of whether or not that formulais a theorem. Presenting the proof here would take us too far away fromthe logic to be worthwhile, however we can give the reader the kernel of theidea behind Urquhart's result.

Urquhart follows work of [Lincoln et al., 1992] by using a propositionallogic to encode the behaviour of a branching counter machines. A countermachine has a �nite number of registers (say, ri for suitable i) which eachhold one non-negative integer, and some �nite set of possible states (say,qj for suitable j). Machines are coded with a list of instructions, whichenable you to increment or decrement registers, and test for registers' beingzero. A branching counter machine dispenses with the test instructions andallows instead for machines to take multiple execution paths, by way offorking instructions. The instruction qi + rjqk means \when in qi, add 1 toregister rj and enter stage qk," and qi� rjqk means \when in qi, subtract 1to register rj (if it is non-empty) and enter stage qk," and qifqjqk is \whenin qi, fork into two paths, one taking state qj and the other taking qk."

A machine con�guration is a state, together with the values of each reg-ister. Urquhart uses the logic LR to simulate the behaviour of a machine.For each register ri, choose a distinct variable Ri, for each state qj choosea distinct variable Qj . The con�guration hqi;n1; : : : ; nli, where ni is thevalue of ri is the formula

Qi ÆRn11 Æ � � � ÆRnl

l

and the instructions are modelled by sequents in the Gentzen system, asfollows:

Instruction Sequentqi + rjqk Qi ` Qk ÆRj

qi � rjqk Qi; Rj ` Qk

qifqjqk Qi ` Qj _Qk

Given a machine program (a set of instructions) we can consider what isprovable from the sequents which code up those instructions. This set ofsequents we can call the theory of the machine. Qi Æ Rn1

1 Æ � � � Æ Rnll `

Qj Æ Rm11 Æ � � � Æ Rml

l is intended to mean that from state con�gurationhqi;n1; : : : ; nli all paths will go through con�guration hqj ;m1; : : : ;mli aftersome number of steps.

A branching counter machine accepts an initial con�guration if when runon that con�guration, all branches terminate at the �nal state qf , with allregisters taking the value zero. The corresponding condition in LR will bethe provability of

Qi ÆRn11 Æ � � � ÆRnl

l ` Qm

RELEVANCE LOGIC 95

This will nearly do to simulate branching counter machines, except for thefact that in LR we have A ` A Æ A. This means that each of our registerscan be incremented as much as you like, provided that they are non-zero tostart with. This means that each of our machines need to be equipped withevery instruction of the form qi>0 + rjqi, meaning \if in state qi, add 1 torj , provided that it is already nonzero, and remain in state qi."

Given these de�nitions, Urquhart is able to prove that a con�gurationis accepted in branching counter machine, if and only if the correspondingsequent is provable from the theory of that machine. But this is equivalentto a formula ^

Theory(M) ^ t! (Q1 ! Qm)

in the language of LR. It is then a short step to our complexity result,given the fact that there is no primitive recursive bound on determiningacceptability for these machines. Once this is done, the translation of LRinto R!^ gives us our complexity result.

It is still unknown if R! has similar complexity or whether it is a moretractable system.

Despite this complexity result, Kripke's algorithm can be implementedwith quite some success. The theorem prover Kripke, written by McRobbie,Thistlewaite and Meyer, implements Kripke's decision procedure, togetherwith some quite intelligent proof-search pruning, by means of �nite models.If a branch is satis�able in RM3, for example, there is no need to extend itto give a contradiction. This implementation works in many cases [Thistle-waite et al., 1988]. Clearly, work must be done to see whether the horri�ccomplexity of this problem in general can be transferred to results aboutaverage case complexity.

Finally, before moving to add distribution, we should mention that LinearLogic (see Section 5.5) also lacks distribution, and the techniques used inthe theorem prover Kripke have application in that �eld also.

4.9 Positive R

In this section we will examine extensions of the Gentzen technique to coverall of positive relevance logic. We know (see Section 4.12) that this willnot provide decidability. However, they provide another angle on R andcousins. Dunn and Minc independently developed a Gentzen-style calcu-lus (with some novel features) for R without negation (LR+).39 Belnap

39Dunn's result was presented by title at a meeting of the Association for SymbolicLogic, December, 1969 (see [Dunn, 1974]), and the full account is to be found in [Andersonand Belnap, 1975, Section 28.5]). Minc [1972, earliest presentation said to there to beFebruary 24] obtained essentially the same results (but for the system with a necessityoperator). See also [Belnap Jr. et al., 1980].


[1960b] had already suggested the idea of a Gentzen system in which an-tecedents were sequences of sequences of formulas, rather than just theusual sequences of formulas (in this section `sequence' always means �nitesequence). The problem was that the Elimination Theorem was not prov-able. LR+ goes a step òr two' further, allowing an antecedent of a sequentinstead to be a sequence of sequence of : : : sequences of formulas. Moreformally, we somehow distinguish two kinds of sequences, ìntensional se-quences' and èxtensional sequences' (say pre�x them with an Ì ' or anÈ'). an antecedent can then be an intensional sequence of formulas, anextensional sequence of the last mentioned, etc. or the same thing but withìntensional' and èxtensional' interchanged. (We do not allow things to `pileup', with, e.g. intensional sequences of intensional sequences|there mustbe alternation).40 Extensional sequences are to be interpreted using ordi-nary èxtensional' conjunction ^, whereas intensional sequences are to beinterpreted using ìntensional conjunction' Æ, which may be de�ned in thefull system R as AÆB = :(A! :B), but here it is taken as primitive|seebelow).

We state the rules, using commas for extensional sequences, semicolonsfor intentional sequences, and asterisks ambiguously for either; we also em-ploy an obvious substitution notation.41

Permutation�[� � ] ` A�[ � �] ` A Contraction

�[� � �] ` A�[�] ` A

Thinning�[�] ` A

;�[�; ] ` A provided � is non-empty:

�;A ` B(`!)

� ` A! B

� ` A �[B] ` C(!`)

�[�;A! B] ` C

� ` A � ` B(` ^)

� ` A ^ B�[A;B] ` C

(^ `)�[A ^B] ` C

40This di�ers from the presentation of [Anderson and Belnap, 1975] which allows such`pile ups', and then adds additional structural rules to eliminate them. Belnap felt thiswas a clearer, more explicit way of handling things and he is undoubtedly right, butDunn has not been able to read his own section since he rewrote it, and so return to thesimpler, more sloppy form here.41With the understanding that substitutions do not produce `pile ups'. Thus, e.g. a

`substitution' of an intensional sequence for an item in an intensional sequence does notproduce an intensional sequence with an element that is an intensional sequence formedby juxtaposition. Again this di�ers from the presentation of [Anderson and Belnap, 1975,cf. note 28].

RELEVANCE LOGIC 97

� ` A(` _)

� ` A _ B� ` B

(` _)� ` A _ B

�[a] ` C �[B] ` C(_ `)

�[A _ B] ` C

� ` A � ` B(` Æ)

�;� ` A ÆB�[a;B] ` C

(Æ `)� ÆB] ` C

For technical reasons (see below) we add the sentential constant t with theaxiom ` t and the rule:

�[B] ` A(t `)

�[�; t] ` AThe point of the two kind of sequences can now be made clear. Let usexamine the classically (and intuitionistically) valid derivation:

(1) A ` A Axiom(2) A;B ` A Thinning(3) A ` B ! A (`!):

It is indi�erent whether (2) is interpreted as

(2^) (A ^ B)! A; or(2!) A! (B ! A);

because of the principles of exportation and importation. In LR+ howeverwe may regard (2) as ambiguous between

(2; ) A;B ` A (extensional), and(2; ) A;B ` A (intensional).

(2,) continues to be interpreted as (2^), but (2;) is interpreted as

(2Æ) (A ÆB)! A:

Now in R, exportation holds for Æ but not for ^ (importation holds forboth). Thus the move from (2;) to (3) is valid, but not from (2,) to (3). Onthe other hand, in R, the inference from A ! C to (A ^ B) ! C is valid,whereas the inference to (A Æ B) ! C is not. Thus the move from (1) to(2,) is valid, but not the move from (1) to (2;). the whole point of LR+ isto allow some thinning, but only in extensional sequences.

This allows the usual classical derivation of the distribution axiom to gothrough, since clearly

A ^ (B _ C) ` (A ^ B) _ Ccan be derived with no need of any but the usual extensional sequence. Thefollowing sketch of a derivation of distribution in the consequent is even


more illustrative of the powers of LR+ (permutations are left implicit; alsothe top half is left to the reader);

X ` XX ` X

A;B ` (A ^ B) _ C A;C ` (A ^ B) _ C(_ `)

A;B _ C ` (A ^ B) _ C(!`)

A; (X ;X ! B _ C) ` (A ^ B) _ C(!`)

(X ;X ! A); (X ;X ! A;X ! B _ C) ` (A ^ B) _ CX ! A;X ! B _ C;X ` (A ^ B) _ C

(X ! A) ^ (X ! B _ C);X ` (A ^ B) _ C` (X ! A) ^ (X ! B _ C)! [X ! (A ^ B) _ C]

LR+ is equivalent to R+ in the sense that for any negation-free sentence Aof R;` A is derivable in LR+ i� A is a theorem of R. The proofs of bothhalves of the equivalence are complicated by technical details. Right-to-left(the interpretation theorem) requires the addition of intensional conjunctionas primitive, and then a lemma, due to R. K. Meyer, to the e�ect thatthis is harmless (a conservative extension). Left-to-right (the EliminationTheorem) is what requires the addition of the constant true sentence t. Thisis because the `Cut' rule is stated as:

� ` A �(A) ` B;

�(�) ` Bwhere �(�) is the result of replacing arbitrarily many occurrences of A in�(A) by � if � is non-empty, and otherwise by t.42 Without this emendationof the Cut rule one could derive B ` A whenever ` A is derivable (forarbitrary B, relevant or not) as follows

` AA ` A

ThinningA;B ` A

CutB ` A

Discussing decidability a bit, one problem seems to be that Kripke'sLemma (appropriately modi�ed) is just plain false. The following is a se-quence of cognate sequents in just the two propositional variables X and Ywhich is irredundant in the sense that structural rules will not get you froma later member to an earlier member:

X ;Y ` X (X ;Y ); X ` X ((X ;Y ); X);Y ` X : : : 43

42Considerations about the eliminability of occurrences of t are then needed to show theadmissibility of modus ponens. This was at least the plan of [Dunn, 1974]. A di�erentplan is to be found in [Anderson and Belnap, 1975, Section 28.5], where things arearranged so that sequents are never allowed to have empty left-hand sides (they have tthere instead).

RELEVANCE LOGIC 99

4.10 Systems Without Contraction

Gentzen systems without the contraction rule tend to be more amenable todecision procedures than those with it. Clearly, all of the work in Kripke'sLemma is in keeping contraction under control. So it comes as no surprisethat if we consider systems without contraction for intensional structure, de-cision procedures are forthcoming. If we remove the contraction rule fromLR we get the system which has been known as LRW (R without W with-out distribution), and which is equivalent to the additive and multiplicativefragment of Girard's linear logic [Girard, 1987]. It is well known that thissystem is decidable. In the Gentzen system, de�ne the complexity of a se-quent to be the number of connectives and commas which appear in it. Itis trivial to show that complexity never increases in a proof and that as aresult, from any given sequent there are only a �nite number of sequentswhich could appear in a proof of the original sequent (if there is one). Thisgives rise to a simple decision procedure for the logic. (Once the work hasalready been done in showing that Cut is eliminable.)

If we add the extensional structure which appears in the proof theories oftraditional relevance logics then the situation becomes more diÆcult. How-ever, work by Giambrone has shown that the Gentzen systems for positiverelevance logics without contraction do in fact yield decision procedures [Gi-ambrone, 1985]. In these systems we do have extensional contraction, sosuch a simple minded measure of complexity as we had before will not yielda result. In the rest of this section we will sketch Giambrone's ideas, andconsider some more recent extensions of them to include negation. For de-tails, the reader should consult his paper. The results are also in the secondvolume of Entailment [Anderson et al., 1992].

Two sequents are equivalent just when you can get from one to the otherby means of the invertible structural rules (intensional commutativity, ex-tensional commutativity, and so on). A sequent is super-reduced if no equiv-alent sequent can be the premise of a rule of extensional contraction. Asequent is reduced if for any equivalent sequents which are the premise of arule of extensional contraction, the conclusion of that rule is super-reduced.So, intuitively, a super-reduced sequent has no duplication in it, and a re-duced sequent can have one part of it `duplicated', but no more. Clearly anysequent is equivalent to a super-reduced sequent. The crucial lemma is thatany super-reduced sequent has a proof in which every sequent appearing is

43Further, this is not just caused by a paucity of structural rules. Interpreting thesequents of formulas of R+ (^ for comma, Æ for semicolon, ! for `) no later formulaprovably implies an earlier formula. Incidentally, one does need at least two variables (cf.R. K. Meyer [1970b]).


reduced. This is clear, for given any proof you can transform it into one inwhich every sequent is reduced without too much fuss.

As a result, we have gained as much control over extensional contractionas we need. Giambrone is able to show that only �nitely many reducedsequent can appear in the proof of a given sequent, and as a result, thesize of the proof-search tree is bounded, and we have decidability. Thistechnique does not work for intensional contraction, as we do not have theresult that every sequent is equivalent to an intensionally super-reducedsequent, in the absence of the mingle rule. While A ^ A ` B is equivalentto A ` B, we do not have the equivalence of A ÆA ` B and A ` B, withoutmingle.

These methods can be extended to deal with negation. Brady [1991]

constructs out of signed formulae TA and FA instead of formulae alone,and this is enough to include negation without spoiling the decidabilityproperty. Restall [1998] uses the techniques of Belnap's Display Logic (seeSection 5.2) to provide an alternate way of modelling negation in sequentsystems. These techniques show that the decidability of systems withoutintensional contraction are decidable, to a large extent independently of theother properties of the intensional structure.

4.11 Various Methods Used to Attack the Decision Problem

Decision procedures can basically be subdivided into two types: syntactic(proof-theoretic) and semantic (model-theoretical). A paradigm of the �rsttype would be the use of Gentzen systems, and a paradigm of the secondwould be the development of the �nite model property. It seems fair to say,looking over the previous sections, that syntactic methods have dominatedthe scene when nested implications have been afoot, and that semanticalmethods have dominated when the issue has been �rst-degree implicationsand �rst-degree formulas.44

There are two well-known model-theoretic decision procedures used forsuch non-classical logics as the intuitionistic and modal logics. One is dueto McKinsey and Tarski and is appropriate to algebraic models (matrices)(cf. [Lemmon, 1966, p. 56 �.]), and the other (often called `�ltration') isdue to Lemmon and Scott and is appropriate to Kripke-style models (cf.[Lemmon, 1966, p. 208 �]). Actually these two methods are closely con-nected (equivalent?) in the familiar situation where algebraic model sandKripke models are duals. The problem is that neither seems to work withE and R. The diÆculty is most clearly stated with R as paradigm. For thealgebraic models the problem is given a de Morgan monoid (M;^;_;�; Æ)44As something like `the exception that proves the rule' it should be noted that Belnap's

[1967a] work on �rst-degree formulas and slightly more complex formulas has actuallybeen a subtle blend of model-theoretic (algebraic) and proof-theoretic methods.

RELEVANCE LOGIC 101

and a �nite de Morgan sublattice (D;^0;_;�0), how to de�ne a new multi-plicative operation Æ0 on D so as to make it a de Morgan monoid and so forx; y 2 D, if x Æ y 2 D then x Æ y = x Æ0 y. the chief diÆcult is in satisfyingthe associative law. For the Kripke-style models (say the Routley{Meyervariety) the problem is more diÆcult to state (especially if the reader hasskipped Section 3.7) but the basic diÆculty is in the satisfying of certain re-quirements on the three-placed accessibility relation once set-ups have beenidenti�ed into a �nite number of equivalence classes by `�ltration'. Thus,e.g. the requirement corresponding to the algebraic requirement of associa-tivity is Rayx & Rbcy ) 9y(Raby & Rycx)45 the problem in a nutshellis that after �ltration one does not know that there exists the appropriateequivalence class �y needed to feed such an existentially hungry postulate.

The McKinsey-Tarski method has been used successfully by Maksimova[1967] with respect to a subsystem of R, which di�ers essentially only inthat it replaces the nested form of the transitivity axiom

(A! B)! [(B ! C)! (A! C)]

by the `conjoined' form

(A! B) ^ (B ! C)! (A! C):46

Perhaps the most striking positive solution to the decision problem fora relevance logic is that provided for RM by Meyer (see [Anderson andBelnap, 1975, Section 29.3], although the result was �rst obtained by Meyer[1968].47 Meyer showed that a formula containing n propositional variablesis a theorem of RM i� it is valid in the `Sugihara matrix' de�ned on the non-zero integers from �n to +n. this result was extended by [Dunn, 1970] toshow that every `normal' extension of RM has some �nite Sugihara matrix(with possibly 0 as an element) as a characteristic matrix. So clearly RMand its extensions have at least the �nite model property. Cf. Section 3.10for further information about RM.

Meyer [private communication] has thought that the fact that the de-cidability of R is equivalent to the solvability of the word problem for deMorgan monoids suggests that R might be shown to be undecidable bysome suitable modi�cation of the proof that the word problem for monoidsis unsolvable. It turns out that this is technique is the one which pays o� |although the proof is very complex. The complexity arises because thereis an important disanalogy between monoids and de Morgan monoids in

45This is suggestively written (following Meyer) as Ra(bc)x) R(ab)cx.47In fact neither McKinsey-Tarski methods nor �ltration was used in this proof. We are

no clearer now that they could not be used, and we think the place to start would be totry to apply �ltration to the Kripke-style semantics for RM of [Dunn, 1976b], which usesa binary accessibility relation and seems to avoid the problems caused by `existentiallyhungry axioms' for the ternary accessibility relation.


that in the latter the multiplicative operation is necessarily commutative(and the word problem for commutative monoids is solvable).48 Still it hasoccurred to both Meyer and Dunn that it might be possible to de�ne a newmultiplication operation � for both Æ and ^ in such a way as to embed thefree monoid into the free de Morgan monoid. This suspicion has turned outto be right, as we shall see in the next section.

4.12 R, E and Related Systems

As is quite well known by now, the principal systems of relevance logic, R,E and others, are undecidable. Alasdair Urquhart proved this in his groundbreaking papers [Urquhart, 1983; Urquhart, 1984]. We have recounted ear-lier attempts to come to a conclusion on the decidability question. Theinsights that helped decide the issue came from an unexpected quarter |projective geometry. To see why projective geometry gave the necessaryinsights, we will �rst consider a simple case, the undecidability of the sys-tem KR. KR is given by adding A ^ :A ! B to R. A KR frame is onesatisfying the following conditions (given by adding the clause that a = a�

to the conditions for an R frame).

R0ab i� a = b Rabc i� Rbac i� Racb (total permutation)Raaa for each a R2abcd only if R2acbd

The clauses for the connectives are standard, with the proviso that a � :Ai� a 6� A, since a = a�.

Urquhart's �rst important insight was that KR frames are quite likeprojective spaces. A projective space P is a set P of points, and a collectionL of subsets of P called lines, such that any two distinct points are onexactly one line, and any two distinct lines intersect in exactly one point.But we can de�ne projective spaces instead through the ternary relationof collinearity. Given a projective space P , its collinearity relation C is aternary relation satisfying the condition:

Cabc i� a = b = c, or a, b and c are distinct and they lie on acommon line.

If P is a projective space, then its collinearity relation C satis�es the fol-lowing conditions,

Caaa for each a. Cabc i� Cbac i� Cacb. C2abcd only if C2acbd.

48In this connection two things should be mentioned. First, Meyer [unpublished type-script, 1973] has shown that not all �nitely generated de Morgan monoids are �nitelypresentable. Second, Meyer and Routley [1973c] have constructed a positive relevancelogic Q+ (the algebraic semantics for which dispenses with commutativity) and shown itundecidable.

RELEVANCE LOGIC 103

provided that every line has at least four points (this last requirement isnecessary to verify the last condition). Conversely, if we have a set with aternary relation C satisfying these conditions, then the space de�ned withthe original set as points and the sets lab = fc : Cabcg [ fa; bg where a 6= bas lines is a projective space.

Now the similarity with KR frames becomes obvious. If P is a projectivespace, the frame F(P) generated by P is given by adjoining a new point 0,adding the conditions C0aa, Ca0a, and Caa0, and by taking the extendedrelation C to be the accessibility relation of the frame.

Projective spaces have a naturally associated undecidable problem. Theproblem arises when considering the linear subspaces of projective spaces.A subspace of a projective space is a subset which is also a projective spaceunder its inherited collinearity relation. Given any two linear subspaces Xand Y , the subspace X + Y is the set of all points on lines through pointsin X and points in Y .

In KR frames there are propositions which play the role of linear sub-spaces in projective spaces. We need a convention to deal with the extrapoint 0, and we simply decree that 0 should be in every \subspace." Thenlinear subspaces are equivalent to the positive idempotents in a frame. Thatis, they are the propositions X which are positive (so 0 2 X) and idempotent(so X = X ÆX). Clearly for any sentence A and any KR model M, theextension of A, jjAjj inM is a positive idempotent i� 0 � A^ (A$ A ÆA).It is then not too diÆcult to show that if A and B are positive idempotents,so are A ÆB and A ^ B, and that t and > are positive idempotents.

Given a projective space P , the lattice algebra hL;\;+i of all linearsubspaces of the projective space, under intersection and + is a modular ge-ometric lattice. That is, it is a complete lattice, satisfying these conditions:

Modularity a � c) (8b)�a \ (b+ c) � (a \ b) + c�

Geometricity Every lattice element is a join of atoms, and if a is an atomand X is a set where a � �X then there's some �nite Y � X , wherea � �Y .

The lattice of linear subspaces of a projective space satis�es these conditions,and that in fact, any modular geometric lattice is isomorphic to the latticeof linear subspaces of some projective space. Furthermore the lattice ofpositive idempotents of any KR frame is also a modular geometric lattice.

The undecidable problem which Urquhart uses to prove the undecidabil-ity of KR is now simple to state. Hutchinson [1973] and Lipshitz [1974]

proved that

The word problem for any class of modular lattices which in-cludes the subspace lattice of an in�nite dimensional projectivespace is undecidable.


Given an in�nite dimensional projective space in which every line includesat least four points P , the logic of the frame (P) is said to be a strong logic.Our undecidability theorem then goes like this:

Any logic between KR and a strong logic is undecidable.

The proof is not too diÆcult. Consider a modular lattice problem

If v1 = w1 : : : vn = wn then v = w

stated in a language with variables xi (i = 1; 2; : : : ) constants 1 and 0,and the lattice connectives \ and +. Fix a map into the language of KRby setting xti = pi for variables, 0t = t, 1t = >, (v \ w)t = vt ^ wt and(v + w)t = vt Æ wt. The translation of our modular lattice problem is thenthe KR sentence�

B ^ (vt1 $ wt1) ^ � � � ^ (vtn $ wt

n) ^ t�! (vt $ wt)

where the sentence B is the conjunction of all sentences pi ^ (pi $ pi Æ pi)for each pi appearing in the formulae vtj or wt

j .We will show that given a particular in�nite dimensional projective space

(with every line containing at least four points) P , then the word problemis valid in the lattice of linear subspaces of P if and only if its translationis provable in L, for any logic L intermediate between KR and the logic ofthe frame F(P).

If the translation of the word problem is valid in L, then it holds in theframe F(P). Consider the word problem. If it were invalid, then therewould be linear subspaces x1; x2; : : : in the space P such that each vi = wiwould be true while v 6= w. Construct a model on the frame F(P) asfollows. Let the extension of pi be the space xi together with the point 0.It is then simple to show that 0 � B, as each pi is a positive idempotent. Inaddition, 0 � t, and 0 � vti $ wt

i , for the extension of each vti and wti will

be the spaces picked out by vi and wi (both with the obligatory 0 added).However, we would have 0 6� vt $ wt, since the extensions of vt and wt

were picked out to di�er. This would amount to a counterexample to thetranslation of the word problem, which we said was valid. As a result, theword problem is valid in the space P . The converse reasoning is similar.

Unfortunately, these techniques do not work for systems weaker than KR.The proof that positive idempotents are modular uses essentially the specialproperties of KR. Not every positive idempotent in R need be modular.But nonetheless, the techniques of the proof can be extended to apply to amuch wider range of systems. Urquhart examined the structure of the of themodular lattice undecidability result, and he showed that you could makedo with much less. You do not need to restrict your attention to modularlattices to construct an undecidable word problem. But to do that, you needto examine Lipshitz and Hutchinson's proof more carefully. In the rest of

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this section, we will sketch the structure of Urquhart's undecidability proof.The techniques are quite involved, so we do not have the space to go intodetail. For that, the reader is referred to Urquhart [1984].

Lipshitz and Hutchinson proved that the word problem for modular lat-tices was undecidable by embedding into that problem the already knownundecidable word problem for semigroups. It is enough to show that a struc-ture can de�ne a \free associative binary operation", for then you will havethe tools for representing arbitrary semigroup problems. (A semigroup is aset with an associative binary operation. An operation is a \free associative"operation if it satis�es those conditions satis�ed by any associative opera-tion but no more.) We will sketch how this can be done without resortingto a modular lattice.

The required structure is what is called a 0-structure, and a modular4-frame de�ned within a 0-structure. An 0-structure is a set equipped withthe following structure

� A semilattice with respect to u.

� With a binary operator + which is associative and commutative.

� And x � y ) x+ z � y + z.

� 0 + x = x.

� y � 0 ) x u (x+ y) = x.

A 4-frame in a 0-structure is a set fa1; a2; a3; a4g [ fcij : i 6= j; i; j =1; : : : ; 4g such that

� The ais are independent. If G;H � fa1; : : : ; a4g then (�G)u (�H) =�(G \H) (where �; = 0)

� If G � fa1; : : : ; a4g then �G is modular

� ai + ai = ai

� cij = cji

� ai + aj = ai + cik; cij u aj = 0, if i 6= j

� (ai + ak) u (cij + cjk) = cik for distinct i; j; k

Now, we are nearly at the point where we can de�ne a semigroup structure.First, for each distinct i; j, we de�ne the set Lij to be fx : x + aj =ai + aj and x u aj = 0g. Then if b 2 Lij and d 2 Ljk where i; j; k aredistinct, then we set b d = (b + d) u (ai + ak), and it is not diÆcult toshow that b d 2 Lik. Then, we can de�ne a semigroup operation `:' onL12 by setting

x:y = (x c23) (c31 y)


Now it is quite an involved operation to show that this is in fact an associa-tive operation, but it can be done. And in fact, in certain circumstances,the operation is a free associative operation. Given a countably in�nite-dimensional vector space V , its lattice of subspaces is a 0-structure, and itis possible to de�ne a modular 4-frame in this lattice of subspaces, such thatany countable semigroup is isomorphic to a subsemigroup of L12 under thede�ned associative operation. (Urquhart gives the complete proof of thisresult [Urquhart, 1984].)

The rest of the work of the undecidability proof involves showing that thisconstruction can be modelled in a logic. Perhaps surprisingly, it can all bedone in a weak logic like TW[^;_;!;>;?]. We can do without negation bypicking out a distinguished propositional atom f , and by de�ning �A to beA! f , t to be �f , and A :B to be �(A! �B). A is a regular propositioni� ��A$ A is provable. The regular propositions form an 0-structure, un-der the assumption of the formula � = fR(t; f;>;?); N(t; f;>;?);�> $?g. where R(A) is �� A$ A, N(A) is (t! A)! A, and R(A;B; : : : ) isR(A) ^ R(B) ^ � � � and similarly for N . In other words, we can show thatthe conditions for an 0-structure hold in the regular propositions, assuming� as an extra premise. To interpret the 0-structure conditions we interpretu by ^, + by : and 0 by t.

Now we need to model a 4-frame in the 0-structure. This can be doneas we get just the modularity we need from another condition which issimple to state. De�ne K(A) to be R(A) ^ (A ^ �A $ ?) ^ (A _ �A $>) ^ (A : �A$ �A) ^ (A$ A :A). Then we can show the following

K(A); R(B;C); C ! A ` A ^ (B :C)$ (A ^ B) :C

In other words, if K(A), then A is modular in the class of regular proposi-tions. Then the conditions for a 4-frame are simple to state. We pick outour atomic propositions A1; : : : ; A4 and C12; : : : ; C34 which will do duty fora1; : : : ; a4 and c12; : : : ; c34. Then, for example, one independence axiom is

(A1 :A2 :A3) ^ (A2 :A3 :A4)$ (A2 :A3)

and one modularity condition is

K(A1 :A3 :A4)

We will let � be the conjunction of the statements that express that thepropositions Ai and Cij form a 4-frame in the 0-structure of regular proposi-tions. So, �[� is a �nite (but complex) set of propositions. In any algebrain which � [ � is true, the lattice of regular propositions is a 0-structure,and the denotations of the propositions Ai and Cij form a 4-frame. Finally,when coding up a semigroup problem with variables x1; x2; : : : ; xm, we willneed formulae in the language which do duty for these variables. Thus we

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need a condition which picks out the fact that pi (standing for xi) is in L12.We de�ne L(p) to be (p :A2 $ A1 :A2)^ (p^A2 $ t). Then the semigroupoperation on elements of L12 can be de�ned in terms of ^ and : and theformulae Ai and Cij . We assume that done, and we will simply take it thatthere is an operation � on formulae which picks out the algebraic operationon L12. This is enough for us to sketch the undecidability argument.

The deducibility problem for any logic between TW[^;_;!;>;?] and KR is undecidable.

Take a semigroup problem which is known to be undecidable. It may bepresented in the following way

If v1 = w1 : : : vn = wn then v = w

where each term vi; wi is a term in the language of semigroups, constructedout of the variables x1; x2; : : : ; xm for some m. The translation of thatproblem into the language of TW[^;_;!;>;?] is the deducibility problem

�;�; L(p1; : : : ; pm); vt1 $ wt1; : : : ; v

tn $ wt

n ` vt $ wt

where each the translation ut of each term u is de�ned recursively by settingxti to be pi, and (u1:u2)

t to be ut1 � ut2.Now the undecidability result will be immediate once we show that for

any logic between TW and KR the word problem in semigroups is valid ifand only if its translation is valid in that logic.

For left to right, if the word problem is valid in the theory of semi-groups, its translation must be valid, for given the truth of � and � andL(p1; : : : ; pm), the operator � is provably a semigroup operation on thepropositions in L12 in the algebra of the logic, and the terms vi and wisatisfy the semigroup conditions. As a result, we must have vt and wt

picking out the same propositions, hence we have a proof of vt $ wt.Conversely, if the word problem is invalid, then it has an interpretation

in the semigroup S de�ned on L12 in the lattice of subspaces of an in�nitedimensional vector space. The lattice of subspaces of this vector space isthe 0-structure in our countermodel. However, we need a countermodel forour | the 0-structure is not a model of the whole of the logic, since it justmodels the regular propositions. How can we construct this? Consider theargument for KR. There, the subspaces were the positive idempotents in theframe. The other propositions in the frame were arbitrary subsets of points.Something similar can work here. On the vector space, consider the subsetsof points which are closed under multiplication (that is, if x 2 �, so is kx,where k is taken from the �eld of the vector space). This is a De Morganalgebra, de�ning conjunction and disjunction by means of intersection andunion as is usual. Negation is modelled by set di�erence. The fusion �Æ� oftwo sets of points is the set fx+ y : x 2 � and y 2 �g. It is not too diÆcult


to show that this is commutative and associative, and square increasing,when the vector space is in a �eld of characteristic other than 2, since ifx 2 � then x = 1

2x + 12x 2 � Æ �. Then � ! � is simply �(� Æ ��). It

is not too diÆcult to show that this is an algebraic model for KR, andthat the regular propositions in this model are exactly the subspaces of thevector space. It follows that our counterexample in the 0-structure is acounterexample in a model of KR to the translation of the word problem.As a result, the translation is not provable in KR or in any weaker logic.

This result applies to systems between TW and KR, and it shows thatthe deducibility problem is undecidable for any of these systems. In thepresence of the modus ponens axiom A^(A! B)^t! B, this immediatelyyields the undecidability of the theoremhood problem, as the deducibilityproblem can be rewritten as a single formula.

�� ^ � ^ L(p1; : : : ; pm) ^ (vt1 $ wt

1) ^ � � � ^ (vtn $ wtn) ^ t�! (vt $ wt)

As a result, the theoremhood problem for logics between T and KR isundecidable. In particular, R, E and T are all undecidable.

The restriction to TW is necessary in the theorem. Without the pre-�xing and suÆxing axioms, you cannot show that the lattice of regularpropositions is closed under the `fusion-like' connective ` : '.

Before moving on to our next section, let us mention that these geometricmethods have been useful not only in proving the undecidability of logics,but also in showing that interpolation fails in R and related logics [Urquhart,1993].

5 LOOKING ABOUT

A lot of the work in relevance logics taking place in the late 1980's and inthe 1990's has not focussed on Anderson's core problems. Now that thesehave been more or less resolved, work has proceeded apace in other direc-tions. In this section we will give an undeniably indiosyncratic and personaloverview of what we think are some of the strategic directions of this recentresearch. The �rst two sections in this part deal with generalisations |�rst of semantics, and second of proof theory | which situate relevancelogic into a wider setting. The next sections deal with neighbouring formaltheories, and we end with one philosophical application of the machinery ofrelevance logics.

5.1 Gaggle Theory

The fusion connective Æ has played an important part in the study of rel-evance logics. This is because fusion and implication are tied together by

RELEVANCE LOGIC 109

the residuation condition

a � b! c i� a Æ b � c

In addition, in the frame semantics, fusion and implication are tied to thesame ternary relation R, implication with the universal condition and fusionwith the existential condition.

This is an instance of a generalised Galois connection. Galois studiedconnections between functions on partially ordered sets. A Galois connec-tion between two partial orders � on A and �0 on B is a pair of functionsf : A! B and g : B ! A such that

b �0 f(a) i� a � g(b)

The condition tying together fusion and implication is akin to that tyingtogether f and g for Galois. So, gaggle theory (for `ggl': generalised Galoislogic) studies these connections in their generality, and it turns out that rele-vance logics like R, E and T are a part of a general structure which not onlyincludes other relevance logics, but also traditional modal logics, J�onssonand Tarski's Boolean algebras with operators [J�onsson and Tarski, 1951] andmany other formal systems. Dunn has shown that if a logic has a family of n-ary connectives which are tied together with a generalised galois connection,then the logic has a frame semantics in which those connectives are mod-elled using the one n+1-ary relation, in the way that fusion and implicationare modelled by the same ternary relation in relevance logics [Dunn, 1991;Dunn, 1993a; Dunn, 1994].

In general, an n-ary connective f has a trace (�1; : : : ; �n) 7! + if

� f(c1; : : : ;1; : : : ; cn) = 1, if �i = + (where the 1 is in position i).

� f(c1; : : : ;0; : : : ; cn) = 1, if �i = � (where the 0 is in position i).

� If a � b, and if �i = + then f(c1; : : : ; a; : : : ; cn) � f(c1; : : : ; b; : : : ; cn).

� If a � b, and if �i = � then f(c1; : : : ; b; : : : ; cn) � f(c1; : : : ; a; : : : ; cn).

We write this as T (f) = (�1; : : : ; �n) 7! +. On the other hand, the connec-tive f has trace (�1; : : : ; �n) 7! � if

� f(c1; : : : ;1; : : : ; cn) = 0, if �i = + (where the 0 is in position i).

� f(c1; : : : ;0; : : : ; cn) = 0, if �i = � (where the 1 is in position i).

� If a � b, and if �i = + then f(c1; : : : ; b; : : : ; cn) � f(c1; : : : ; a; : : : ; cn).

� If a � b, and if �i = � then f(c1; : : : ; a; : : : ; cn) � f(c1; : : : ; b; : : : ; cn).


We write this as T (c) = (�1; : : : ; �n) 7! �. Here are a few examples oftraces of connectives. Conjunction-like connectives tend to be (�;�) 7! �,disjunction-like connectives tend to be (+;+) 7! +, necessity-like connec-tives tend to be + 7! +, possibility-like connectives tend to be � 7! �, andnegations can be either + 7! � or � 7! + (and in many cases they areboth).

Now we are nearly able to state the abstract law of residuation. First, wede�ne S(f; a1; : : : ; an; b) as follows. If T (f) = (� � � ) 7! +, then S(f; a1; : : : ;an; b) is the condition f(a1; : : : ; an) � b. If, on the other hand, T (f) =(� � � ) 7! �, then S(f; a1; : : : ; an; b) is b � f(a1; : : : ; an). Then, two connec-tives f and g are contrapositives in place j i�, if T (f) = (�1; : : : ; �j ; : : : ; �n)7! � , then T (g) = (�1; : : : ;��; : : : ; �n) 7! ��j . (Where we de�ne �+ as �and �� as +.) Two operators f and g satisfy the abstract law of residuationi� f and g are contrapositives in place j, and S(f; a1; : : : ; aj ; : : : ; an; b) i�S(g; a1; : : : ; b; : : : ; an; aj).

A collection of connectives in which there is some connective f such thatevery element of the collection satis�es the abstract law of residuation withf , is called a founded family of connectives. Dunn's major result is that ifyou have an algebra in which every connective is in a founded family, thenthe algebra is isomorphic to a subalgebra of the collection of propositions ina model in which each founded family of n-ary connectives shares an n+ 1-ary relation. The soundness and completeness of the Routley{Meyer ternaryrelational semantics is for the implication-fusion fragment of relevance logicsis an instance of this more general result.

The gaggle theoretic account of negation in relevance logics is interest-ing. We do not automatically get negation modelled by the Routley star |instead, being a unary connective, negation is modelled with a binary re-lation. One way negation can be modelled along gaggle theoretic lines isas follows. The De Morgan negation connective has trace � 7! +, so thegaggle theoretic result is that there is a binary relation C between set-upssuch that

� x � :A i� for each y where xCy, y 6� AThis is the general semantic structure which models negation connectiveswith trace � 7! +. Given a relation C, which we may read as `compatibil-ity', we can de�ne another negation connective �, using C's converse:

� x � �A i� for each y where yCx, y 6� AThen it follows that A ` �B i� B ` :A. For the De Morgan negation ofrelevance logics, � and : are the same, for the compatibility relation C issymmetric. But in more general settings, this need not hold.

The general perspective of gaggle theory not only opens up new formalsystems to study | it also helps with interpreting the semantics. The

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condition for : above can be read as follows: :A is true at x i� for eachy compatible with x, A is not true at y. This certainly sounds like a morepalatable condition for negation than that using Routley star. We have anunderstanding of what it is for two set-ups (theories, worlds or situations)to be compatible, and the notion of compatibility is tied naturally to thatof negation. Furthermore, the Routley star condition is an instance of thismore general `compatibility' condition. For any set-up a, a� can be seen asthe set-up which `wraps up' all set-ups compatible with a. We can arguewhether there is such an all-encompassing set-up, but if there is, then thesemantics for negation in terms of the compatibility relation is equivalentto that of the Routley star. And in addition, we have another means ofexplaining it.

Furthermore, once we have this generalised position from which to viewnegation, we can tinker with the binary accessibility relation in just the sameway that modal logics are studied. Clearly if Boolean negation (written`�') is present, then :A is simply ��A for the positive modal operator �which uses C as its accessibility relation; and the study of these negationis dealt with using the techniques of modal logic. However, in relevancelogics and other related systems, boolean negation is not present. Andin this case the theory of negations arising from compatibility clauses likethe one we have seen is a young and interesting subject in its own right.This perspective is pursued in Dunn [1994], and Restall [1999] develops aphilosophical interpretation of the semantics.

5.2 Display Logic

Nuel Belnap has developed proof theoretical techniques which are quite sim-ilar to those from gaggle theory. Consider the general problem of providinga sequent calculus for logics like R and others. We have the choice of howto formulae sequents. If they are of the form X ` A, where X is a struc-tured collection of formulae, and A is a formula, then we have the problemof how to state the introduction and elimination of negation rules in sucha way as to make ::A equivalent to A. It is unclear how to do this whilemaintaining that the succedent of every sequent is a single formula. On theother hand, if we allow that sequents are of the form X ` Y , where nowboth X and Y are structured complexes of formulae, it is unclear how tostate a cut rule which is both valid and admits of a cut-elimination proof inthe style of Gentzen. If we are restricted to single formulae in the succedentposition the rule is easy to state:

X ` A Y (A) ` BY (X) ` B

but in the presence of multiple succedents it is unclear how to state therule generally enough to be eliminable yet strictly enough to be valid under


interpretation. If there is only one sort of structuring in the consequent thismight be possible, in the way used in the proof theories of classical or linearlogic, for example:

X ` A; Y X 0; A ` Y 0X;X 0 ` Y; Y 0

But if we have X ` Y (A) and X 0(A) ` Y 0 where the indicated instances ofA are buried under multiple sorts of structure, then what is the appropriateconclusion of a cut rule? X 0(X) ` Y (Y 0) will not do in general, for it isinvalid in many instances. For example, in R if Y (A) is A ^ B and X 0(A)is A ÆD, then we have A^B ` A^B and A ÆD ` A ÆD, but we don't have(A ^ B) Æ D ` (A Æ D) ^ B in general. (Consider the case where B = A.A ÆD needn't imply A.)

The alternative examined by Belnap is to make do with Cut where thecut formula is \displayed" in both premises of the rule.

X ` A A ` YX ` Y

In order to get away with this, a system needs to be such that whenever youneed to use a cut you can. The way Belnap does this is by requiring what hecalls the \display condition". The display condition is satis�ed i� for everyformula, every sequent including that formula is equivalent (using invertiblerules) to one in which that formula is either the entire antecedent or theentire succedent of the sequent. For Belnap's original formulation, this isachieved by having a binary structuring connective Æ (not to be confusedwith the sentential connective Æ) and a unary connective �. The displayrules were as follows:

X Æ Y ` Z () X ` �Y Æ ZX ` Y Æ Z () X Æ �Y ` Z () X ` Z Æ Y

X ` Y () �Y ` �X () � �X ` YA structure is in antecedent position if it is in the left under an even numberof stars, or in the right under an odd number of stars. If it is not inantecedent position, it is in succedent position. The star is read as negation,and the circle is read as conjunction in antecedent position, and disjunctionin succedent position. The display postulates are a reworking of conditionslike the residuation condition for fusion and implication. Here we have theconditions that a Æ b � c i� a � �b+ c (where x+ y is the �ssion of x andy).

Belnap's system allows that di�erent families of structural connectivescan be used for di�erent families of connectives in the language. For exam-ple, when Æ and � are read intensionally, we can have the following rules for

RELEVANCE LOGIC 113

implication:

X ÆA ` BX ` A! B

X ` A B ` YA! B ` �X Æ Y

If the properties of Æ vary, so do the properties of the connective !. Wecan give Æ properties of extensional conjunction in order to get a materialconditional. Or conditions can be tightened, to give ! modal properties.It is clear that the family of structural connectives (here Æ and �) act inanalogously to accessibility relations on frames. However, the connectionswith gaggle theory run deeper, however. It can be shown a connective in-troduced in with rules without side conditions, and in a way which `mimics'structural connectives (just as here A ! B mimics �X Æ Y in consequentposition) must have a de�nable trace. Any implication satisfying those ruleswill have trace (�;+) 7! +, for example. For more details of this connectionand a general argument, see Restall's paper [Restall, 1995a].

Display logic gives these systems a natural cut-free proof theory, for Bel-nap has shown that under a broad set of conditions, any proof theory withthis structure will satisfy cut-elimination. So again, just as with gaggle the-ory, we have an example of the way that the study of relevance logics likeR and E have opened up into a more general theory of logics with similarstructures.

5.3 Paraconsistency

Relevance logics are paraconsistent, in that argument forms such asA^:A `B are taken to be invalid. As a result, relevance logics have been seen tobe important for the study of paraconsistent theories. [[See Priest's articlein this volume]]. Relevance logics are suited to applications for which aparaconsistent notion of consequence is needed however, not all logics areequal in this regard. For example, paraconsistentists have often consideredthe topic of na��ve theories of sets and of truth (any predicate yields the setof things satisfying that predicate, the proposition p is true if and only ifp). With a relevance logic at hand, you can avoid the inference to trivialityfrom contradictions such as that arising from the liar

This proposition is not true.

(from which you can deduce that it is true, and hence that it isn't) andRussell's paradox (fx : x 62 xg both is and is not a member of itself).However, the Curried forms of these paradoxes

If this proposition is true then there is a Santa Claus.

and fx : (x 2 x) ! Pg are more diÆcult to deal with. These yield ar-guments for the existence of Santa Claus and the truth of P (which was


arbitrary) in logics like R, or any others with theorems related to the ruleof contraction. The theoremhood of propositions such as

�A ! (A !

B)� ! (A ! B) and A ^ (A ! B) ! B rule out a logic for service in the

cause of paraconsistent theories like these [Meyer et al., 1979].However, this has not deterred some hardier souls in considering weaker

relevance logics which do not allow one to deduce triviality in these theo-ries. Some work has been done to show that in some logics these theoriesare consistent, and in others, though inconsistent, not everything is a theo-rem [Brady, 1989].

Another direction of paraconsistency in which techniques of relevancelogics have borne fruit is in the more computational area of reasoning withinconsistent information. The techniques of �rst degree entailment havefound a home in the study of \bilattices" by Melvin Fitting and others,who seen in them a suitable framework for reasoning under the possibilityof inconsistent information [Fitting, 1989].

5.4 Semantic Neighbours

Another area in which research has grown in the recent years has beentoward connections with other �elds. It has turned out that seeminglycompletely unrelated �elds have studied structures remarkably like thosestudied in relevance logics. These neighbours are helpful, not only for giv-ing independent evidence for the fact that relevance logicians have beenstudying something worthwhile, but also because of the di�erent insightsthey can bring to bear on theorising. In this section we will see just threeof the neighbours which can shed light on work in relevance logics.

The �rst connection comes with Barwise and Perry's situation semantics[1983]. For Barwise and Perry, utterances classify situations (parts of theworld) which may be incomplete with regard to their semantic `content'.Consider the claim that Max saw Queensland win the SheÆeld Shield".How is this to be understood? For the Barwise and Perryof Situations andAttitudes [Barwise and Perry, 1983], this was to be parsed as expressinga relationship between Max and a situation, where a situation is simplya restricted part of the world. Situations are parts of the world and theysupport information. Max saw a situation and in this situation, Queenslandwon the SheÆeld Shield. If, in this very situation, Queensland beat SouthAustralia, then Max saw Queensland beat South Australia.

This shows why for this account situations have to be (in general) re-stricted bits of the world. The situation Max saw had better not be onein which Paul Keating lost the 1996 Federal Election, lest it follow fromthe fact that Max witnessed Queensland's victory that he also witnessedKeating's defeat, and surely that would be an untoward conclusion. Let'sdenote this relationship between situations an the information they supportas follows. We'll abbreviate the claim that the situation s supports the

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information that A by writing `s � A', and we'll write its negation, that sdoesn't support the information that A by writing `s 6� A'. This is standardin the situation theoretic literature. The information carried by these situ-ations has, according to Barwise and Perry, a kind of logical coherence. Forthem, infons are closed under conjunction and disjunction, and s � A ^ Bif and only if s � A and s � B, and s � A _ B if and only if s � A ors � B. However, negation is a di�erent story | clearly situations don'tsupport the traditional equivalence between s � :A and s 6� A (where :Ais the negation of A), for our situation witnessed by Max supports neitherthe infon \Keating won the 1996 election" nor its negation.

What to do? Well, Barwise and Perry suggest that negation inter-acts with conjunction and disjunction in the familiar ways | :(A _ B)is (equivalent to) :A ^ :B, and :(A ^ B) is (equivalent to) :A _ :B.And similarly, ::A is (equivalent to) A. This gives us a logic of sortsof negation | it is �rst degree entailment. Now for Barwise and Perry,there are no actual situations in which s � A ^ :A (the world is notself-contradictory). However, they agree that it is helpful to consider ab-stract situations which allow this sort of inconsistency. So, Barwise andPerry have an independent motivation for a semantic account of �rst-degreeentailment. (More work has gone on to consider other connections be-tween situation theory and relevance logics [Mares, 1997; Restall, 1994;Restall, 1995b].)

Another connection with a parallel �eld has come from completely di�er-ent areas of research. The semantic structures of relevance logics have closecousins in the models for the Lambek Calculus and in Relation algebras.Let's consider relation algebras �rst.

A relation algebra is a Boolean algebra with some extra operations, abinary operation which denotes composition of relation, a unary operation^,for the converse of a relation, and a constant 1 for the identity relation.There is a widely accepted axiomatisation of the variety RA of relationalgebras. A relation algebra is set R with operations ^;_;�; 1; Æ;^ suchthat

� hR;^;_;�i is a boolean algebra.

� ^ is an automorphism on the algebra, satisfying a^^ = a, (a^b)^ =a^ ^ b^, �(a^) = (�a)^.

� Æ is associative, with a left and right identity 1, satisfying (a_ b) Æ c =(a Æ c) _ (b Æ c), a Æ (b _ c) = (a Æ b) _ (a Æ c).� ^ and Æ are connected by setting (a Æ b)^ = b^ Æ a^.

These conditions are satis�ed by the class of relations on any base set (thatis, by any concrete relation algebra). However, not every algebra satisfyingthese equations is isomorphic to a subalgebra of a concrete relation algebra.


These algebras are quite similar to de Morgan monoids. If we de�ne :Ato be �(a)^ or (�a)^ then the conjunction, disjunction, :, 1 fragment isthat of �rst degree entailment. We do not have a � b _ :b, and nor do wehave a ^ :a � b. Consider the relation a:

a x yx 1 1y 0 1

Then :a is the following relation

a x yx 0 1y 0 0

So we don't have b � a _ :a for every b, and nor do we have a ^ :a _ b.(However, we do have 1 � a _ :a.)

The class of relation algebras have a natural form of implication to goalong with the fusionlike connective Æ. If we de�ne a ! b to be :(:b Æ a),then we have the residuation condition a Æ b � c i� a � b ! c. However,that is not the only implication-like connective we may de�ne. If we setb a to be :(aÆ:b), then aÆ b � c i� b � c a. Since Æ is not, in general,commutative, we have two residuals.

In logics like R this is not possible, for the left and the right residuals offusion are the same connective. However, in systems in the vicinity of E,these implication operations come apart. This is mirrored by the behaviouron frames, since we can de�ne B A by setting x � B A i� for each y; zwhere Ryxz if y � A then z � B. This will be another residual for fusion,and it will not agree with ! in the absence of commutativity of R (if Rxyzthen Ryxz).49

It was hoped for some time that relation algebras would give an inter-esting model for logics like R. However, there does not seem to be a nat-ural class of relations for which composition is commutative and squareincreasing. (The class of symmetric relations will not do. Even if a = a^

and b = b^, it does not follow that a Æ b = b Æ a. You merely get thata Æ b = a^ Æ b^ = (b Æ a)^.) Considered as a logic, RA is a sublogic of R(ignoring boolean negation for the moment). It is not a sublogic of E, sincein RA, a = 1 ! a. Another di�erence between RA and typical relevancelogics is the behaviour of contraposition. We do not have a! b = :b! :a.Instead, a! b = :a :b.

A �nal connection between RA and relevance logics is in the issue ofsemantics. As we stated earlier, not all relation algebras are representableas subalgebras of concrete relation algebras. However, Dunn has shown

49We should ag here that in the relevance logic literature, [Meyer and Routley, 1972]

seems to have been the �rst to consider both left- and right-residuals for fusion.

RELEVANCE LOGIC 117

that all relation algebras are representable by algebras of propositions ona particular class of Routley{Meyer frames [Dunn, 1993b]. This is the �rstrepresentation theorem for RA, and it shows that the semantical techniquesof relevance logics have a wider scope than applications to R, E and theirimmediate neighbours.

In a similar vein, Dunn and Meyer [1997] have provided a Routley{Meyerstyle frame semantics for combinatory logic. The key idea here is that theternary relation R satis�es no special conditions, but these properties areencoded by combinators, which are modelled by special propositions onframes.

Lambek's categorial grammar is also similar to relevance logics, thoughthis time it is introduced with frames, not algebras [Lambek, 1958; Lambek,1961]. Here, the points in frames are pieces of syntax, and the `propositions'are syntactic classi�cations of various kinds. For example, the classi�cationsinto noun phrases, verbs, and sentences. The interest comes with the wayin which these classi�cations can be combined. For example A Æ B can bede�ned, where we say x � A Æ B i� x is a concatenation of two strings yand z, where y � A and z � B. We can also de�ne `slicing' operations,setting x � AnB i� for each y where y � A, yx � B; and x � B=A i�for each y where y � A, xy � B. These are obviously analogues for Æ and! in relevance logics, and again, we have a `left' and `right' residuals forfusion. In these frames Rxyz i� xy = z. So the Lambek calculus gives us anindependently motivated interpretation of a class of Routley{Meyer frames.This connection has been explored by Kurtonina [1995], which is a helpfulsourcebook of some recent work on ternary frames in connection with theLambek calculus and related logics.

If you like, you can enrich the logic of strings with conjunction and dis-junction, and if you do it in the obvious way (using the same clauses as inrelevance logics) you get a formal logic quite like RA [Restall, 1994]. Butmore importantly, the conditions for conjunction and disjunction may beindependently motivated. A string is of type A _ B just when it is of typeA or of type B. A string is of type A ^ B just when it is of type A and oftype B. The resulting logic is clearly interpretable, but it was a number ofyears before a proof theory was found for it. Here the techniques for theGentzenisation for positive relevance logics are appropriate, and the prooftheory can be found by utilising the proof theory for R+, and removingthe commutativity and contraction of the intensional bunching operation.The resulting proof theory captures exactly the Lambek calculus enrichedwith conjunction and disjunction. In addition, the techniques of Giambroneshow that the resulting logic is decidable [Restall, 1994].


5.5 Linear Logic

The burgeoning phenomenon of linear logic is one which has a number offormal similarities to relevance logics [Girard, 1987; Troelstra, 1992]. Linearlogic is the study of systems in the vicinity of LRW (R without contraction,without distribution). This is proof-theoretically a very stable system. Itis simple to show that it is decidable. Girard's innovation, however, is toextend the proof theory with a modal operator ! which allows intuitionisticlogic to be modelled inside linear logic. This operation in given as follows,in single-succedent Gentzen systems.

X ` BX; !A ` B

X ;A ` BX ; !A ` B

!X ` B!X ` !B

X; !A; !A ` BX; !A ` B

Given this proof theory it is possible to show that A) B de�ned as !A!B is an intuitionistic implication. This is similar to Meyer's result thatA ^ t! B is an intuitionistic implication in R (indeed, !A de�ned as A ^ tsatis�es each of the conditions for ! above in R, but not in systems withoutcontraction). However, nothing like it holds in relevance logics withoutcontraction.

Linear logic also brings with it many new algebraic structures and modelsin category theory. None of these models have been mined to see if they canbring any `relevant' insight. However, some transfer has gone on in the otherdirection | Allwein and Dunn [1993] have shown that the multiplicativeand additive fragment of linear logic can be given a Routley{Meyer stylesemantics. This is not a simple job, as the absence of the distribution of(additive) conjunction over disjunction means that at least one of these con-nectives (in this case, disjunction) must take a non-standard interpretation.

5.6 Relevant Predication

There has been one major way in which relevance logics have been used inapplication to philosophical issues, and this application makes a good topicto end this article. The topic is Dunn's work on relevant predication [Dunn,1987].50 The guiding idea is that a theory of relevant implication will giveyou some way of marking out the distinction between the way that Socrates'wisdom is a property of Socrates, in the way that Socrates' wisdom is nota property of Bill Clinton.

Classical �rst order logic is not good at marking out such a distinction,for if Wx stands for `x is wise', and s stands for Socrates, and c stands for

50The reference [Dunn, 1987] is of course \Relevant Predication": Of course all workhas precursors, in this instance (largely unpublished) thoughts in the 1970's by N. Belnap,J. Freeman, and most importantly R. K. Meyer and A. Urquhart (and Dunn). Cf. Sec. 9of [Dunn, 1987] for some history.

RELEVANCE LOGIC 119

Bill Clinton, then Wx is true of x i� it is wise, and (Ws ^ x = x) _Wsis true of something i� Socrates is wise. Why is one a `real' property andthe other not? The guiding idea for relevant predication is the followingdistinction. It is true that if x is Socrates then x is wise. However, it is nottrue that if x is Bill Clinton then Socrates is wise. At least, it is plausiblethat this conditional fail, when read `relevantly'. This can be cashed outformally as follows. F is a relevant property of a (written (�xFx)a) if andonly if (8x)(x = a! Fx).

Given this de�nition, if F is a relevant property of a then Fa holds(quite clearly) and if F and G are relevant properties of a then so is theirconjunction, and the disjunction of any relevant property with anything atall is still a relevant property.

Furthermore, one can de�ne what it is for a relation to truly be a relationbetween objects. If Hx is `x's height is over 1 meter', and Ly is `y is alogician' then, it is true that Greg's height is over 1 meter and Mike is alogician. However, it would be bizarre to hold that in this there is a realrelation that holds between Greg and Mike because of this fact. We wouldhave the following

8x8y(x = g ^ y = m ! Hx ^ Ly)

(assuming that (�xHx)g and (�yLy)m) but it need not follow that

8x8y(x = g ! (y = m ! Hx ^ Ly))

for there is no reason that Hx should follow from y = m, even given thatx = g holds. There is no connection between `y's being m' and Hg.This latter proposition is a good candidate for expressing that there is areal relationship holding between g and m. In other words, we can de�ne(�xyLxy)ab to be

8x8y�x = a! (y = b! Lxy)�

to express the holding of a relevant relation. For more on relevant predi-cation, consult Dunn's series of papers [Dunn, 1987; Dunn, 1990a; Dunn,1990b]

Relevance logics are very good at telling you what follows from whatas a matter of logic | and in this case, the logical structure of relevantpredication and relations. However, more work needs to be done to see inwhat it consists to say that a relevant implication is true. For that, weneed a better grip on how to understand the models of relevance logics. Itis our hope that this chapter will help people in this aim, ant to bring thetechnique of relevance logics to a still wider audience.


ACKNOWLEDGEMENTS

Dunn's acknowledgements from the �rst edition: I wish to express my thanksand deep indebtedness to a number of fellow toilers in the relevant vineyards,for information and discussion over the years. These include Richard Rout-ley and Alasdair Urquhart and especially Nuel D. Belnap, Jr. and RobertK. Meyer, and of course Alan Ross Anderson, to whose memory I dedicatethis essay. I also wish to thank Yong Auh for his patient and skilful help inpreparing this manuscript, and to thank Nuel Belnap, Lloyd Humberstone,and Allen Hazen for corrections, although all errors and infelicities are tobe charged to me.

Our acknowledgements from the second edition: Thanks to Bob Meyer, JohnSlaney, Graham Priest, Nuel Belnap, Richard Sylvan, Ed Mares, RajeevGor�e, Errol Martin, Chris Mortensen, Uwe Petersen and Pragati Jain forhelpful conversations and correspondence on matters relevant to what isdiscussed here. Thanks too to Jane Spurr, who valiantly typed in the �rstedition of the article, to enable us to more easily create this version. Thisedition is dedicated to Richard Sylvan, who died while this essay was beingwritten. Relevance (and relevant) logic has lost one of its most original andproductive proponents.

J. Michael DunnIndiana University

Greg RestallMacquarie University

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[Meyer, 1979b] Robert K. Meyer. Arithmetic formulated relevantly. Unpublishedmanuscript. Some of the results appear in Entailment Volume 2 [Anderson et al.,1992], 1979.

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MARIA LUISA DALLA CHIARA AND ROBERTOGIUNTINI

QUANTUM LOGICS

1 INTRODUCTION

The oÆcial birth of quantum logic is represented by a famous article ofBirkho� and von Neumann \The logic of quantum mechanics" [Birkho�and von Neumann, 1936]. At the very beginning of their paper, Birkho�and von Neumann observe:

One of the aspects of quantum theory which has attracted themost general attention, is the novelty of the logical notions whichit presupposes .... The object of the present paper is to discoverwhat logical structures one may hope to �nd in physical theorieswhich, like quantum mechanics, do not conform to classical logic.

In order to understand the basic reason why a non classical logic arisesfrom the mathematical formalism of quantum theory (QT), a comparisonwith classical physics will be useful.

There is one concept which quantum theory shares alike withclassical mechanics and classical electrodynamics. This is theconcept of a mathematical \phase-space". According to thisconcept, any physical system S is at each instant hypotheticallyassociated with a \point" in a �xed phase-space �; this pointis supposed to represent mathematically, the \state" of S, andthe \state" of S is supposed to be ascertainable by \maximal"observations.

Maximal pieces of information about physical systems are called also purestates . For instance, in classical particle mechanics, a pure state of a singleparticle can be represented by a sequence of six real numbers hr1; : : : ; r6iwhere the �rst three numbers correspond to the position-coordinates, where-as the last ones are the momentum-components.

As a consequence, the phase-space of a single particle system can beidenti�ed with the set IR6, consisting of all sextuples of real numbers. Sim-ilarly for the case of compound systems, consisting of a �nite number n ofparticles.

Let us now consider an experimental proposition P about our system,asserting that a given physical quantity has a certain value (for instance:\the value of position in the x-direction lies in a certain interval"). Such


130 MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

a proposition P will be naturally associated with a subset X of our phase-space, consisting of all the pure states for which P holds. In other words,the subsets of � seem to represent good mathematical representatives ofexperimental propositions. These subsets are called by Birkho� and vonNeumann physical qualities (we will say simply events). Needless to say,the correspondence between the set of all experimental propositions andthe set of all events will be many-to-one. When a pure state p belongs toan event X , we will say that our system in state p veri�es both X and thecorresponding experimental proposition.

What about the structure of all events? As is well known, the power-setof any set is a Boolean algebra. And also the set F(�) of all measurablesubsets of � (which is more tractable than the full power-set of �) turns outto have a Boolean structure. Hence, we may refer to the following Booleanalgebra:

B = hF(�);�;\;[;�;1;0i;

where:

1) � ;\ ;[ ; � are, respectively, the set-theoretic inclusion relation andthe operations intersection, union, relative complement;

2) 1 is the total space �, while 0 is the empty set.

According to a standard interpretation, \ ;[ ; � can be naturally re-garded as a set-theoretic realization of the classical logical connectives and ,or , not . As a consequence, we will obtain a classical semantic behaviour:

� a state p veri�es a conjunction X \ Y i� p 2 X \ Y i� p veri�es bothmembers;

� p veri�es a disjunction X [ Y i� p 2 X [ Y i� p veri�es at least onemember;

� p veri�es a negation �X i� p =2 X i� p does not verify X .

To what extent can such a picture be adequately extended to QT? Birkho�and von Neumann observe:

In quantum theory the points of � correspond to the so called\wave-functions" and hence � is : : : a function-space, usuallyassumed to be Hilbert space.

As a consequence, we immediately obtain a basic di�erence between thequantum and the classical case. The excluded middle principle holds in

QUANTUM LOGICS 131

classical mechanics. In other words, pure states semantically decide anyevent: for any p and X ,

p 2 X or p 2 �X:

QT is, instead, essentially probabilistic. Generally, pure states assign onlyprobability-values to quantum events. Let represent a pure state (a wavefunction) of a quantum system and let P be an experimental proposition(for instance \the spin value in the x-direction is up"). The following casesare possible:

(i) assigns to P probability-value 1 ( (P) = 1);

(ii) assigns to P probability-value 0 ( (P) = 0);

(iii) assigns to P a probability-value di�erent from 1 and from 0 ( (P) 6=0; 1).

In the �rst two cases, we will say that P is true (false) for our system instate . In the third case, P will be semantically indeterminate.

Now the question arises: what will be an adequate mathematical repre-sentative for the notion of quantum experimental proposition? The mostimportant novelty of Birkho� and von Neumann's proposal is based on thefollowing answer: \The mathematical representative of any experimentalproposition is a closed linear subspace of Hilbert space" (we will say simplya closed subspace).1 Let H be a (separable) Hilbert space, whose unitaryvectors correspond to possible wave functions of a quantum system. Theclosed subspaces ofH are particular instances of subsets of H that are closedunder linear combinations and Cauchy sequences. Why are mere subsets ofthe phase-space not interesting in QT? The reason depends on the super-position principle, which represents one of the basic dividing line betweenthe quantum and the classical case. Di�erently from classical mechanics,in quantum mechanics, �nite and even in�nite linear combinations of purestates give rise to new pure states (provided only some formal conditionsare satis�ed). Suppose three pure states ; 1 ; 2 and let be a linearcombination of 1 ; 2:

= c1 1 + c2 2:

1A Hilbert space is a vector space over a division ring whose elements are the real orthe complex or the quaternionic numbers such that

(i) An inner product ( : ; :) that transforms any pair of vectors into an element of thedivision ring is de�ned;

(ii) the space is metrically complete with respect to the metrics induced by the innerproduct ( : ; :).

A Hilbert space H is called separable i� H admits a countable basis.


According to the standard interpretation of the formalism, roughly thismeans that a quantum system in state might verify with probability jc1j2those propositions that are certain for state 1 (and are not certain for )and might verify with probability jc2j2 those propositions that are certainfor state 2 (and are not certain for ). Suppose now some pure states 1; 2; : : : each assigning probability 1 to a given experimental propositionP, and suppose that the linear combination

=Xi

ci i (ci 6= 0)

is a pure state. Then also will assign probability 1 to our propositionP. As a consequence, the mathematical representatives of experimentalpropositions should be closed under �nite and in�nite linear combinations.The closed subspaces ofH are just the mathematical objects that can realizesuch a role.

What about the algebraic structure that can be de�ned on the set C(H)of all mathematical representatives of experimental propositions (let us callthem quantum events)? For instance, what does it mean negation, con-junction and disjunction in the realm of quantum events? As to negation,Birkho� and von Neumann's answer is the following:

The mathematical representative of the negative of any experi-mental proposition is the orthogonal complement of the math-ematical representative of the proposition itself.

The orthogonal complement X 0 of a subspace X is de�ned as the setof all vectors that are orthogonal to all elements of X . In other words, 2 X 0 i� ? X i� for any � 2 X : ( ; �) = 0 (where ( ; �) is the innerproduct of and �). From the point of view of the physical interpretation,the orthogonal complement (called also orthocomplement) is particularlyinteresting, since it satis�es the following property: for any event X andany pure state ,

(X) = 1 i� (X 0) = 0;

(X) = 0 i� (X 0) = 1;

In other words, assigns to an event X probability 1 (0, respectively)i� assigns to the orthocomplement of X probability 0 (1, respectively).As a consequence, one is dealing with an operation that inverts the twoextreme probability-values, which naturally correspond to the truth-valuestruth and falsity (similarly to the classical truth-table of negation).

As to conjunction, Birkho� and von Neumann notice that this can be stillrepresented by the set-theoretic intersection (like in the classical case). For,

QUANTUM LOGICS 133

the intersection X \ Y of two closed subspaces is again a closed subspace.Hence, we will obtain the usual truth-table for the connective and :

veri�es X \ Y i� veri�es both members.

Disjunction, however, cannot be represented here as a set-theoretic union.For, generally, the union X [Y of two closed subspaces is not a closed sub-space. In spite of this, we have at our disposal another good representativefor the connective or : the supremum X t Y of two closed subspaces, thatis the smallest closed subspace including both X and Y . Of course, X t Ywill include X [ Y .

As a consequence, we obtain the following structure

C(H) = hC(H) ;v ;u ;t ; 0 ;1 ;0i ;

where v ;u are the set-theoretic inclusion and intersection; t ; 0 are de�nedas above; while 1 and 0 represent, respectively, the total space H and thenull subspace (the singleton of the null vector, representing the smallestpossible subspace). An isomorphic structure can be obtained by using asa support, instead of C(H), the set P (H) of all projections P of H. As iswell known projections (i.e. idempotent and self-adjoint linear operators)and closed subspaces are in one-to-one correspondence, by the projectiontheorem. Our structure C(H) turns out to simulate a \quasi-Boolean be-haviour"; however, it is not a Boolean algebra. Something very essential ismissing. For instance, conjunction and disjunction are no more distributive.Generally,

X u (Y t Z) 6= (X u Y ) t (X u Z):

It turns out that C(H) belongs to the variety of all orthocomplemented or-thomodular lattices , that are not necessarily distributive.

The failure of distributivity is connected with a characteristic property ofdisjunction in QT. Di�erently from classical (bivalent) semantics, a quantumdisjunction X t Y may be true even if neither member is true. In fact, itmay happen that a pure state belongs to a subspace X t Y , even if belongs neither to X nor to Y (see Figure 1).

Such a semantic behaviour, which may appear prima facie somewhatstrange, seems to re ect pretty well a number of concrete quantum situa-tions. In QT one is often dealing with alternatives that are semanticallydetermined and true, while both members are, in principle, strongly inde-terminate. For instance, suppose we are referring to some one-half spinparticle (say an electron) whose spin may assume only two possible values:either up or down. Now, according to one of the uncertainty principles , thespin in the x direction (spinx) and the spin in the y direction (spiny) rep-resent two strongly incompatible quantities that cannot be simultaneously


OO

��

oo //

X

jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj 77ooooooooooooooo

Y

��

Figure 1. Failure of bivalence in QT

measured. Suppose an electron in state veri�es the proposition \spinx isup". As a consequence of the uncertainty principle both propositions \spinyis up" and \spiny is down" shall be strongly indeterminate. However thedisjunction \either spiny is up or spiny is down" must be true.

Birkho� and von Neumann's proposal did not arouse any immediate in-terest, either in the logical or in the physical community. Probably, thequantum logical approach appeared too abstract for the foundational de-bate about QT, which in the Thirties was generally formulated in a moretraditional philosophical language. As an example, let us only think of thefamous discussion between Einstein and Bohr. At the same time, the workof logicians was still mainly devoted to classical logic.

Only twenty years later, after the appearance of George Mackey's bookMathematical Foundations of Quantum Theory [Mackey, 1957], one has wit-nessed a \renaissance period\ for the logico-algebraic approach to QT. Thishas been mainly stimulated by the contributions of Jauch, Piron, Varadara-jan, Suppes, Finkelstein, Foulis, Randall, Greechie, Gudder, Beltrametti,Cassinelli, Mittelstaedt and many others. The new proposals are charac-terized by a more general approach, based on a kind of abstraction fromthe Hilbert space structures. The starting point of the new trends can besummarized as follows. Generally, any physical theory T determines a class

QUANTUM LOGICS 135

of event-state systems hE ; Si, where E contains the events that may occurto our system, while S contains the states that a physical system describedby the theory may assume. The question arises: what are the abstract con-ditions that one should postulate for any pair hE ; Si? In the case of QT,having in mind the Hilbert space model, one is naturally led to the followingrequirement:

� the set E of events should be a good abstraction from the structureof all closed subspaces in a Hilbert space. As a consequence E shouldbe at least a �-complete orthomodular lattice (generally non distribu-tive).

� The set S of states should be a good abstraction from the statisticaloperators in a Hilbert space, that represent possible states of physicalsystems. As a consequence, any state shall behave as a probabilitymeasure, that assigns to any event in E a value in the interval [0; 1].Both in the concrete and in the abstract case, states may be eitherpure (maximal pieces of information that cannot be consistently ex-tended to a richer knowledge) or mixtures (non maximal pieces ofinformation).

In such a framework two basic problems arise:

I) Is it possible to capture, by means of some abstract conditions that arerequired for any event-state pair hE ; Si, the behaviour of the concreteHilbert space pairs?

II) To what extent should the Hilbert space model be absolutely binding?

The �rst problem gave rise to a number of attempts to prove a kind ofrepresentation theorem. More precisely, the main question was: what arethe necessary and suÆcient conditions for a generic event-state pair hE ; Sithat make E isomorphic to the lattice of all closed subspaces in a Hilbertspace?

Our second problem stimulated the investigation about more and moregeneral quantum structures. Of course, looking for more general structuresseems to imply a kind of discontent towards the standard quantum logicalapproach, based on Hilbert space lattices. The fundamental criticisms thathave been moved concern the following items:

1) The standard structures seem to determine a kind of extensional col-lapse. In fact, the closed subspaces of a Hilbert space represent atthe same time physical properties in an intensional sense and the ex-tensions thereof (sets of states that certainly verify the properties inquestion). As happens in classical set theoretical semantics, there is nomathematical representative for physical properties in an intensional


sense. Foulis and Randall have called such an extensional collapse \themetaphysical disaster" of the standard quantum logical approach.

2) The lattice structure of the closed subspaces automatically renders thequantum proposition system closed under logical conjunction. Thisseems to imply some counterintuitive consequences from the physicalpoint of view. Suppose two experimental propositions that concerntwo strongly incompatible quantities, like \the spin in the x directionis up", \the spin in the y direction is down". In such a situation,the intuition of the quantum physicist seems to suggest the followingsemantic requirement: the conjunction of our propositions has no def-inite meaning; for, they cannot be experimentally tested at the sametime. As a consequence, the lattice proposition structure seems to betoo strong.

An interesting weakening can be obtained by giving up the lattice condi-tion: generally the in�mum and the supremum are assumed to exist only forcountable sets of propositions that are pairwise orthogonal. In the recentquantum logical literature an orthomodular partially ordered set that satis-�es the above condition is simply called a quantum logic. At the same time,by standard quantum logic one usually means the complete orthomodularlattice based on the closed subspaces in a Hilbert space. Needless to ob-serve, such a terminology that identi�es a logic with a particular example ofan algebraic structure turns out to be somewhat misleading from the strictlogical point of view. As we will see in the next sections, di�erent forms ofquantum logic, which represent \genuine logics" according to the standardway of thinking of the logical tradition, can be characterized by convenientabstraction from the physical models.

2 ORTHOMODULAR QUANTUM LOGIC AND ORTHOLOGIC

We will �rst study two interesting examples of logic that represent a natu-ral logical abstraction from the class of all Hilbert space lattices.These arerepresented respectively by orthomodular quantum logic (OQL) and by theweaker orthologic (OL), which for a long time has been also termed min-imal quantum logic. In fact, the name \minimal quantum logic" appearstoday quite inappropriate, since a number of weaker forms of quantum logichave been recently investigated. In the following we will use QL as an ab-breviation for both OL and OQL.

The language of QL consists of a denumerable set of sentential literalsand of two primitive connectives: : (not), ^ (and). The notion of formulaof the language is de�ned in the expected way. We will use the followingmetavariables: p; q; r; : : : for sentential literals and �, �, ; : : : for formulas.

QUANTUM LOGICS 137

The connective disjunction (_ ) is supposed de�ned via de Morgan's law:

� _ � := : (:� ^ :�) :

The problem concerning the possibility of a well behaved conditional con-nective will be discussed in the next Section. We will indicate the basicmetalogical constants as follows: not, and, or, y (if...then), i� (if and onlyif), 8 (for all ), 9 (for at least one).

Because of its historical origin, the most natural characterization of QLcan be carried out in the framework of an algebraic semantics. It will beexpedient to recall �rst the de�nition of ortholattice:

DEFINITION 1 (Ortholattice). An ortholattice is a structure B = hB ;v ;0 ;1 ;0i, where

(1.1) hB ;v ;1 ;0i is a bounded lattice, where 1 is the maximum and0 is the minimum. In other words:

(i) v is a partial order relation on B (re exive, antisymmetricand transitive);

(ii) any pair of elements a; b has an in�mum aub and a supremuma t b such that:a u b v a; b and 8c: c v a; b y c v a u b;a; b v a t b and 8c: a; b v c y a t b v c;

(iii) 8a: 0 v a; a v 1.

(1.2) the 1-ary operation 0 (called orthocomplement) satis�es the fol-lowing conditions:

(i) a00 = a (double negation);

(ii) a v b y b0 v a0 (contraposition);

(iii) a u a0 = 0 (non contradiction).

Di�erently from Boolean algebras, ortholattices do not generally satisfythe distributive laws of u and t. There holds only

(a u b) t (a u c) v a u (b t c)

and the dual form

a t (b u c) v(a t b) u (a t c):The lattice hC(H) ;v ; 0 ;1 ;0i of all closed subspaces in a Hilbert space

H is a characteristic example of a non distributive ortholattice.

DEFINITION 2 (Algebraic realization for OL). An algebraic realization forOL is a pair A = hB ; vi, consisting of an ortholattice B = hB ;v ; 0 ;1 ;0iand a valuation-function v that associates to any formula � of the languagean element (truth-value) in B, satisfying the following conditions:


(i) v(:�) = v(�)0;

(ii) v(� ^ ) = v(�) u v( ).

DEFINITION 3 (Truth and logical truth). A formula � is true in a real-ization A = hB ; vi (abbreviated as j=A �) i� v(�) = 1; � is a logical truthof OL (j=

OL�) i� for any algebraic realization A = hB ; vi, j=A �.

When j=A �, we will also say that A is a model of �; A will be called amodel of a set of formulas T (j=A T ) i� A is a model of any � 2 T .

DEFINITION 4 (Consequence in a realization and logical consequence).Let T be a set of formulas and let A = hB ; vi be a realization. A formula� is a consequence in A of T (T j=A �) i� for any element a of B:if for any � 2 T , a v v(�) then a v v(�).A formula � is a logical consequence of T (T j=

OL�) i� for any algebraic

realization A: T j=A �.

Instead of f�g j=OL� we will write � j=

OL�. If T is �nite and equal to

f�1; : : : ; �ng, we will obviously have: T j=OL� i� v(�1)u � � �uv(�n) v v(�).

One can easily check that j=OL� i� for any T , T j=

OL�.

OL can be equivalently characterized also by means of a Kripke-stylesemantics, which has been �rst proposed by [Dishkant, 1972]. As is wellknown, the algebraic semantic approach can be described as founded on thefollowing intuitive idea: interpreting a language essentially means associat-ing to any sentence � an abstract truth-value or, more generally, an abstractmeaning (an element of an algebraic structure). In the Kripkean semantics,instead, one assumes that interpreting a language essentially means associ-ating to any sentence � the set of the possible worlds or situations where� holds. This set, which represents the extensional meaning of �, is calledthe proposition associated to � (or simply the proposition of �). Hence,generally, a Kripkean realization for a logic L will have the form:

K =DI ;�!Ri ;�!oj ;� ; �

E;

where

(i) I is a non-empty set of possible worlds possibly correlated by relations

in the sequence�!Ri and operations in the sequence �!oj . In most cases,

we have only one binary relation R, called accessibility relation.

(ii) � is a set of sets of possible worlds, representing possible propositionsof sentences. Any proposition and the total set of propositions � mustsatisfy convenient closure conditions that depend on the particularlogic.

(iii) � transforms sentences into propositions preserving the logical form.

QUANTUM LOGICS 139

The Kripkean realizations that turn out to be adequate for OL haveonly one accessibility relation, which is re exive and symmetric. As is wellknown, many logics, that are stronger than positive logic, are instead char-acterized by Kripkean realizations where the accessibility relation is at leastre exive and transitive. As an example, let us think of intuitionistic logic.From an intuitive point of view, one can easily understand the reason whysemantic models with a re exive and symmetric accessibility relation maybe physically signi�cant. In fact, physical theories are not generally con-cerned with possible evolutions of states of knowledge with respect to a con-stant world, but rather with sets of physical situations that may be similar ,where states of knowledge must single out some invariants . And similarityrelations are re exive and symmetric, but generally not transitive.

Let us now introduce the basic concepts of a Kripkean semantics for OL.

DEFINITION 5 (Orthoframe). An orthoframe is a relational structure F =hI; R i, where I is a non-empty set (called the set of worlds) and R (theaccessibility relation) is a binary re exive and symmetric relation on I .

Given an orthoframe, we will use i; j; k; : : : as variables ranging over theset of worlds. Instead of Rij (not Rij) we will also write i ?= j (i ? j).DEFINITION 6 (Orthocomplement in an orthoframe). Let F = hI; R i bean orthoframe. For any set of worlds X � I , the orthocomplement X 0 of Xis de�ned as follows:

X 0 = fi j 8j(j 2 X y j ? i)g :

In other words, X is the set of all worlds that are unaccessible to allelements of X . Instead of i 2 X 0, we will also write i ? X (and we will readit as \i is orthogonal to the set X"). Instead of i =2 X 0, we will also writei ?= X .

DEFINITION 7 (Proposition). Let F = hI; R i be an orthoframe. A set ofworlds X is called a proposition of F i� it satis�es the following condition:

8i [i 2 X i� 8j(i ?= j y j ?= X)] :

In other words, a proposition is a set of worlds X that contains all andonly the worlds whose accessible worlds are not unaccessible to X . Noticethat the conditional i 2 X y 8j(i ?= j y j ?= X) trivially holds for any setof worlds X .

Our de�nition of proposition represents a quite general notion of \possiblemeaning of a formula", that can be signi�cantly extended also to otherlogics. Suppose for instance, a Kripkean frame F = hI; R i, where theaccessibility relation is at least re exive and transitive (as happens in theKripkean semantics for intuitionistic logic). Then a set of worlds X turnsout to be a proposition (in the sense of De�nition 7) i� it is R-closed (i.e.,


8ij(i 2 X and Rij y j 2 X)). And R-closed sets of worlds representprecisely the possible meanings of formulas in the Kripkean characterizationof intuitionistic logic.

LEMMA 8. Let F be an orthoframe and X a set of worlds of F .

(8.1) X is a proposition of F i� 8i [i =2 X y 9j(i ?= j and j ? X)]

(8.2) X is a proposition of F i� X = X 00.

LEMMA 9. Let F = hI; R i be an orthoframe.

(9.1) I and ; are propositions.

(9.2) If X is any set of worlds, then X 0 is a proposition.

(9.3) If C is a family of propositions, thenTC is a proposition.

DEFINITION 10 (Kripkean realization for OL). A Kripkean realization forOL is a system K = hI; R ;� ; �i, where:

(i) hI; Ri is an orthoframe and � is a set of propositions of the framethat contains ;; I and is closed under the orthocomplement 0 andthe set-theoretic intersection \;

(ii) � is a function that associates to any formula � a proposition in�, satisfying the following conditions:

�(:�) = �(�)0;

�(� ^ ) = �(�) \ �( ).

Instead of i 2 �(�), we will also write i j= � (or, i j=K �, in case ofpossible confusions) and we will read: \� is true in the world i". If T is aset of formulas, i j= T will mean i j= � for any � 2 T .

THEOREM 11. For any Kripkean realization K and any formula �:

i j= � i� 8j ?= i9k ?= j (k j= �):

Proof. Since the accessibility relation is symmetric, the left to right im-plication is trivial. Let us prove i j==� y not8j ?= i9k ?= j (k j= �),which is equivalent to i =2 �(�) y 9j ?= i8k ?= j (k =2 �(�)). Supposei =2 �(�). Since �(�) is a proposition, by Lemma 8.1 there holds for a cer-tain j: j ?= i and j ? �(�). Let k ?= j, and suppose, by contradiction,k 2 �(�). Since j ? �(�), there follows j ? k, against k ?= j. Consequently,9j ?= i8k ?= j (k =2 �(�)). �

LEMMA 12. In any Kripkean realization K:

(12.1) i j= :� i� 8j ?= i (j j==�);

QUANTUM LOGICS 141

(12.2) i j= � ^ i� i j= � and i j= .

DEFINITION 13 (Truth and logical truth). A formula � is true in arealization K = hI; R ;� ; �i (abbreviated j=K �) i� �(�) = I ; � is a logicaltruth of OL (j=

OL�) i� for any realization K, j=K �.

When j=K �, we will also say that K is a model of �. Similarly in thecase of a set of formulas T .

DEFINITION 14 (Consequence in a realization and logical consequence).Let T be a set of formulas and let K be a realization. A formula � is a

consequence in K of T (T j=K �) i� for any world i of K, i j= T y i j= �.A formula � is a logical consequence of T (T j=

OL�) i� for any realization K:

T j=K �. When no confusion is possible we will simply write T j= �.

Now we will prove that the algebraic and the Kripkean semantics for OLcharacterize the same logic. Let us abbreviate the metalogical expressions\� is a logical truth of OL according to the algebraic semantics", \� isa logical consequence in OL of T according to the algebraic semantics",\� is a logical truth of OL according to the Kripkean semantics", \� is alogical consequence in OL of T according to the Kripkean semantics", by

j=AOL� ; T j=A

OL� , j=K

OL� ; T j=K

OL�, respectively.

THEOREM 15. j=AOL� i� j=K

OL�; for any �.

The Theorem is an immediate corollary of the following Lemma:

LEMMA 16.

(16.1) For any algebraic realization A there exists a Kripkean realiza-tion KA such that for any �, j=A � i� j=KA �.

(16.2) For any Kripkean realization K there exists an algebraic real-ization AK such that for any �, j=K � i� j=AK �.

Sketch of the proof(16.1) The basic intuitive idea of the proof is the following: any algebraic

realization can be canonically transformed into a Kripkean realization byidentifying the set of worlds with the set of all non-null elements of thealgebra, the accessibility-relation with the non-orthogonality relation in thealgebra, and �nally the set of propositions with the set of all principalquasi-ideals (i.e., the principal ideals, devoided of the zero-element). Moreprecisely, given A = hB ; vi, the Kripkean realization KA = hI; R ;� ; �i isde�ned as follows:

I = fb 2 B j b 6= 0g;Rij i� i 6v j0;� = ffb 2 B j b 6= 0 and b v ag j a 2 Bg;


�(p) = fb 2 I j b v v(p)g.One can easily check that KA is a \good" Kripkean realization; further,there holds, for any � : �(�) = fb 2 B j b 6= 0 and b v v(�)g. Conse-quently, j=A � i� j=KA �.

(16.2) Any Kripkean realization K = hI; R ;� ; �i can be canonicallytransformed into an algebraic realization AK = hB ; vi by putting:

B = �;

for any a; b 2 B: a v b i� a � b;a0 = fi 2 I j i ? ag;1 = I ; 0 = ;;v(p) = �(p).

It turns out that B is an ortholattice. Further, for any �, v(�) = �(�).Consequently: j=K � i� j=AK �. �

THEOREM 17. T j=AOL� i� T j=K

OL�.

Proof. In order to prove the left to right implication, suppose by con-

tradiction: T j=AOL� and T j=K

OL=�. Hence there exists a Kripkean realization

K = hI; R ;� ; �i and a world i of K such that i j= T and i j==�. One caneasily see that K can be transformed into KÆ = hI; R ;�Æ ; �i where �Æ isthe smallest subset of the power-set of I , that includes � and is closed un-der in�nitary intersection. Owing to Lemma 9.3, KÆ is a \good" Kripkeanrealization for OL and for any �, �(�) turns out to be the same propositionin K and in KÆ. Consequently, also in KÆ, there holds: i j= T and i j==�.Let us now consider AKÆ . The algebra B of AKÆ is complete, because �Æ isclosed under in�nitary intersection. Hence,

T f�(�) j � 2 Tg is an elementof B. Since i j= � for any � 2 T , we will have i 2 T f�(�) j � 2 Tg. Thusthere is an element of B, which is less or equal than v(�)(= �(�)) for any� 2 T , but is not less or equal than v(�)(= �(�)), because i =2 �(�). This

contradicts the hypothesis T j=AOL�.

The right to left implication is trivial. �

Let us now turn to a semantic characterization of OQL. We will �rstrecall the de�nition of orthomodular lattice.

DEFINITION 18 (Orthomodular lattice). An orthomodular lattice is anortholattice B = hB ;v ;0 ;1 ;0i such that for any a; b 2 B:

a u (a0 t (a u b)) v b:

Orthomodularity clearly represents a weak form of distributivity.

LEMMA 19. Let B be an ortholattice. The following conditions are equiv-alent:

QUANTUM LOGICS 143

(i) B is orthomodular.

(ii) For any a; b 2 B: a v b y b = a t (a0 u b).(iii) For any a; b 2 B: a v b i� a u (a u b)0 = 0.

(iv) For any a; b 2 B: a v b and a0 u b = 0 y a = b.

The property considered in (19(iii)) represents a signi�cant weakening ofthe Boolean condition:

a v b i� a u b0 = 0:

DEFINITION 20 (Algebraic realization for OQL). An algebraic realizationfor OQL is an algebraic realization A = hB; vi for OL, where B is anorthomodular lattice.

The de�nitions of truth, logical truth and logical consequence in OQLare analogous to the corresponding de�nitions of OL.

Like OL, also OQL can be characterized by means of a Kripkean seman-tics.

DEFINITION 21 (Kripkean realization for OQL). A Kripkean realizationfor OQL is a Kripkean realization K = hI; R ;� ; �i for OL, where the setof propositions � satis�es the orthomodular property :

X 6� Y y X \ (X \ Y )0 6= ;:

The de�nitions of truth, logical truth and logical consequence in OQL areanalogous to the corresponding de�nitions of OL. Also in the case of OQLone can show:

THEOREM 22. j=AOQL

� i� j=KOQL

�.

The Theorem is an immediate corollary of Lemma 16 and of the followinglemma:

LEMMA 23.

(23.1) If A is orthomodular then KA is orthomodular;

(23.2) If K is orthomodular then AK is orthomodular.

Proof. (23.1) We have to prove X 6� Y y X \ (X \ Y )0 6= ; for anypropositions X;Y of KA. Suppose X 6� Y . By de�nition of proposition inKA:

X = fb j b 6= 0 and b v xg for a given x;

Y = fb j b 6= 0 and b v yg for a given y;

Consequently, x 6v y, and by Lemma 19: x u (x u y)0 6= 0 , because A isorthomodular. Hence, x u (x u y)0 is a world in KA. In order to prove


X \ (X \ Y )0 6= ;, it is suÆcient to prove x u (x u y)0 2 X \ (X \ Y )0.There holds trivially x u (x u y)0 2 X . Further, x u (x u y)0 2 (X \ Y )0,because (xu y)0 is the generator of the quasi-ideal (X \ Y )0 . Consequently,x u (x u y)0 2 X \ (X \ Y )0.(23.2) Let K be orthomodular. Then for any X;Y 2 �:

X 6� Y y X \ (X \ Y )0 6= ;:

One can trivially prove:

X \ (X \ Y )0 6= ; y X 6� Y:

Hence, by Lemma 19, the algebra B of AK is orthomodular. �

As to the concept of logical consequence, the proof we have given for OL(Theorem 17) cannot be automatically extended to the case of OQL. Thecritical point is represented by the transformation of K into KÆ whose set ofpropositions is closed under in�nitary intersection: KÆ is trivially a \good"OL-realization; at the same time, it is not granted that KÆ preserves theorthomodular property. One can easily prove:

THEOREM 24. T j=KOQL

� y T j=AOQL

�:

The inverse relation has been proved by [Minari, 1987]:

THEOREM 25. T j=AOQL

� y T j=KOQL

�:

Are there any signi�cant structural relations between A and KAK and

between K and AKA? The question admits a very strong answer in the case

of A and KAK .

THEOREM 26. A = hB; vi and AKA = hB�; v�i are isomorphic realiza-tions.

Sketch of the proof Let us de�ne the function : B ! B� in the followingway:

(a) = fb j b 6= 0 and b v ag for any a 2 B.

One can easily check that: (1) is an isomorphism (from B onto B�); (2)v�(p) = (v(p)) for any atomic formula p. �

At the same time, in the case of K and KAK

, there is no natural cor-respondence between I and �. As a consequence, one can prove only theweaker relation:

THEOREM 27. Given K = hI ; R ;� ; �i and KAK

= hI� ; R� ;�� ; ��i,there holds:

��(�) = fX 2 � j X � �(�)g ; for any �:

QUANTUM LOGICS 145

In the class of all Kripkean realizations for QL, the realizationsKA (whichhave been obtained by canonical transformation of an algebraic realizationA) present some interesting properties, which are summarized by the fol-lowing theorem.

THEOREM 28. In any KA = hI ; R ;� ; �i there is a one-to-one correspon-dence � between the set of worlds I and the set of propositions ��f;g suchthat:

(28.1) i 2 �(i);

(28.2) i ?= j i� �(i) 6� �(j)0;

(28.3) 8X 2 �: i 2 X i� 8k 2 �(i)(k 2 X).

Sketch of the proof Let us take as �(i) the quasi-ideal generated by i. �

Theorem 28 suggests to isolate, in the class of all K, an interesting sub-class of Kripkean realizations, that we will call algebraically adequate.

DEFINITION 29. A Kripkean realization K is algebraically adequate i� itsatis�es the conditions of Theorem 28.

When restricting to the class of all algebraically adequate Kripkean real-izations one can prove:

THEOREM 30. K = hI ; R ;� ; �i and KAK

= hI� ; R� ;�� ; ��i are isomor-phic realizations; i.e., there exists a bijective function from I onto I� suchthat:

(30.1) Rij i� R� (i) (j), for any i; j 2 I;

(30.2) �� = f (X) j X 2 �g, where (X) := f (i) j i 2 Xg;(30.3) ��(p) = (�(p)), for any atomic formula p.

One can easily show that the class of all algebraically adequate Krip-kean realizations determines the same concept of logical consequence thatis determined by the larger class of all possible realizations.

The Kripkean characterization of QL turns out to have a quite naturalphysical interpretation. As we have seen in the Introduction, the mathemat-ical formalism of quantum theory (QT) associates to any physical systemS a Hilbert space H, while the pure states of S are mathematically rep-resented by unitary vectors of H. Let us now consider an elementarysublanguage LQ of QT, whose atomic formulas represent possible measure-ment reports (i.e., statements of the form \the value for the observable Qlies in the Borel set �") and suppose LQ closed under the quantum logicalconnectives. Given a physical system S (whose associated Hilbert space isH), one can de�ne a natural Kripkean realization for the language LQ asfollows:

KS = hI ; R ;� ; �i ;


where:

� I is the set of all pure states of S.

� R is the non-orthogonality relation between vectors (in other words,two pure states are accessible i� their inner product is di�erent fromzero).

� � is the set of all propositions that is univocally determined by theset of all closed subspaces of H (one can easily check that the set ofall unitary vectors of any subspace is a proposition).

� For any atomic formula p, �(p) is the proposition containing all thepure states that assign to p probability-value 1.

Interestingly enough, the accessibility relation turns out to have the fol-lowing physical meaning: Rij i� j is a pure state into which i can betransformed after the performance of a physical measurement that concernan observable of the system.

3 THE IMPLICATION PROBLEM

Di�erently from most weak logics, QL gives rise to a critical \implication-problem". All conditional connectives one can reasonably introduce in QLare, to a certain extent, anomalous; for, they do not share most of the char-acteristic properties that are satis�ed by the positive conditionals (whichare governed by a logic that is at least as strong as positive logic). Justthe failure of a well-behaved conditional led some authors to the conclusionthat QL cannot be a \real" logic. In spite of these diÆculties, these daysone cannot help recognizing that QL admits a set of di�erent implicationalconnectives, even if none of them has a positive behaviour. Let us �rst pro-pose a general semantic condition for a logical connective to be classi�ed asan implication-connective.

DEFINITION 31. In any semantics, a binary connective�! is called an

implication-connective i� it satis�es at least the two following conditions:

(31.1) ��! � is always true (identity);

(31.2) if � is true and ��! � is true then � is true (modus ponens).

In the particular case of QL, one can easily obtain:

LEMMA 32. A suÆcient condition for a connective�! to be an implication-

connective is:

(i) in the algebraic semantics: for any realization A = hA; vi, j=A ��! �

i� v(�) v v(�);

QUANTUM LOGICS 147

(ii) in the Kripkean semantics: for any realization K = hI; R;�; �i,j=K �

�! � i� �(�) � �(�).

In QL it seems reasonable to assume the suÆcient condition of Lemma32 as a minimal condition for a connective to be an implication-connective.

Suppose we have independently de�ned two di�erent implication-connect-ives in the algebraic and in the Kripkean semantics. When shall we admitthat they represent the \same logical connective"? A reasonable answer tothis question is represented by the following convention:

DEFINITION 33. LetA� be a binary connective de�ned in the algebraic

semantics andK� a binary connective de�ned in the Kripkean semantics:

A�and

K� represent the same logical connective i� the following conditions aresatis�ed:

(33.1) given any A = hB; vi and given the corresponding KA =

hI; R;�; �i, �(�K� �) is the quasi-ideal generated by v(�

A� �);

(33.2) given any K = hI; R;�; �i and given the corresponding AK =

hB ; vi, there holds: v(�A� �) = �(�

K� �).

We will now consider di�erent possible semantic characterizations of animplication-connective in QL. Di�erently from classical logic, in QL a mate-rial conditional de�ned by Philo-law (�! � := :�_�), does not give rise toan implication-connective. For, there are algebraic realizations A = hB; visuch that v(:� _ �) = 1, while v(�) 6v v(�). Further, ortholattices andorthomodular lattices are not, generally, pseudocomplemented lattices: inother words, given a; b 2 B, the maximum c such that a u c v b does notnecessarily exist in B. In fact, one can prove [Birkho�, 1995] that anypseudocomplemented lattice is distributive.

We will �rst consider the case of polynomial conditionals , that can bede�ned in terms of the connectives ^ ;_ ;:. In the algebraic semantics, theminimal requirement of Lemma 32 restricts the choice only to �ve possiblecandidates [Kalmbach, 1983]. This result follows from the fact that in theorthomodular lattice freely generated by two elements there are only �vepolynomial binary operations Æ satisfying the condition a v b i� a Æ b = 1.These are our �ve candidates:

(i) v(�!1 �) = v(�)0 t (v(�) u v(�)).

(ii) v(�!2 �) = v(�) t (v(�)0 u v(�)0).

(iii) v(�!3 �) = (v(�)0 u v(�)) t (v(�) u v(�)) t (v(�)0 u v(�)0).

(iv) v(�!4 �) = (v(�)0 uv(�))t (v(�)uv(�))t ((v(�)0 tv(�))uv(�)0).


(v) v(�!5 �) = (v(�)0uv(�))t(v(�)0 uv(�)0)t(v(�)u(v(�)0 tv(�))).

The corresponding �ve implication-connectives in the Kripkean semanticscan be easily obtained. It is not hard to see that for any i (1 � i � 5), !i

represents the same logical connective in both semantics (in the sense ofDe�nition 33).

THEOREM 34. The polynomial conditionals!i (1 � i � 5) are implication-connectives in OQL; at the same time they are not implication-connectivesin OL.

Proof. Since !i represent the same connective in both semantics, it willbe suÆcient to refer to the algebraic semantics. As an example, let us provethe theorem for i = 1 (the other cases are similar). First we have to provev(�) v v(�) i� 1 = v(�!1 �) = v(�)0 t (v(�) u v(�)), which is equivalentto v(�) v v(�) i� v(�)u (v(�)uv(�))0 = 0. From Lemma 19, we know thatthe latter condition holds for any pair of elements of B i� B is orthomodular.This proves at the same time that!1 is an implication-connective in OQL,but cannot be an implication-connective in OL. �

Interestingly enough, each polynomial conditional !i represents a goodweakening of the classical material conditional. In order to show this result,let us �rst introduce an important relation that describes a \Boolean mutualbehaviour" between elements of an orthomodular lattice.

DEFINITION 35 (Compatibility).Two elements a; b of an orthomodular lattice B are compatible i�

a = (a u b0) t (a u b):

One can prove that a; b are compatible i� the subalgebra of B generatedby fa; bg is Boolean.

THEOREM 36. For any algebraic realization A = hB; vi and for any �; �:

v(�!i �) = v(�)0 t v(�) i� v(�) and v(�) are compatible.

As previously mentioned, Boolean algebras are pseudocomplemented lat-tices. Therefore they satisfy the following condition for any a; b; c:

c u a v b i� c v a b;

where: a b := a0 t b.An orthomodular lattice B turns out to be a Boolean algebra i� for any

algebraic realization A = hB; vi, any i (1 � i � 5) and any �; � the followingimport-export condition is satis�ed:

v( ) u v(�) v v(�) i� v( ) v v(�!i �):

QUANTUM LOGICS 149

In order to single out a unique polynomial conditional, various weaken-ings of the import-export condition have been proposed. For instance thefollowing condition (which we will call weak import-export):

v( ) u v(�) v v(�) i� v( ) v v(�) !i v(�);whenever v(�) and v(�) are compatible.

One can prove [Hardegree, 1975; Mittelstaedt, 1972] that a polynomialconditional !i satis�es the weak import-export condition i� i = 1. Asa consequence, we can conclude that !1 represents, in a sense, the bestpossible approximation for a material conditional in quantum logic. Thisconnective (often called Sasaki-hook) was originally proposed by [Mittel-staedt, 1972] and [Finch, 1970], and was further investigated by [Hardegree,1976] and other authors. In the following, we will usually write ! insteadof !1 and we will neglect the other four polynomial conditionals.

Some important positive laws that are violated by our quantum logicalconditional are the following:

�! (� ! �);

(�! (� ! ))! ((�! �)! (�! ));

(�! �)! ((� ! )! (�! ));

(� ^ � ! )! (�! (� ! ));

(�! (� ! ))! (� ! (�! )):

This somewhat \anomalous" behaviour has suggested that one is deal-ing with a kind of counterfactual conditional . Such a conjecture seems tobe con�rmed by some important physical examples. Let us consider againthe class of the Kripkean realizations of the sublanguage LQ of QT (whoseatomic sentences express measurement reports). And let KS = hI; R;�; �irepresent a Kripkean realization of our language, which is associated to aphysical system S. As [Hardegree, 1975] has shown, in such a case the con-ditional! turns out to receive a quite natural counterfactual interpretation(in the sense of Stalnaker). More precisely, one can de�ne, for any formula�, a partial Stalnaker-function f� in the following way:

f� : Dom(f�)! I;

where:

Dom(f�) = fi 2 I j i ?= �(�)g :In other words, f� is de�ned for all and only the states that are not orthog-onal to the proposition of �.


If i 2 Dom(f�), then:

f�(i) = jP�(�)i;

where P�(�) is the projection that is uniquely associated with the closed sub-

space determined by �(�), and jP�(�)i is the normalized vector determinedby P�(�)i. There holds:

i j= �! � i� either 8j ?= i(j j==�) or f�(i) j= �:

From an intuitive point of view, one can say that f�(i) represents the \purestate nearest" to i, that veri�es �, where \nearest" is here de�ned in termsof the metrics of the Hilbert space H. By de�nition and in virtue of one ofthe basic postulates of QT (von Neumann's collapse of the wave function),f�(i) turns out to have the following physical meaning: it represents thetransformation of state i after the performance of a measurement concerningthe physical property expressed by �, provided the result was positive. As aconsequence, one obtains: �! � is true in a state i i� either � is impossiblefor i or the state into which i has been transformed after a positive �-test,veri�es �.

Another interesting characteristic of our connective !, is a weak nonmonotonic behaviour. In fact, in the algebraic semantics the inequality

v(�! ) v v(� ^ � ! )

can be violated (a counterexample can be easily obtained in the orthomod-ular lattice based on IR3). As a consequence:

�! j==� ^ � ! :

Polynomial conditionals are not the only signi�cant examples of implication-connectives in QL. In the framework of a Kripkean semantic approach, itseems quite natural to introduce a conditional connective(, that representsa kind of strict implication. Given a Kripkean realization K = hI; R;�; �ione would like to require:

i j= �( � i� 8j ?= i (j j= � y j j= �):

However such a condition does not automatically represent a correct se-mantic de�nition, because it is not granted that �(�( �) is an element of�. In order to overcome this diÆculty, let us �rst de�ne a new operation inthe power-set of an orthoframe hI; Ri.DEFINITION 37 (Strict-implication operation ( ( ). Given an orthoframehI; Ri and X;Y � I :

X ( Y := fi j 8j (i ?= j and j 2 X y j 2 Y )g :

QUANTUM LOGICS 151

If X and Y are sets of worlds in the orthoframe, then X ( Y turns outto be a proposition of the frame.

When the set � of K is closed under ( , we will say that K is a realizationfor a strict-implication language.

DEFINITION 38 (Strict implication ((). If K = hI; R;�; �i is a realizationfor a strict-implication language, then

�(�( �) := �(�) ( �(�):

One can easily check that ( is a \good" conditional. There followsimmediately:

i j= �( � i� 8j ?= i (j j= � y j j= �):

Another interesting implication that can be de�ned in QL is represented byan entailment-connective.

DEFINITION 39 (Entailment (�). Given K = hI; R;�; �i,

�(�� ) :=

(I; if �(�) � �(�);

;; otherwise:

Since I; ; 2 �, the de�nition is correct. One can trivially check that� isa \good" conditional. Interestingly enough, our strict implication and ourentailment represent \good" implications also for OL.

The general relations between!;( and� are described by the followingtheorem:

THEOREM 40. For any realization K for a strict-implication language ofOL:

j=K (�� )� (�( �):

For any realization K for a strict-implication language of OQL:

j=K (�� )� (�! �); j=K (�( �)� (�! �):

But the inverse relations do not generally hold!

Are the connectives( and � de�nable also in the algebraic semantics?The possibility of de�ning � is straightforward.

DEFINITION 41 (Entailment in the algebraic semantics). GivenA = hB; vi,

v(�� ) :=

(1; if v(�) v v(�);

0; otherwise:


One can easily check that � represents the same connective in the twosemantics. As to (, given A = hB; vi, one would like to require:

v(�( �) =Ffb 2 B j b 6= 0 and 8c(c 6= 0 and b 6v c0 and

c v v(�) y c v v(�))g:However such a de�nition supposes the algebraic completeness of B. Fur-

ther we can prove that ( represents the same connective in the two se-mantics only if we restrict our consideration to the class of all algebraicallyadequate Kripkean realizations.

4 METALOGICAL PROPERTIES AND ANOMALIES

Some metalogical distinctions that are not interesting in the case of a num-ber of familiar logics weaker than classical logic turn out to be signi�cantfor QL (and for non distributive logics in general).

We have already de�ned (both in the algebraic and in the Kripkean se-mantics) the concepts of model and of logical consequence. Now we willintroduce, in both semantics, the notions of quasi-model , weak consequenceand quasi-consequence. Let T be any set of formulas.

DEFINITION 42 (Quasi-model).

Algebraic semantics Kripkean semanticsA realization A = hB; vi A realization K = hI; R;�; �iis a quasi-model of T i� is a quasi-model of T i�9a[a 2 B and a 6= 0 and 9i(i 2 I and i j= T ).8� 2 T (a v v(�))].

The following de�nitions can be expressed in both semantics.

DEFINITION 43 (Realizability and veri�ability). T is realizable (RealT )i� it has a quasi-model; T is veri�able (VerifT ) i� it has a model.

DEFINITION 44 (Weak consequence). A formula � is a weak consequenceof T (T j� �) i� any model of T is a model of �.

DEFINITION 45 (Quasi-consequence). A formula � is a quasi-consequenceof T (T j� �) i� any quasi-model of T is a quasi-model of �.

One can easily check that the algebraic notions of veri�ability, realizabil-ity, weak consequence and quasi-consequence turn out to coincide with thecorresponding Kripkean notions. In other words, T is Kripke-realizable i�T is algebraically realizable. Similarly for the other concepts.

In both semantics one can trivially prove the following lemmas.

LEMMA 46. Verif T y Real T.

LEMMA 47. Real T i� for any contradiction � ^ :�, T j==� ^ :�.

QUANTUM LOGICS 153

LEMMA 48. T j= � y T j� �; T j= � y T j� �.

LEMMA 49. � j� � i� :� j� :�.

Most familiar logics, that are stronger than positive logic, turn out tosatisfy the following metalogical properties, which we will call Herbrand{Tarski , veri�ability and Lindenbaum, respectively.

� Herbrand{Tarski

T j= � i� T j� � i� T j� �� Veri�ability

VerT i� RealT

� Lindenbaum

RealT y 9T � [T � T � and ComplT �], where

ComplT i� 8� [� 2 T or :� 2 T ].

The Herbrand{Tarski property represents a semantic version of the de-duction theorem. The Lindenbaum property asserts that any semanticallynon-contradictory set of formulas admits a semantically non-contradictorycomplete extension. In the algebraic semantics, canonical proofs of theseproperties essentially use some versions of Stone-theorem, according towhich any proper �lter F in an algebra B can be extended to a propercomplete �lter F � (such that 8a(a 2 F � or a0 2 F �)). However, Stone-theorem does not generally hold for non distributive orthomodular lattices!In the case of ortholattices, one can still prove that every proper �lter canbe extended to an ultra�lter (i.e., a maximal �lter that does not admitany extension that is a proper �lter). However, di�erently from Booleanalgebras, ultra�lters need not be complete.

A counterexample to the Herbrand{Tarski property in OL can be ob-tained using the \non-valid" part of the distributive law. We know that(owing to the failure of distributivity in ortholattices):

� ^ (� _ ) j== (� ^ �) _ (� ^ ):

At the same time

� ^ (� _ ) j� (� ^ �) _ (� ^ );

since one can easily calculate that for any realization A = hB; vi the hypoth-esis v(� ^ (� _ )) = 1, v((� ^ �) _ (� ^ )) 6= 1 leads to a contradiction 2.

2In OQL a counterexample in two variables can be obtained by using the failureof the contraposition law for !. One has: � ! � j==:� ! :�. At the same time� ! � j� :� ! :�; since for any realization A = hB; vi the hypothesis v(� ! �) = 1,implies v(�) v v(�) and therefore v(:� ! :�) = v(�)t(v(�)0uv(�)0) = v(�)tv(�)0 = 1.


A counterexample to the veri�ability-property is represented by the nega-tion of the a fortiori principle for the quantum logical conditional !:

:= :(�! (� ! �)) = :(:� _ (� ^ (:� _ (� ^ �)))):

This has an algebraic quasi-model. For instance the realization A =hB; vi, where B is the orthomodular lattice determined by all subspaces ofthe plane (as shown in Figure 2). There holds: v( ) = v(�) 6= 0. But onecan easily check that cannot have any model, since the hypothesis thatv( ) = 1 leads to a contradiction in any algebraic realization of QL.

OO

��

oo //

v(�)

jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

v(�)��

��

Figure 2. Quasi-model for

The same also represents a counterexample to the Lindenbaum-property.Let us �rst prove the following lemma.

LEMMA 50. If T is realizable and T � T �, where T � is realizable andcomplete, then T is veri�able.

Sketch of the proof Let us de�ne a realization A = hB; vi such that

(i) B = f1; 0g;(ii)

v(�) =

(1; if T � j= �;

0; otherwise:

QUANTUM LOGICS 155

Since T � is realizable and complete, A is a good realization and is triviallya model of T . �

Now, one can easily show that violates Lindenbaum. Suppose, by con-tradiction, that has a realizable and complete extension. Then, by Lemma50, must have a model, and we already know that this is impossible.

The failure of the metalogical properties we have considered represents,in a sense, a relevant \anomaly" of quantum logics. Just these anomaliessuggest the following conjecture: the distinction between epistemic logics(characterized by Kripkean models where the accessibility relation is at leastre exive and transitive) and similarity logics (characterized by Kripkeanmodels where the accessibility relation is at least re exive and symmetric)seems to represent a highly signi�cant dividing line in the class of all logicsthat are weaker than classical logic.

5 A MODAL INTERPRETATION OF OL AND OQL

QL admits a modal interpretation [Goldblatt, 1974; Dalla Chiara, 1981]

which is formally very similar to the modal interpretation of intuitionisticlogic. Any modal interpretation of a given non-classical logic turns out tobe quite interesting from the intuitive point of view, since it permits usto associate a classical meaning to a given system of non-classical logicalconstants. As is well known, intuitionistic logic can be translated into themodal system S4. The modal basis that turns out to be adequate for OL isinstead the logic B. Such a result is of course not surprising, since both theB-realizations and the OL-realizations are characterized by frames wherethe accessibility relation is re exive and symmetric.

Suppose a modal language LM whose alphabet contains the same senten-tial literals as QL and the following primitive logical constants: the classicalconnectives � (not), f (and) and the modal operator � (necessarily). Atthe same time, the connectives g (or), � (if ... then), � (if and only if ),and the modal operator � (possibly) are supposed de�ned in the standardway.

The modal logic B is semantically characterized by a class of Kripkeanrealizations that we will call B-realizations.

DEFINITION 51. A B-realization is a system M = hI; R;�; �i where:

(i) hI; Ri is an orthoframe;

(ii) � is a subset of the power-set of I satisfying the following con-ditions:

I; ; 2 �;

� is closed under the set-theoretic relative complement �,the set-theoretic intersection \ and the modal operation �,which is de�ned as follows:


for any X � I; �X := fi j 8j (Rij y j 2 X)g;(iii) � associates to any formula � of LM a proposition in � satisfying

the conditions: �(� �) = ��(�); �(�f ) = �(�)\�( ); �(��) =��(�).

Instead of i 2 �(�), we will write i j= �. The de�nitions of truth, logicaltruth and logical consequence for B are analogous to the correspondingde�nitions in the Kripkean semantics for QL.

Let us now de�ne a translation � of the language of QL into the languageLB.

DEFINITION 52 (Modal translation of OL).

� �(p) = ��p;

� �(:�) = � � �(�);

� �(� ^ ) = �(�) f �( ).

In other words, � translates any atomic formula as the necessity of thepossibility of the same formula; further, the quantum logical negation is in-terpreted as the necessity of the classical negation, while the quantum logicalconjunction is interpreted as the classical conjunction. We will indicate theset f�(�) j � 2 Tg by �(T ).

THEOREM 53. For any � and T of OL: T j=OL� i� �(T ) j=

B�(�)

Theorem 53 is an immediate corollary of the following Lemmas 54 and55.

LEMMA 54. Any OL-realization K = hI; R;�; �i can be transformed intoa B-realization MK = hI�; R�;��; ��i such that: I� = I; R� = R;8i (i j=K � i� i j=MK �(�)).

Sketch of the proof Take �� as the smallest subset of the power-set ofI that contains �(p) for any atomic formula p and that is closed underI; ;;�;\;�. Further, take ��(p) equal to �(p). �

LEMMA 55. Any B-realization M = hI; R;�; �i can be transformed into aOL-realization KM = hI�; R�;��; ��i such that: I� = I; R� = R;8i (i j=KM � i� i j=M �(�)).

Sketch of the proof Take �� as the smallest subset of the power-set ofI that contains �(��p) for any atomic formula p and that is closed underI; ;;0 ;\ (where for any set X of worlds, X 0 := fj j notRijg). Further take��(p) equal to �(��p). The set ��(p) turns out to be a proposition in theorthoframe hI�; R�i, owing to the B-logical truth: �� . �

QUANTUM LOGICS 157

The translation of OL into B is technically very useful, since it permitsus to transfer to OL some nice metalogical properties such as decidabilityand the �nite-model property .

Does also OQL admit a modal interpretation? The question has a some-what trivial answer. It is suÆcient to apply the technique used for OLby referring to a convenient modal system Bo (stronger than B) which isfounded on a modal version of the orthomodular principle. SemanticallyBo can be characterized by a particular class of realizations. In order todetermine this class, let us �rst de�ne the concept of quantum propositionin a B-realization.

DEFINITION 56. Given a B-realization M = hI; R;�; �i the set �Q ofall quantum propositions of M is the smallest subset of the power-set of Iwhich contains �(��p) for any atomic p and is closed under 0 and \.

LEMMA 57. In any B-realization M = hI; R;�; �i, there holds �Q � �.

Sketch of the proof The only non-trivial point of the proof is representedby the closure of � under 0. This holds since one can prove: 8X 2 � (X 0 =��X). �

LEMMA 58. Given M = hI; R;�; �i and KM = hI; R;��; ��i, there holds�Q = ��.

LEMMA 59. Given K = hI; R;�; �i and MK = hI; R;��; ��i, there holds� � ��Q.

DEFINITION 60. A Bo-realization is a B-realization hI; R;�; �i that sat-is�es the orthomodular property:

8X;Y 2 �Q : X 6� Y y X \ (X \ Y )0 6= ;:

We will also call the Bo-realizations orthomodular realizations .

THEOREM 61. For any T and � of OQL: T j=OL� i� �(T ) j=

Bo�(�).

The Theorem is an immediate corollary of Lemmas 54, 55 and of thefollowing Lemma:

LEMMA 62.

(62.1) If K is orthomodular then MK is orthomodular.

(62.2) If M is orthomodular then KM is orthomodular.

Unfortunately, our modal interpretation of OQL is not particularly in-teresting from a logical point of view. Di�erently from the OL-case, Bo

does not correspond to a familiar modal system with well-behaved metalog-ical properties. A characteristic logical truth of this logic will be a modalversion of orthomodularity:

�f � � � � [� f� � (� f �)] ;


where �; � are modal translations of formulas of OQL into the languageLM.

6 AN AXIOMATIZATION OF OL AND OQL

QL is an axiomatizable logic. Many axiomatizations are known: both inthe Hilbert-Bernays style and in the Gentzen-style (natural deduction andsequent-calculi).3 We will present here a QL-calculus (in the natural deduc-tion style) which is a slight modi�cation of a calculus proposed by [Gold-blatt, 1974]. The advantage of this axiomatization is represented by thefact that it is formally very close to the algebraic de�nition of ortholattice;further it is independent of any idea of quantum logical implication.

Our calculus (which has no axioms) is determined as a set of rules . LetT1; : : : ; Tn be �nite or in�nite (possibly empty) sets of formulas. Any rulehas the form

T1 j��1; : : : ; Tn j��nT j��

( if �1 has been inferred from T1; : : : ; �n has been inferred from Tn, then �can be inferred from T ). We will call any T j�� a con�guration. The con-�gurations T1 j��1; : : : ; Tn j��n represent the premisses of the rule, whileT j�� is the conclusion. As a limit case, we may have a rule, where theset of premisses is empty; in such a case we will speak of an improper rule.Instead of ;

T j��we will write T j��; instead of ; j��, we will write j��.

Rules of OL

(OL1) T [ f�g j�� (identity)

(OL2)T j��; T � [ f�g j��

T [ T � j�� (transitivity)

(OL3) T [ f� ^ �g j�� (^-elimination)

(OL4) T [ f� ^ �g j�� (^-elimination)

(OL5)T j��; T j��T j�� ^ � (^-introduction)

(OL6)T [ f�; �g j� T [ f� ^ �g j� (^-introduction)

3Sequent calculi for di�erent forms of quantum logic will be described in Section 17.

QUANTUM LOGICS 159

(OL7)f�g j��; f�g j�:�

:� (absurdity)

(OL8) T [ f�g j�::� (weak double negation)

(OL9) T [ f::�g j�� (strong double negation)

(OL10) T [ f� ^ :�g j�� (Duns Scotus)

(OL11)f�g j��f:�g j�:� (contraposition)

DEFINITION 63 (Derivation). A derivation of OL is a �nite sequence ofcon�gurations T j��, where any element of the sequence is either the conclu-sion of an improper rule or the conclusion of a proper rule whose premissesare previous elements of the sequence.

DEFINITION 64 (Derivability). A formula � is derivable from T (T j�OL�)

i� there is a derivation such that the con�guration T j�� is the last elementof the derivation.

Instead of f�g j�OL� we will write � j�

OL�. When no confusion is possible,

we will write T j�� instead of T j�OL�.

DEFINITION 65 (Logical theorem). A formula � is a logical theorem ofOL ( j�

OL�) i� ; j�

OL�.

One can easily prove the following syntactical lemmas.

LEMMA 66. �1; : : : ; �n j�� i� �1 ^ � � � ^ �n j��.

LEMMA 67. Syntactical compactness.T j�� i� 9T � � T (T � is �nite and T � j��).

LEMMA 68. T j�� i� 9�1; : : : ; �n (�1 2 T and : : : and �n 2 T and�1 ^ � � � ^ �n j��).

DEFINITION 69 (Consistency). T is an inconsistent set of formulas if9� (T j�� ^ :�); T is consistent , otherwise.

DEFINITION 70 (Deductive closure). The deductive closure T of a set offormulas T is the smallest set which includes the set f� j T j��g. T iscalled deductively closed i� T = T .

DEFINITION 71 (Syntactical compatibility). Two sets of formulas T1 andT2 are called syntactically compatible i�

8� (T1 j�� y T2 j�=:�):

The following theorem represents a kind of \weak Lindenbaum theorem".


THEOREM 72. Weak Lindenbaum theorem.If T j�=:�, then there exists a set of formulas T � such that T � is compatiblewith T and T � j��.

Proof. Suppose T j�=:�. Take T � = f�g. There holds trivially: T � j��.Let us prove the compatibility between T and T � . Suppose, by contradic-tion, T and T � incompatible. Then, for a certain �, T � j�� and T j�:�.Hence (by de�nition of T �), � j�� and by contraposition, :� j�:�. Conse-quently, because T j�:�, one obtains by transitivity: T j�:�, against ourhypothesis. �

We will now prove a soundness and a completeness theorem with respectto the Kripkean semantics.

THEOREM 73. Soundness theorem.

T j�� y T j= �:

Proof. Straightforward. �

THEOREM 74. Completeness theorem.

T j= � y T j��:

Proof. It is suÆcient to construct a canonical model K = hI; R;�; �i suchthat:

T j�� i� T j=K �:

As a consequence we will immediately obtain:

T j�=� y T j==K� y T j==�:De�nition of the canonical model

(i) I is the set of all consistent and deductively closed sets of formulas;

(ii) R is the compatibility relation between sets of formulas;

(iii) � is the set of all propositions in the frame hI; Ri;(iv) �(p) = fi 2 I j p 2 ig.

In order to recognize that K is a \good" OL-realization, it is suÆcientto prove that: (a) R is re exive and symmetric; (b) �(p) is a propositionin the frame hI; Ri.The proof of (a) is immediate (re exivity depends on the consistency of anyi, and symmetry can be shown using the weak double negation rule).

QUANTUM LOGICS 161

In order to prove (b), it is suÆcient to show (by Lemma 8.1): i =2 �(p) y9j ?= i (j ? �(p)). Let i =2 �(p). Then (by de�nition of �(p)): p =2 i; and,since i is deductively closed, i j�= p. Consequently, by the weak Lindenbaumtheorem (and by the strong double negation rule), for a certain j: j ?= i and:p 2 j. Hence, j ? �(p).

LEMMA 75. Lemma of the canonical model.

For any � and any i 2 I, i j= � i� � 2 i.

Sketch of the proof By induction on the length of �. The case � = pholds by de�nition of �(p). The case � = :� can be proved by using Lemma12.1 and the weak Lindenbaum theorem. The case � = � ^ can be provedusing the ^-introduction and the ^-elimination rules. �

Finally we can show that T j�� i� T j=K �. Since the left to right impli-cation is a consequence of the soundness-theorem, it is suÆcient to prove:T j�=� y T j==K�. Let T j�=�; then, by Duns Scotus, T is consistent. Takei := T . There holds: i 2 I and T � i. As a consequence, by the Lemma ofthe canonical model, i j= T . At the same time i j==�. For, should i j= � bethe case, we would obtain � 2 i and by de�nition of i, T j��, against ourhypothesis. �

An axiomatization of OQL can be obtained by adding to the OL-calculusthe following rule:

(OQL) � ^ :(� ^ :(� ^ �)) j� �. (orthomodularity)

All the syntactical de�nitions we have considered for OL can be extendedto OQL. Also Lemmas 66, 67, 68 and the weak Lindenbaum theorem can beproved exactly in the same way. Since OQL admits a material conditional,we will be able to prove here a deduction theorem:

THEOREM 76. � j�OQL

� i� j�OQL

�! �.

This version of the deduction-theorem is obviously not in contrast withthe failure in QL of the semantical property we have called Herbrand{Tarski. For, di�erently from other logics, here the syntactical relation j�does not correspond to the weak consequence relation!

The soundness theorem can be easily proved, since in any orthomodularrealization K there holds:

� ^ :(� ^ :(� ^ �)) j=K �:

As to the completeness theorem, we need a slight modi�cation of theproof we have given for ` OL. In fact, should we try and construct thecanonical model K, by taking � as the set of all possible propositions of the


frame, we would not be able to prove the orthomodularity of K. In orderto obtain an orthomodular canonical model K = fI; R;�; �g, it is suÆcientto de�ne � as the set of all propositions X of K such that X = �(�)for a certain �. One immediately recognizes that �(p) 2 � and that �is closed under 0 and \. Hence K is a \good" OL-realization. Also forthis K one can easily show that i j= � i� � 2 i. In order to prove theorthomodularity of K, one has to prove for any propositions X;Y 2 �,X 6� Y y X \ (X \ Y )0 6= ;; which is equivalent (by Lemma 19) toX\ (X \ (X \Y )0)0 � Y . By construction of �, X = �(�) and Y = �(�) forcertain �; �. By the orthomodular rule there holds �^:(�^:(�^�)) j� �.Consequently, for any i 2 I; i j= � ^ :(� ^ :(� ^ �)) y i j= �. Hence,�(�) \ (�(�) \ (�(�) \ �(�))0)0 � �(�).

Of course, also the canonical model of OL could be constructed by taking� as the set of all propositions that are \meanings" of formulas. Neverthe-less, in this case, we would lose the following important information: thecanonical model of OL gives rise to an algebraically complete realization(closed under in�nitary intersection).

7 THE INTRACTABILITY OF ORTHOMODULARITY

As we have seen, the proposition-ortholattice in a Kripkean realizationK = hI; R;� ; �i does not generally coincide with the (algebraically) com-plete ortholattice of all propositions of the orthoframe hI; Ri.4 When �is the set of all propositions, K will be called standard . Thus, a standardorthomodular Kripkean realization is a standard realization, where � is or-thomodular. In the case of OL, every non standard Kripkean realizationcan be naturally extended to a standard one (see the proof of Theorem 17).In particular, � can be always embedded into the complete ortholattice ofall propositions of the orthoframe at issue. Moreover, as we have learntfrom the completeness proof, the canonical model of OL is standard. In thecase of OQL, instead, there are variuos reasons that make signi�cant thedistinction between standard and non standard realizations:

(i) Orthomodularity is not elementary [Goldblatt, 1984]. In other words,there is no way to express the orthomodular property of the ortholat-tice � in an orthoframe hI; Ri as an elementary (�rst-order) property.

(ii) It is not known whether every orthomodular lattice is embeddable intoa complete orthomodular lattice.

(iii) It is an open question whether OQL is characterized by the class ofall standard orthomodular Kripkean realizations.

4For the sake of simplicity, we indicate brie y by � the ortholattice h� ;v ; 0 ;1 ;0i.Similarly, in the case of other structures dealt with in this section.

QUANTUM LOGICS 163

(iv) It is not known whether the canonical model of OQL is standard. Tryand construct a canonical realization for OQL by taking � as the setof all possible propositions (similarly to the OL-case). Let us call sucha realization a pseudo canonical realization. Do we obtain in this wayan OQL-realization, satisfying the orthomodular property? In otherwords, is the pseudo canonical realization a model of OQL?

In order to prove that OQL is characterized by the class of all standardKripkean realizations it would be suÆcient to show that the canonical modelbelongs to such a class. Should orthomodularity be elementary, then, bya general result proved by Fine, this problem would amount to showingthe following statement: there is an elementary condition (or a set thereof)implying the orthomodularity of the standard pseudo canonical realization.Result (i), however, makes this way de�nitively unpracticable.

Notice that a positive solution to problem (iv) would automatically pro-vide a proof of the full equivalence between the algebraic and the Kripkean

consequence relation (T j=AOQL

� i� T j=KOQL

�). If OQL is characterized by a

standard canonical model, then we can apply the same argument used in thecase of OL, the ortholattice � of the canonical model being orthomodular.By similar reasons, also a positive solution to problem (ii) would provide adirect proof of the same result. For, the orthomodular lattice � of the (notnecessarily standard) canonical model of OQL would be embeddable intoa complete orthomodular lattice.

We will now present Goldblatt's result proving that orthomodularity isnot elementarity. Further, we will show how orthomodularity leaves de-feated one of the most powerful embedding technique: the MacNeille com-pletion method.

Orthomodularity is not elementaryLet us consider a �rst-order language L2 with a single predicate denotinga binary relation R. Any frame hI; Ri (where I is a non-empty set and Rany binary relation) will represent a classical realization of L2.

DEFINITION 77 (Elementary class).

(i) Let � be a class of frames. A possible property P of the elements of� is called �rst-order (or elementary) i� there exists a sentence � ofL2 such that for any hI; Ri 2 �:

hI; Ri j= � i� hI; Ri has the property P :

(ii) � is said to be an elementary class i� the property of being in � is anelementary property of �.


Thus, � is an elementary class i� there is a sentence � of L2 such that

� = fhI; Ri j hI; Ri j= �g :DEFINITION 78 (Elementary substructure). Let hI1; R1i ; hI2; R2i be twoframes.

(a) hI1; R1i is a substructure of hI2; R2i i� the following conditions aresatis�ed:

(i) I1 � I2;(ii) R1 = R2 \ (I1 � I1);

(b) hI1; R1i is an elementary substructure of hI2; R2i i� the following con-ditions hold:

(i) hI1; R1i is a substructure of hI2; R2i;(ii) For any formula �(x1; : : : ; xn) of L2 and any i1; : : : ; in of I1:

hI1; R1i j= �[i1; : : : in] i� hI2; R2i j= �[i1; : : : in]:

In other words, the elements of the \smaller" structure satisfy exactly thesame L2-formulas in both structures. The following Theorem [Bell andSlomson, 1969] provides an useful criterion to check whether a substructureis an elementary substructure.

THEOREM 79. Let hI1; R1i be a substructure of hI2; R2i. Then, hI1; R1iis an elementary substructure of hI2; R2i i� whenever �(x1; � � � ; xn; y) is aformula of L2 (in the free variables x1; � � � ; xn; y) and i1; � � � ; in are elementsof I1 such that for some j 2 I2, hI2; R2i j= �[i1; � � � ; in; j], then there is somei 2 I1 such that hI2; R2i j= �[i1; � � � ; in; i]:

Let us now consider a pre-Hilbert space 5 H and letH+ := f 2 H j 6= 0g,where 0 is the null vector. The pairH+;?= �is an orthoframe, where 8 ; � 2 H+: ?= � i� the inner product of and � is di�erent from the null vector 0 (i.e., ( ; �) 6= 0). Let �(H) be theortholattice of all propositions of hH+;?= i, which turns out to be isomorphicto the ortholattice C(H) of all (not necessarily closed) subspaces of H (aproposition is simply a subspace devoided of the null vector). The followingdeep Theorem, due to Amemiya and Halperin [Varadarajan, 1985] permits

5A pre-Hilbert space is a vector space over a division ring whose elements are the realor the complex or the quaternionic numbers such that an inner product (which transformsany pair of vectors into an element of the ring) is de�ned. Di�erently from Hilbert spaces,pre-Hilbert spaces need not be metrically complete.

QUANTUM LOGICS 165

us to characterize the class of all Hilbert spaces in the larger class of allpre-Hilbert spaces, by means of the orthomodular property.

THEOREM 80 (Amemiya{Halperin Theorem). C(H) is orthomodular i� His a Hilbert space.

In other words, C(H) is orthomodular i� H is metrically complete.As is well known [Bell and Slomson, 1969], the property of \being metri-

cally complete" is not elementary. On this basis, it will be highly expectedthat also the orthomodular property is not elementary. The key-lemma inGoldblatt's proof is the following:

LEMMA 81. Let Y be an in�nite-dimensional (not necessarily closed) sub-space of a separable Hilbert space H. If � is any formula of L2 and 1; � � � ; nare vectors of Y such that for some � 2 H, hH+;?= i j= �[ 1; � � � ; n; �], then there exists a vector 2 Y such that hH+;?= i j= �[ 1; � � � ; n; ].

As a consequence one obtains:

THEOREM 82. The orthomodular property is not elementary.

Proof. Let H be any metrically incomplete pre-Hilbert space. Let H beits metric completion. Thus H is an in�nite-dimensional subspace of theHilbert space H. By Lemma 81 and by Theorem 79, hH+;?= i is an elemen-

tary substructure ofDH+

;?=E

. At the same time, by Amemiya{Halperin's

Theorem, C(H) cannot be orthomodular, because H is metrically incom-plete. However, C(H) is orthomodular. As a consequence, orthomodularitycannot be expressed as an elementary property. �

The embeddability problemAs we have seen in Section 2, the class of all propositions of an orthoframeis a complete ortholattice. Conversely, the representation theorem for or-tholattices states that every ortholattice B = hB;v ; 0 ;1 ;0i is embeddableinto the complete ortholattice of all propositions of the orthoframe hB+;?= i,where: B+ := B � f0g and 8a; b 2 B: a ?= b i� a 6v b0. The embedding isgiven by the map

h : a 7! ha ];

where ha ] is the quasi-ideal generated by a. In other words: ha ] = fb 6= 0 jb v ag.

One can prove the following Theorem:

THEOREM 83. Let B = hB;v ; 0 ;1 ;0i be an ortholattice. 8X � B, X isa proposition of hB+;?= i i� X = l(u(X)), where:

u(Y ) :=�b 2 B+ j 8a 2 Y : a v b and l(Y ) :=

�b 2 B+ j 8a 2 Y : b v a :


Accordingly, the complete ortholattice of all propositions of the orthoframehB+;?= i is isomorphic to the MacNeille completion (or completion by cuts)of B [Kalmbach, 1983].6 At the same time, orthomodularity (similarly todistributivity and modularity) is not preserved by the MacNeille completion,as the following example shows [Kalmbach, 1983].

Let C0(2)(IR) be the class of all continuous complex-valued functions f onIR such that Z +1

�1

j f(x) j2 dx <1

Let us de�ne the following bilinear form (: ; :) : C0(2)(IR) � C0(2)(IR) ! jC

(representing an inner product):

(f; g) =

Z +1

�1f�(x)g(x)dx;

where f�(x) is the complex conjugate of f(x). It turns out that C0(2)(IR),

equipped with the inner product (: ; :), gives rise to a metrically incom-plete in�nite-dimensional pre-Hilbert space. Thus, by Amemiya{Halperin'sTheorem (Theorem 80), the algebraically complete ortholattice C(C0(2)(IR))

of all subspaces of C0(2)(IR) cannot be orthomodular. Now consider the

sublattice FI of C(C0(2)(IR)), consisting of all �nite or co�nite dimensionalsubspaces. It is not hard to see that FI is orthomodular. One can provethat C(C0(2)(IR)) is sup-dense in FI; in other words, any X 2 C(C0(2)(IR)) isthe sup of a set of elements of FI. Thus, by a theorem proved by McLaren[Kalmbach, 1983], the MacNeille completion of C(C0(2)(IR)) is isomorphic to

the MacNeille completion of FI. Since C(C0(2)(IR)) is algebraically complete,

the MacNeille completion of C(C0(2)(IR)) is isomorphic to C(C0(2)(IR)) itself.As a consequence, FI is orthomodular, while its MacNeille completion isnot.

8 HILBERT QUANTUM LOGIC AND THE ORTHOMODULAR LAW

As we have seen, the prototypical models of OQL that are interesting fromthe physical point of view are based on the class H of all Hilbert lattices,whose support is the set C(H) of all closed subspaces of a Hilbert spaceH. Let us call Hilbert quantum logic (HQL) the logic that is semanticallycharacterized by H . A question naturally arises: do OQL and HQL repre-sent one and the same logic? As proved by [Greechie, 1981],7 this question

6The MacNeille completion of an ortholattice B = hB;v ; 0 ;1 ;0i is the latticewhose support consists of all X � B such that X = l(u(X)), where: u(Y ) :=fb 2 B j 8a 2 Y : a v bg and l(Y ) := fb 2 B j 8a 2 Y : b v ag. Clearly the only di�er-ence between the proposition-lattice of the frame

B+;?=

�and the Mac Neille completion

of B is due to the fact that propositions do not contain 0.7See also [Kalmbach, 1983].

QUANTUM LOGICS 167

has a negative answer: there is a lattice-theoretical equation (the so-calledorthoarguesian law) that holds in H , but fails in a particular orthomodu-lar lattice. As a consequence, OQL does not represent a faithful logicalabstraction from its quantum theoretical origin.

DEFINITION 84. Let � be a class of orthomodular lattices. We say thatOQL is characterized by � i� for any T and any � the following conditionis satis�ed:

T j=OQL

� i� for any B 2 � and any A = hB; vi : T j=A �:

In order to formulate the orthoarguesian law in an equational way, let us�rst introduce the notion of Sasaki projection.

DEFINITION 85 (The Sasaki projection).Let B be an orthomodular lattice and let a; b be any two elements of B.The Sasaki projection of a onto b, denoted by a e b, is de�ned as follows:

a e b := (a t b0) u b:

It is easy to see that two elements a; b of an orthomodular lattice arecompatible (a = (a u b0) t (a u b)) i� a e b = a u b. Consequently, in anyBoolean lattice, e coincides with u.

DEFINITION 86 (The orthoarguesian law).

a v b t f(a e b0) u [(a e c0) t ((b t c) u ((a e b0) t (a e c0)))]g (OAL)

Greechie has proved that (OAL) holds in H but fails in a particular �niteorthomodular lattice. In order to understand Greechie's counterexample, itwill be expedient to illustrate the notion of Greechie diagram.

Let us �rst recall the de�nition of atom.

DEFINITION 87 (Atom). Let B = hB;v;1;0i any bounded lattice. Anatom is an element a 2 B � f0g such that:

8b 2 B : 0 v b v a y b = 0 or a = b:

Greechie diagrams are hypergraphs that permits us to represent particularorthomodular lattices. The representation is essentially based on the factthat a �nite Boolean algebra is completely determined by its atoms. AGreechie diagram of an orthomodular lattice B consists of points and lines.Points are in one-to-one correspondence with the atoms of B; lines are in one-to-one correspondence with the maximal Boolean subalgebras8 of B. Twolines are crossing in a common atom. For example, the Greechie diagrampictured in Figure 3. represents the orthomodular lattice G12 (Figure 4).


Æa

Æb??

????

??

Æc??

????

??Æd

��

Æe��

Figure 3. The Greechie diagram of G12

�1

�a0oooooooooooooo

�a

OOOOOOOOOOOOOOO �b??

????

??

????

????

�cOOOOOOOOOOOOOO

�0

�b0��

��

��

��

��

????

????

�c0

ooooooooooooooo

��

��

�d??

????

??

�eOOOOOOOOOOOOOO �d0

????

????

��

��

????

????

ooooooooooooooo

�e0OOOOOOOOOOOOOO

ooooooooooooooo

��

��

��

��

�

Figure 4. The orthomodular lattice G12

QUANTUM LOGICS 169

Æa

Æo��

��

��

Æn��

��

��

Æc

Æi

Æh

��

Æg�� Æ

f

????

????

Æ e??

????

??Æ b

Æm

????????

Æl

????????Æk

��

��

��

��

Æ s

Æ r????????

????????

Figure 5. The Greechie diagram of B30

Let us now consider a particular �nite orthomodular lattice, called B30,whose Greechie diagram is pictured in Figure 3.

THEOREM 88. (OAL) fails in B30.Proof. There holds: ae b0 = (at b)u b0 = s0 u b0 = e; ae c0 = (at c)u c0 =n0 u c0 = i and b t c = l0. Thus,

b t f(a e b0) u [(a e c0) t ((b t c) u ((a e b0) t (a e c0)))]g= b t fe u [i t (l0 u (e t i))]g= b t fe u [i t (l0 u g0)]g= b t (e u (i t 0))

= b t (e u i)= b

6w a:�

Hence, there are two formulas � and � (whose valuations in a convenientrealization represent the left- and right- hand side of (OAL), respectively)such that � j=

OQL= �. At the same time, for any C(H) 2 H and for any

realization A = hC(H); vi, there holds: � j=A �.As a consequence, OQL is not characterized by H . Accordingly, HQL

is de�nitely stronger than OQL. We are faced with the problem of �nding

8A maximal Boolean subalgebra of an ortholattice B is a Boolean subalgebra of B,that is not a proper subalgebra of any Boolean subalgebra of B.


out a calculus, if any, that turns out to be sound and complete with respectto H . The main question is whether the class of all formulas valid in H isrecursively enumerable. In order to solve this problem, it would be suÆcient(but not necessary) to show that the canonical model of HQL is isomorphicto the subdirect product of a class of Hilbert lattices. So far, very little isknown about this question.

Lattice characterization of Hilbert latticesAs mentioned in the Introduction, the algebraic structure of the set E ofall events in an event-state system hE ; Si is usually assumed to be a �-complete orthomodular lattice. Hilbert lattices, however, satisfy furtherimportant structural properties. It will be expedient to recall �rst somestandard lattice theoretical de�nitions. Let B = hB;v;1;0i be any boundedlattice.

DEFINITION 89 (Atomicity). A bounded lattice B is atomic i� 8a 2B � f0g there exists an atom b such that b v a.

DEFINITION 90 (Covering property). Let a; b be two elements of a latticeB. We say that b covers a i� a v b ; a 6= b, and 8c 2 B : a v c v b y

a = c or b = c.A lattice B satis�es the covering property i� 8a; b 2 B: a covers a u b y

a t b covers b.

DEFINITION 91 (Irreducibility). Let B be an orthomodular lattice. B issaid to be irreducible i�fa 2 B j 8b 2 B : a is compatible with bg = f0;1g.One can prove the following theorem:

THEOREM 92. Any Hilbert lattice is a complete, irreducible, atomic or-thomodular lattice, which satis�es the covering property.

Are these conditions suÆcient for a lattice B to be isomorphic to (or em-beddable into) a Hilbert lattice? In other words, is it possible to capturelattice-theoretically the structure of Hilbert lattices? An important resultalong these lines is represented by the so-called Piron{McLaren's coordina-tization theorem [Varadarajan, 1985].

THEOREM 93 (Piron{McLaren coordinatization theorem). Any orthomod-ular lattice B (of length9 at least 4) that is complete, irreducible, atomic withthe covering property, is isomorphic to the orthomodular lattice of all (: ; :)-closed subspaces of a Hilbertian space hV ; �; (: ; :);Di.10

9The length of a lattice B is the supremum over the numbers of elements of all thechains of B, minus 1.10A Hilbertian space is a 4-tuple hV; �; (: ; :);Di, where V is a vector space over a

division ring D, � is an involutive antiautomorphism on D, and (: ; :) (to be interpreted

QUANTUM LOGICS 171

Do the properties of the coordinatized lattice B restrict the choice to oneof the real, the complex or the quaternionic numbers ( jQ) and thereforeto a classical Hilbert space? Quite unexpectedly, [Keller, 1980] proved anegative result: there are lattices that satisfy all the conditions of Piron-McLaren's Theorem; at the same time, they are coordinatized by Hilbertianspaces over non-archimedean division rings. Keller's counterexamples havebeen interpreted by some authors as showing the de�nitive impossibility forthe quantum logical approach to capture the Hilbert space mathematics.This impossibility was supposed to demonstrate the failure of the quantumlogic approach in reaching its main goal: the \bottom-top" reconstructionof Hilbert lattices. Interestingly enough, such a negative conclusion hasbeen recently contradicted by an important result proved by Sol�er [1995]:Hilbert lattices can be characterized in a lattice-theoretical way. Sol�er'sresult is essentially based on the following Theorem:

THEOREM 94. Let hV ; �; (: ; :);Di be an in�nite-dimensional Hilbertianspace over a division ring D. Suppose our space includes a k-orthogonalset f igi2IN, i.e., a family of vectors of V such that 8i : ( i; i) = k and8i; j(i 6= j) : ( i; j) = 0. Then hV ; �; (: ; :);Di is a classical Hilbert space.

As a consequence, the existence of k-orthogonal sets characterizes Hilbertspaces in the class of all Hilbertian spaces. The point is that the existenceof such sets admits of a purely lattice-theoretic characterization, by meansof the so-called angle bisecting condition [Morash, 1973]. Accordingly, everylattice which satis�es the angle bisecting condition (in addition to the usualconditions of Piron{McLaren's Theorem) is isomorphic to a classical Hilbertlattice.

9 FIRST-ORDER QUANTUM LOGIC

The most signi�cant logical and metalogical peculiarities of QL arise at thesentential level. At the same time the extension of sentential QL to a �rst-order logic seems to be quite natural. Similarly to the case of sententialQL, we will characterize �rst-order QL both by means of an algebraic anda Kripkean semantics.

Suppose a standard �rst-order language with predicates Pnm and individ-

ual constants am.11 The primitive logical constants are the connectives :;^and the universal quanti�er 8. The concepts of term, formula and sentence

as an inner product) is a de�nite symmetric �-bilinear form on V. Let X be any subset ofV and let X0 := f 2 V j 8� 2 X; ( ; �) = 0g; X is called (: ; :)-closed i� X = X00. Thefollowing condition is required to hold: for any (: ; :)-closed set X of V;V = X + X0 :=f + � : 2 X;� 2 X0g.If D is either IR or jC or jQ and the antiautomorphism � is continuous, then hV; �; (: ; :);Diturns out to be a classical Hilbert space.11For the sake of simplicity, we do not assume functional symbols.


are de�ned in the usual way. We will use x; y; z; x1; � � � ; xn; � � � as metavari-ables ranging over the individual variables, and t; t1; t2; � � � as metavariablesranging over terms. The existential quanti�er 9 is supposed de�ned by ageneralized de Morgan law:

9x� := :8x:�:

DEFINITION 95 (Algebraic realization for �rst-order OL). An algebraicrealization for (�rst-order) OL is a system A =

BC ; D ; v�

where:

(i) BC =BC ;v ; 0 ;1 ;0� is an ortholattice closed under in�nitary in�-

mum ( F) and supremum (F

) for any F � BC such that F 2 C (Cbeing a particular family of subsets of BC).

(ii) D is a non-empty set (disjoint from B) called the domain of A.

(iii) v is the valuation-function satisfying the following conditions:

� for any constant am: v(am) 2 D; for any predicate Pnm, v(Pn

m)is an n-ary attribute in A, i.e., a function that associates to anyn-tuple hd1; � � � ;dni of elements of D an element (truth-value)of Bc;

� for any interpretation � of the variables in the domain D (i.e., forany function from the set of all variables into D) the pair hv; �i(abbreviated by v� and called generalized valuation) associatesto any term an element in D and to any formula a truth-valuein Bc, according to the conditions:

v�(am) = v(am)

v�(x) = �(x)

v�(Pnmt1; � � � ; tn) = v(Pn

m)(v�(t1); � � � ; v�(tn))

v�(:�) = v�(�)0

v�(� ^ ) = v�(�) u v�( )

v�(8x�) = F�v�[x=d](�) j d 2 D ; where�v�[x=d](�) j d 2 D 2 C

(�[x=d] is the interpretation that associates to x the individual dand di�ers from � at most in the value attributed to x).

DEFINITION 96 (Truth and logical truth). A formula � is true in A =BC ; D; v� (abbreviated as j=A �) i� for any interpretation of the variables�, v�(�) = 1; � is a logical truth of OL (j=

OL�) i� for any A, j=A �.

DEFINITION 97 (Consequence in a realization and logical consequence).Let A =

BC ; D; v� be a realization. A formula � is a consequence of T inA (abbreviated T j=A � ) i� for any element a of Bc and any interpretation

QUANTUM LOGICS 173

�: if for any � 2 T , a v v�(�), then a v v�(�); � is a logical consequenceof T (T j=

OL� ) i� for any realization A: T j=A �.

DEFINITION 98 (Kripkean realization for (�rst-order) OL). A Kripkeanrealization for (�rst-order) OL is a system K =

I ; R ;�C ; U ; �

�where:

(i)I ; R ;�C

�satis�es the same conditions as in the sentential case; fur-

ther �C is closed under in�nitary intersection for any F � �c suchthat F 2 C (where C is a particular family of subsets of �C);

(ii) U , called the domain of K, is a non-empty set, disjoint from the setof worlds I . The elements of U are individual concepts u such thatfor any world i: u(i) is an individual (called the reference of u in theworld i). An individual concept u is called rigid i� for any pairs ofworlds i, j: u(i) = u(j). The set Ui = fu(i) j u 2 Ug represents thedomain of individuals in the world i . Whenever Ui = Uj for all i,j wewill say that the realization K has a constant domain.

(iii) � associates a meaning to any individual constant am and to anypredicate Pn

m according to the following conditions:

�(am) is an individual concept in U .

�(Pnm) is a predicate-concept , i.e. a function that associates to

any n-tuple of individual concepts hu1; � � � ;uni a proposition in�C ;

(iv) for any interpretation of the variables � in the domain U , the pairh� ; �i (abbreviated as �� and called valuation) associates to any termt an individual concept in U and to any formula a proposition in �C

according to the conditions:

��(x) = �(x)

��(am) = �(am)

��(Pnmt1; � � � ; tn) = �(Pn

m)(��(t1); � � � ; ��(tn))

��(:�) = ��(�)0

��(� ^ ) = ��(�) \ ��( )

��(8x�) =T�

��[x=u](�) j u 2 U ; where��[x=u](�) j u 2 U 2 C.

For any world i and any interpretation � of the variables, the triplet h� ; i ; �i(abbreviated as ��i ) will be called a world-valuation.

DEFINITION 99 (Satisfaction). ��i j= � (��i satis�es �) i� i 2 ��(�).

DEFINITION 100 (Veri�cation). i j= � (i veri�es �) i� for any �: ��i j= �.


DEFINITION 101 (Truth and logical truth). j=K � (� is true in K) i� forany i: i j= �;

j=OL� (� is a logical truth of OL) i� for any K: j=K �.

DEFINITION 102 (Consequence in a realization and logical consequence).T j=K � i� for any i of K and any �: ��i j= T y ��i j= �;

T j=OL� i� for any realization K: T j=K �.

The algebraic and the Kripkean characterization for �rst-order OQL canbe obtained, in the obvious way, by requiring that any realization be ortho-modular.

In both semantics for �rst-order QL one can prove a coincidence lemma:

LEMMA 103. Given A =BC ; D ; v

�and K =

I ; R ;�C ; U; �

�:

(103.1) If � and �� coincide in the values attributed to the variables oc-curring in a term t, then v�(t) = v�

�

(t); ��(t) = ��

(t).

(103.2) If � and �� coincide in the values attributed to the free variablesoccurring in a formula �, then v�(�) = v�

�

(�); ��(�) = ��

(�).

One can easily prove, like in the sentential case, the following lemma:

LEMMA 104.

(104.1) For any algebraic realization A there exists a Kripkean realizationKA such that for any �: j=A � i� j=KA �. Further, if A isorthomodular then KA is orthomodular.

(104.2) For any Kripkean realization K, there exists an algebraic realiza-tion AK such that for any for any �: j=K � i� j=AK �. Further,if K is orthomodular then AK is orthomodular.

An axiomatization of �rst-order OL (OQL) can be obtained by addingto the rules of our OL (OQL)-sentential calculus the following new rules:

(PR1) T [ f8x�g j��(x=t), where �(x=t) indicates a legitimate sub-stitution).

(PR2)T j��T j� 8x� (provided x is not free in T ).

All the basic syntactical notions are de�ned in the expected way. Onecan prove that any consistent set of sentences T admits of a consistentinductive extension T �, such that T � j�8x�(t) whenever for any closed termt, T � j��(t). The \weak Lindenbaum theorem" can be strengthened asfollows: for any sentence �, if T j�=:� then there exists a consistent and

QUANTUM LOGICS 175

inductive T � such that:

T is syntactically compatible with T � and T � j��:12

One can prove a soundness and a completeness theorem of our calculuswith respect to the Kripkean semantics.

THEOREM 105. Soundness.T j�� y T j= �:

Proof. Straightforward. �

THEOREM 106. Completeness.T j= � y T j��:

Sketch of the proof Like in the sentential case, it is suÆcient to constructa canonical model K =

I ; R ;�C ; U ; �

�such that T j�� i� T j=K �.

De�nition of the canonical model

(i) I is the set of all consistent, deductively closed and inductive sets ofsentences expressed in a common language LK , which is an extensionof the original language;

(ii) R is determined like in the sentential case;

(iii) U is a set of rigid individual concepts that is naturally determined bythe set of all individual constants of the extended language LK. Forany constant c of LK, let uc be the corresponding individual conceptin U . We require: for any world i, uc(i) = c. In other words, thereference of the individual concept uc is in any world the constant c.We will indicate by cu the constant corresponding to u.

(iv) �(am) = uam ;

�(Pnm)(uc11 ; : : : ;u

cnn ) = fi j Pn

mc1; : : : ; cn 2 ig :Our � is well de�ned since one can prove for any sentence � of LK:

i j�=� y 9j ?= i : j j�:�:

As a consequence, ��(Pnmt1; : : : ; tn) is a possible proposition.

(v) �C is the set of all \meanings" of formulas (i.e., X 2 �C i� 9�9�(X =��(�)); C is the set of all sets

��[x=u](�) j u 2 U for any formula

�.

12By De�nition 71, T is syntactically compatible with T � i� there is no formula � suchthat T j�� and T � j�:�.


One can easily check that K is a \good" realization with a constantdomain.

LEMMA 107. Lemma of the canonical model.For any �, any i 2 I and any �:

��i j= � i� �� 2 i;

where �� is the sentence obtained by substituting in � any free variable xwith the constant c�(x) corresponding to the individual concept �(x).

Sketch of the proof. By induction on the length of �. The cases � =Pnmt1; � � � ; tn, � = :�, � = � ^ are proved by an obvious transformation

of the sentential argument. Let us consider the case � = 8x� and supposex occurring in � (otherwise the proof is trivial). In order to prove the leftto right implication, suppose ��i j= 8x�. Then, for any u in U , ��[x=u] j=�(x). Hence, by inductive hypothesis, 8u 2 U , [�(x)]�[x=u] 2 i. In otherwords, for any constant cu of i: [�(x)]�(x=cu) 2 i. And, since i is inductiveand deductively closed: 8x�(x)� 2 i. In order to prove the right to leftimplication, suppose [8x�(x)]� 2 i. Then, [by (PR1)], for any constant c of

i: [�(x=c)]� 2 i. Hence by inductive hypothesis: for any uc 2 U , ��[x=uc]i j=

�(x), i.e., ��i j= 8x�(x). On this ground, similarly to the sentential case,one can prove T j�� i� T j=K �. �

First-order QL can be easily extended (in a standard way) to a �rst-orderlogic with identity. However, a critical problem is represented by the possi-bility of developing, within this logic, a satisfactory theory of descriptions .The main diÆculty can be sketched as follows. A natural condition to berequired in any characterization of a �-operator is obviously the following:

9x f�(x) ^ 8y [(�(y) ^ x = y) _ (:�(y) ^ :x = y)] ^ �(x)gis true y �(�x�(x)) is true:

However, in QL, the truth of the antecedent of our implication does notgenerally guarantee the existence of a particular individual such that �x� canbe regarded as a name for such an individual. As a counterexample, let usconsider the following case (in the algebraic semantics): let A be hB ; D ; viwhere B is the complete orthomodular lattice based on the set of all closedsubspaces of the plane IR2, and D contains exactly two individuals d1;d2.Let P be a monadic predicate and X;Y two orthogonal unidimensionalsubspaces of B such that v(P )(d1) = X , v(P )(d2) = Y . If the equalitypredicate = is interpreted as the standard identity relation (i.e., v�(t1 =t2) = 1, if v�(t1) = v�(t2); 0, otherwise), one can easily calculate:

v (9x [Px ^ 8y((Py ^ x = y) _ (:Py ^ :x = y))]) = 1:

QUANTUM LOGICS 177

However, for both individuals d1;d2 of the domain, we have:

v�[x=d1](Px) 6= 1; v�[x=d2](Px) 6= 1:

In other words, there is no precise individual in the domain that satis�esthe property expressed by our predicate P !

10 QUANTUM SET THEORIES AND THEORIES OF QUASISETS

An important application of QL to set theory has been developed by [Takeuti,1981]. We will sketch here only the fundamental idea of this application. LetL be a standard set-theoretical language. One can construct ortho-valuedmodels for L, which are formally very similar to the usual Boolean-valuedmodels for standard set-theory, with the following di�erence: the set oftruth-values is supposed to have the algebraic structure of a complete or-thomodular lattice, instead of a complete Boolean algebra. Let B be acomplete orthomodular lattice, and let �, �,... represent ordinal numbers.An ortho-valued (set-theoretical) universe V is constructed as follows:

V B =S�2On V (�) , where:

V (0) = ;.V (�+1) = fg j g is a function andDom(g) � V (�) andRang(g) � Bg.V (�) =

S�<� V (�), for any limit-ordinal �.

( Dom(g) and Rang(g) are the domain and the range of function g,respectively).

Given an orthovalued universe V B one can de�ne for any formula of L thetruth-value [[�]]� in B induced by any interpretation � of the variables inthe universe V B.

[[x 2 y]]� =Fg2Dom(�(y))

��(y)(g) u [[x = z]]�[z=g]

[[x = y]]� = Fg2Dom(�(x))

��(x)(g) [[z 2 y]]�[z=g]

uFg2Dom(�(y))

��(y)(g) [[z 2 x]]�[z=g]

.

where is the quantum logical conditional operation (a b := a0t (au b),for any a; b 2 B).

A formula � is called true in the universe V B (j=V B �) i� [[�]]� = 1, forany �.

Interestingly enough, the segment V (!) of V B turns out to contain someimportant mathematical objects, that we can call quantum-logical naturalnumbers .


The standard axioms of set-theory hold in B only in a restricted form.An extremely interesting property of V B is connected with the notion ofidentity. Di�erently from the case of Boolean-valued models, the identityrelation in V B turns out to be non-Leibnizian. For, one can choose anorthomodular lattice B such that:

6j=V B x = y ! 8z(x 2 z $ y 2 z):

According to our semantic de�nitions, the relation = represents a kindof \extensional equality". As a consequence, one may conclude that twoquantum-sets that are extensionally equal do not necessarily share all thesame properties. Such a failure of the Leibniz-substitutivity principle inquantum set theory might perhaps �nd interesting applications in the �eldof intensional logics.

A completely di�erent approach is followed in the framework of the the-ories of quasisets (or quasets). The basic aim of these theories is to providea mathematical description for collections of microobjects, which seem toviolate some characteristic properties of the classical identity relation.

In some of his general writings, Schr�odinger discussed the inconsistencybetween the classical concept of physical object (conceived as an individualentity) and the behaviour of particles in quantum mechanics. Quantumparticles { he noticed { lack individuality and the concept of identity cannotbe applied to them, similarly to the case of classical objects.

One of the aims of the theories of quasisets (proposed by [da Costa et al.,1992]) is to describe formally the following idea defended by Schr�odinger:identity is generally not de�ned for microobjects. As a consequence, onecannot even assert that an \electron is identical with itself". In the realmof microobjects only an indistinguishability relation (an equivalence relationthat may violate the substitutivity principle) makes sense.

On this basis, di�erent formal systems have been proposed. Generally,these systems represent convenient generalizations of a Zermelo{Fraenkellike set theory with urelements . Di�erently from the classical case, an ure-lement may be either a macro or a microobject . Collections are representedby quasisets and classical sets turn out to be limit cases of quasisets.

A somewhat di�erent approach has been followed in the theory of quasets(proposed in [Dalla Chiara and Toraldo di Francia, 1993]).

The starting point is based on the following observation: physical kindsand compound systems in QM seem to share some features that are charac-teristic of intensional entities. Further, the relation between intensions andextensions turns out to behave quite di�erently from the classical semanticsituations. Generally, one cannot say that a quantum intensional notionuniquely determines a corresponding extension. For instance, take the no-tion of electron, whose intension is well de�ned by the following physicalproperty: mass = 9:1 � 10�28g, electron charge = 4:8 � 10�10e.s.u., spin

QUANTUM LOGICS 179

= 1=2. Does this property determine a corresponding set , whose elementsshould be all and only the physical objects that satisfy our property at acertain time interval? The answer is negative. In fact, physicists have thepossibility of recognizing, by theoretical or experimental means, whether agiven physical system is an electron system or not. If yes, they can alsoenumerate all the quantum states available within it. But they can do soin a number of di�erent ways. For example, take the spin. One can choosethe x-axis and state how many electrons have spin up and how many havespin down. However, we could instead refer to the z-axis or any other direc-tion, obtaining di�erent collections of quantum states, all having the samecardinality. This seems to suggest that microobject systems present an ir-reducibly intensional behaviour: generally they do not determine preciseextensions and are not determined thereby. Accordingly, a basic feature ofthe theory is a strong violation of the extensionality principle.

Quasets are convenient generalizations of classical sets, where both theextensionality axiom and Leibniz' principle of indiscernibles are violated.Generally a quaset has only a cardinal but not an ordinal number, since itcannot be well ordered.

11 THE UNSHARP APPROACHES

The unsharp approaches to QT (�rst proposed by [Ludwig, 1983] andfurther developed by Kraus, Davies, Mittelstaedt, Busch, Lahti, Bugajski,Beltrametti, Cattaneo and many others) have been suggested by some deepcriticism of the standard logico-algebraic approach. Orthodox quantumlogic (based on Birkho� and von Neumann's proposal) turns out to be atthe same time a total and a sharp logic. It is total because the meaning-ful propositions are represented as closed under the basic logical operations:the conjunction (disjunction) of two meaningful propositions is a meaningfulproposition. Further, it is also sharp, because propositions, in the standardinterpretation, correspond to exact possible properties of the physical sys-tem under investigation. These properties express the fact that \the valueof a given observable lies in a certain exact Borel set".

As we have seen, the set of the physical properties, that may hold fora quantum system, is mathematically represented by the set of all closedsubspaces of the Hilbert space associated to our system. Instead of closedsubspaces, one can equivalently refer to the set of all projections , that is inone-to-one correspondence with the set of all closed subspaces. Such a cor-respondence leads to a collapse of di�erent semantic notions, which Foulisand Randall described as the \metaphysical disaster" of orthodox QT. Thecollapse involves the notions of \experimental proposition", \physical prop-erty", \physical event" (which represent empirical and intensional con-cepts), and the notion of proposition as a set of states (which corresponds


to a typical extensional notion according to the tradition of standard se-mantics).

Both the total and the sharp character of QL have been put in questionin di�erent contexts. One of the basic ideas of the unsharp approaches is a\liberalization" of the mathematical counterpart for the intuitive notion of\experimental proposition". Let P be a projection operator in the Hilbertspace H, associated to the physical system under investigation. Suppose Pdescribes an experimental proposition and let W be a statistical operatorrepresenting a possible state of our system. Then, according to one of theaxioms of the theory (the Born rule), the number Tr(WP ) (the trace ofthe operator WP ) will represent the probability-value that our system instate W veri�es P . This value is also called Born probability . However,projections are not the only operators for which a Born probability can bede�ned. Let us consider the class E(H) of all linear bounded operators Esuch that for any statistical operator W ,

Tr(WE) 2 [0; 1]:

It turns out that E(H) properly includes the set P (H) of all projections onH. The elements of E(H) represent, in a sense, a \maximal" mathemati-cal representative for the notion of experimental proposition, in agreementwith the probabilistic rules of quantum theory. In the framework of theunsharp approach, E(H) has been called the set of all e�ects .13 An im-portant di�erence between projections and proper e�ects is the following:projections can be associated to sharp propositions having the form \thevalue for the observable A lies in the exact Borel set �", while e�ects mayrepresent also fuzzy propositions like \the value of the observable A lies inthe fuzzy Borel set �". As a consequence, there are e�ects E, di�erent fromthe null projection jO, such that no state W can verify E with probability1. A limit case is represented by the semitransparent e�ect 1

21I (where 1I isthe identity operator), to which any state W assigns probability-value 1

2 .From the intuitive point of view, one could say that moving to an unsharp

approach represents an important step towards a kind of \second degree offuzziness". In the framework of the sharp approach, any physical event Ecan be regarded as a kind of \clear" property. Whenever a state W assignsto E a probability value di�erent from 1 and 0, one can think that thesemantic uncertainty involved in such a situation totally depends on theambiguity of the state (�rst degree of fuzziness). In other words, even apure state in QT does not represent a logically complete information, thatis able to decide any possible physical event. In the unsharp approaches,instead, one take into account also \genuine ambiguous properties". Thissecond degree of fuzziness may be regarded as depending on the accuracy

13It is easy to see that an e�ect E is a projection i� E2 := EE = E. In other words,projections are idempotent e�ects.

QUANTUM LOGICS 181

of the measurement (which tests the property), and also on the accuracyinvolved in the operational de�nition for the physical quantities which ourproperty refers to.

12 EFFECT STRUCTURES

Di�erent algebraic structures can be induced on the class E(H) of all e�ects.Let us �rst recall some de�nitions.

DEFINITION 108 (Involutive bounded poset (lattice)). An involutivebounded poset (lattice) is a structure B = hB ;v ; 0 ;1 ;0i, where hB ;v ;1 ;0iis a partially ordered set (lattice) with maximum (1) and minimum (0); 0

is a 1-ary operation on B such that the following conditions are satis�ed:(i) a00 = a; (ii) a v b y b0 v a0.DEFINITION 109 (Orthoposet). An orthoposet is an involutive boundedposet that satis�es the non contradiction principle:

a u a0 = 0:

DEFINITION 110 (Orthomodular poset). An orthomodular poset is an or-thoposet that is closed under the orthogonal sup (a v b0 y a t b exists)and satis�es the orthomodular property:

a v b y 9c such that a v c0 and b = a t c.

DEFINITION 111 (Regularity). An involutive bounded poset (lattice) B isregular i� a v a0 and b v b0 y a v b0.

Whenever an involutive bounded poset B is a lattice, then B is regulari� it satis�es the Kleene condition:

a u a0 v b t b0:

The set E(H) of all e�ects can be naturally structured as an involutivebounded poset:

E(H) = hE(H) ;v ; 0 ;1 ;0i ;where

(i) E v F i� for any state (statistical operator) W , Tr(WE) � Tr(WE)(in other words, any state assigns to E a probability-value that is lessor equal than the probability-value assigned to F );

(ii) 1, 0 are the identity (1I) and the null ( jO) projection, respectively;

(iii) E0 = 1�E.


One can easily check that v is a partial order, 0 is an order-reversing invo-lution, while 1 and 0 are respectively the maximum and the minimum withrespect to v. At the same time this poset fails to be a lattice. Di�erentlyfrom projections, some pairs of e�ects have no in�mum and no supremumas the following example shows [Greechie and Gudder, n.d.]:

EXAMPLE 112. Let us consider the following e�ects (in the matrix-repre-sentation) on the Hilbert space IR2:

E =

�12 00 1

2

�F =

�34 00 1

4

�G =

�12 00 1

4

�

It is not hard to see that G v E;F . Suppose, by contradiction, that L =E u F exists in E(IR2). An easy computation shows that L must be equalto G. Let

M =

�716

18

18

316

�

Then M is an e�ect such that M v E;F ; however, M 6v L, which is acontradiction.

In order to obtain a lattice structure, one has to embed E(H) into itsMacNeille completion E(H).

The MacNeille completion of an involutive bounded poset

Let hB ;v ;1 ;0i be an involutive bounded poset. For any non-emptysubset X of B, let l(X) and u(X) represent respectively the set of alllower bounds and the set of all upper bounds of X . Let MC(B) :=fX � B j X = u(l(X))g. It turns out that X 2MC(B) i� X = X 00,where X 0 := fa 2 B j 8b 2 X : a v b0g. Moreover, the structure

B = hMC(B) ;� ; 0 ; f0g ; Bi

is a complete involutive bounded lattice (which is regular if B is reg-ular), where X u Y = X \ Y and X t Y = (X [ Y )00.

It turns out that B is embeddable into B, via the map h : a ! ha],where ha] is the principal ideal generated by a. Such an embeddingpreserves the in�mum and the supremum, when existing in B.

The Mac Neille completion of an involutive bounded poset does not gener-ally satis�es the non contradiction principle (a u a0 = 0 ) and the excludedmiddle principle (a t a0 = 1 ). As a consequence, di�erently from theprojection case, the Mac Neille completion of E(H) is not an ortholattice.Apparently, our operation 0 turns out to behave as a fuzzy negation, bothin the case of E(H) and of its Mac Neille completion. This is one of thereasons why proper e�ects (that are not projections) may be regarded as

QUANTUM LOGICS 183

representing unsharp physical properties , possibly violating the non contra-diction principle.

The e�ect poset E(H) can be naturally extended to a richer structure,equipped with a new complement �, that has an intuitionistic-like be-haviour:

E� is the projection operator PKer(E) whose range is the kernelKer(E)of E, consisting of all vectors that are transformed by the operator Einto the null vector.

By de�nition, the intuitionistic complement of an e�ect is always a pro-jection. In the particular case, where E is a projection, it turns out that:E0 = E�. In other words, the fuzzy and the intuitionistic complementcollapse into one and the same operation.

The structure hE(H) ;v ; 0 ; � ;1 ;0i turns out to be a particular exampleof a Brouwer Zadeh poset [Cattaneo and Nistic�o, 1986].

DEFINITION 113. A Brouwer{Zadeh poset (simply a BZ-poset) is a struc-ture hB ;v ; 0 ; � ;1 ;0i, where

(113.1) hB ;v ; 0 ;1 ;0i is a regular involutive bounded poset;

(113.2) � is a 1-ary operation on B, which behaves like an intuitionisticcomplement:

(i) a u a� = 0.

(ii) a v a��.

(iii) a v b y b� v a�.

(113.3) The following relation connects the fuzzy and the intuitionisticcomplement:

a�0 = a��.

DEFINITION 114. A Brouwer Zadeh lattice is a BZ-poset that is also alattice.

The Mac Neille completion of a BZ-poset

Let B = hB ;v ; 0 ; � ;1 ;0i be a BZ-poset and let B the Mac Neillecompletion of the regular involutive bounded poset hB ;v ; 0 ; 1 ;0i.For any non-empty subset X of B, let

X� := fa 2 B j 8b 2 X : a v b�g :

It turns out that B = hMC(B);� ; 0 ; � ; f0g ; Bi is a complete BZ-lattice [Giuntini, 1991], which B can be embedded into, via the maph de�ned above.


Another interesting way of structuring the set of all e�ects can be ob-tained by using a particular kind of partial structure, that has been called ef-fect algebra [Foulis and Bennett, 1994] or unsharp orthoalgebra [Dalla Chiaraand Giuntini, 1994]. Abstract e�ect algebras are de�ned as follows:

DEFINITION 115. An e�ect algebra is a partial structure A = hA ;� ;1 ;0iwhere � is a partial binary operation on A. When � is de�ned for a paira ; b 2 A, we will write 9 (a� b). The following conditions hold:

(i) Weak commutativity

9(a� b) y 9(b� a) and a� b = b� a.

(ii) Weak associativity

[9(b� c) and 9(a� (b� c))] y [9(a � b) and 9((a � b) � c)and a� (b� c) = (a� b)� c].

(iii) Strong excluded middle

For any a, there exists a unique x such that a� x = 1.

(iv) Weak consistency

9(a� 1) y a = 0.

From an intuitive point of view, our operation � can be regarded as anexclusive disjunction (aut), which is de�ned only for pairs of logically in-compatible events.

An orthogonality relation ?, a partial order relation v and a generalizedcomplement 0 can be de�ned in any e�ect algebra.

DEFINITION 116. Let A = hA ;� ;1 ;0i be an e�ect algebra and let a; b 2A.

(i) a ? b i� a� b is de�ned in A.

(ii) a v b i� 9c 2 A such that a ? c and b = a� c.

(iii) The generalized complement of a is the unique element a0 suchthat a� a0 = 1 (the de�nition is justi�ed by the strong excludedmiddle condition).

The category of all e�ect algebras turns out to be (categorically) equiva-lent to the category of all di�erence posets , which have been �rst studied in[Kopka and Chovanec, 1994] and further investigated in [Dvure�censkij andPulmannov�a, 1994].

E�ect algebras that satisfy the non contradiction principle are calledorthoalgebras :

DEFINITION 117. An orthoalgebra is an e�ect algebra B = hB ;� ;1 ;0isuch that the following condition is satis�ed:

QUANTUM LOGICS 185

Strong consistency

9 (a� a) y a = 0.

In other words: 0 is the only element that is orthogonal to itself.

In order to induce the structure of an e�ect algebra on E(H), it is suÆ-cient to de�ne a partial sum � as follows:

9 (E � F ) i� E + F 2 E(H);

where + is the usual sum-operator. Further:

9 (E � F ) y E � F = E + F:

It turns out that the structure hE(H) ;� ; 1I ; jOi is an e�ect algebra, wherethe generalized complement of any e�ect E is just 1I�E. At the same time,this structure fails to be an orthoalgebra.

Any abstract e�ect algebra

A = hA ;� ;1 ;0ican be naturally extended to a kind of total structure, that has been termedquantum MV-algebra(abbreviated as QMV-algebra) [Giuntini, 1996].

Before introducing QMV-algebras, it will be expedient to recall the de�ni-tion of MV-algebra. As is well known, MV-algebras (multi-valued algebras)have been introduced by Chang [1957] in order to provide an algebraic proofof the completeness theorem for Lukasiewicz' in�nite-many-valued logic [email protected] \privileged" model of this logic is based on the real interval [0; 1], whichgives rise to a particular example of a totally ordered (or linear) MV-algebra.

Both MV-algebras and QMV-algebras are total structures having thefollowing form:

M = (M ;� ; � ;1;0)

where:

(i) 1 ;0 represent the certain and the impossible propositions (or alter-natively the two extreme truth values);

(ii) � is the negation-operation;

(iii) � represents a disjunction (or) which is generally non idempotent(a� a 6= a).

A (generally non idempotent) conjunction (and) is then de�ned via deMorgan law:

a� b := (a� � b�)� :


On this basis, a pair consisting of an idempotent conjunction et (e) andof an idempotent disjunction vel (d) is then de�ned:

a e b := (a� b�)� b

a d b := (a� b�)� b:In the concrete MV-algebra based on [0; 1], the operations are de�ned as

follows:

(i) 1 = 1; 0 = 0;

(ii) a� = 1� a;

(iii) � is the truncated sum:

a� b =

(a+ b; if a+ b � 1;

1; otherwise:

In this particular case, it turns out that:

a e b = Minfa; bg(a et b is the minimum between a and b):

a d b = Maxfa; bg(a vel b is the maximum between a and b):

A standard abstract de�nition of MV-algebras is the following [Mangani,1973]:

DEFINITION 118. An MV-algebra is a structure M = (M ;� ; � ;1;0),where � is a binary operation, � is a unary operation and 0 and 1 arespecial elements of M , satisfying the following axioms:

(MV1) (a� b)� c = a� (b� c)(MV2) a� 0 = a

(MV3) a� b = b� a(MV4) a� 1 = 1

(MV5) (a�)� = a

(MV6) 0� = 1

(MV7) a� a� = 1

(MV8) (a� � b)� � b = (a� b�)� � a

QUANTUM LOGICS 187

In other words, an MV-algebra represents a particular weakening of aBoolean algebra, where � and � are generally non idempotent.

A partial order relation can be de�ned in any MV-algebra in the followingway:

a � b i� a e b = a:

Some important properties of MV-algebras are the following:

(i) the structure hM ;� ;� ;1 ;0i is a bounded involutive distributive lat-tice, where a e b (a d b) is the inf (sup) of a; b;

(ii) the non-contradiction principle and the excluded middle principlesfor �;e;d are generally violated: a d a� 6= 1 and a e a� 6= 0 arepossible. As a consequence, MV algebras permit us to describe fuzzyand paraconsistent situations;

(iii) a� � b = 1 i� a � b. In other words: similarly to the Boolean case,\not-a or b" represents a good material implication;

(iv) every MV-algebra is a subdirect product of totally ordered MV-algebras[Chang, 1958];

(v) an equation holds in the class of all MV-algebras i� it holds in theconcrete MV-algebra based on [0; 1] [Chang, 1958].

Let us now go back to our e�ect-structure hE(H) ;� ;1 ;0i. The partialoperation � can be extended to a total operation � that behaves like atruncated sum. For any E;F 2 E(H):

E � F =

(E + F; if 9(E � F );

1; otherwise:

Further, let us put:

E� = 1I�E:The structure E(H) = hE(H) ;� ;� ;1 ;0i turns out to be \very close" to anMV-algebra. However, something is missing: E(H) satis�es the �rst sevenaxioms of our de�nition (MV1-MV7); at the same time one can easily checkthat the axiom (MV8) (usually called \ Lukasiewicz axiom") is violated. Forinstance, let us consider two non trivial projections P;Q such that P is notorthogonal to Q� and Q is not orthogonal to P �. Then, by de�nition of �,we have that P �Q� = 1I and Q� P � = 1I. Hence: (P � �Q)

� �Q = Q 6=P = (P �Q�)� � P .

As a consequence, Lukasiewicz axiom must be conveniently weakened toobtain a representation for our concrete e�ect structure. This can be doneby means of the notion of QMV-algebra.


DEFINITION 119. A quantum MV-algebra (QMV-algebra) is a structureM = (M ;� ; � ;1;0) where � is a binary operation, � is a 1-ary operation,and 0;1 are special elements of M . For any a; b 2M : a�b := (a��b�)� ; aeb := (a� b�)� a ; a d b := (a� b�)� b. The following axioms are required:

(QMV1) a� (b� c) = (b� a)� c,(QMV2) a� a� = 1,

(QMV3) a� 0 = a,

(QMV4) a� 1 = 1,

(QMV5) a�� = a,

(QMV6) 0� = 1,

(QMV7) a� [(a� e b) e (c e a�)] = (a� b) e (a� c).

The operations e and d of a QMV-algebra M are generally non com-mutative. As a consequence, they do not represent lattice-operations. It isnot diÆcult to prove that a QMV-algebra M is an MV-algebra i� for alla; b 2M : a e b = b e a.

At the same time, any QMV-algebraM = (M ;� ; � ;1;0) gives rise to aninvolutive bounded poset hM ;� ; � ;1 ;0i, where the partial order relationis de�ned like in the MV case.

One can easily show that QMV-algebras represent a \good abstraction"from the e�ect-structures:

THEOREM 120. The structure E(H) = hE(H) ;� ;� ;1 ;0i (where � ;� ;1 ;0are the operations and the special elements previously de�ned) is a QMV-algebra.

The QMV-algebra E(H) cannot be linear. For, one can easily check thatany linear QMV-algebra collapses into an MV-algebra.

In spite of this, our algebra of e�ects turns out to satisfy some weak formsof linearity.

DEFINITION 121. A QMV-algebraM is called weakly linear i� 8a; b 2M :a e b = b or b e a = a.

DEFINITION 122. A QMV-algebraM is called quasi-linear i� 8a; b 2M :a e b = a or a e b = b.

It is easy to see that every quasi-linear QMV-algebra is weakly linear,but not the other way around (because e is not commutative).

A very strong relation connects the class of all e�ect algebras with theclass of all quasi-linear QMV-algebras: every e�ect algebra can be uniquelytransformed into a quasi-linear QMV-algebra and viceversa.

Let B = hB ;� ;1 ;0i be an e�ect algebra. The operation � can beextended to a total operation

� : B �B ! B

QUANTUM LOGICS 189

in the following way:

a� b :=

(a� b; if 9(a� b);

1; otherwise:

The resulting structureB ;� ; 0 ; 1 ;0

�will be denoted by Bqmv.

Viceversa, let M = (M ;� ; � ;1;0) be a QMV-algebra. Then, one cande�ne a partial operation � on M such that

Dom(�) := fha; bi 2M �M j a � b�g :9(a� b) y a� b = a� b:

The resulting structure hM ;� ;1 ;0i will be denoted by Mea.

THEOREM 123. [Gudder, 1995; Giuntini, 1995] Let B = hB ;� ;1 ;0i bean e�ect algebra and let M = (M ;� ; � ;1;0) be a QMV-algebra.

(i) Bqmv is a quasi-linear QMV-algebra;

(ii) Mea is an e�ect algebra;

(iii) (Bqmv)ea = B;

(iv) M is quasi-linear i� (Mea)qmv =M;

(v) Bqmv is the unique quasi-linear QMV-algebra such that � extends� and a � b in Bqmv implies a v b in B.

As a consequence, the e�ect algebra E(H) of all e�ects on a Hilbert spaceH determines a quasi-linear QMV-algebra E(H)qmv = hE(H) ;� ;� ;1 ;0i,where

E � F =

(E + F; if 9(E � F );

1; otherwise;

andE� = 1�E = E0:

These di�erent ways of inducing a structure on the set of all unsharpphysical properties have suggested di�erent logical abstractions. In thefollowing sections, we will investigate some interesting examples of unsharpquantum logics.

13 PARACONSISTENT QUANTUM LOGIC

Paraconsistent quantum logic (PQL) represents the most obvious unsharpweakening of orthologic. In the algebraic semantics, this logic is character-ized by the class of all realizations based on an involutive bounded lattice,where the non contradiction principle (a u a0 = 0) is possibly violated.


In the Kripkean semantics, instead, PQL is characterized by the class ofall realizations hI ; R ;� ; �i, where the accessibility relation R is symmetric(but not necessarily re exive), while � behaves like in the OL - case. Anypair hI; Ri, where R is a symmetric relation on I , will be called symmetricframe. Di�erently from OL and OQL, a world i of a PQL realization mayverify a contradiction. Since R is generally not re exive, it may happenthat i 2 � (�) and i ? � (�). Hence: i j= � ^ :�.

All the other semantic de�nitions are given like in the case of OL, mutatismutandis . On this basis, one can show that our algebraic and Kripkeansemantics characterize the same logic.

An axiomatization of PQL can be obtained by dropping the absurdityrule and the Duns Scotus rule in the OL calculus. Similarly to OL, ourlogic PQL satis�es the �nite model property and is consequently decidable.

Hilbert-space realizations for PQL can be constructed, in a natural way,both in the algebraic and in the Kripkean semantics. In the algebraic se-mantics, take the realizations based on the Mac Neille completion of aninvolutive bounded poset having the form

hE(H) ;v ; 0 ;1 ;0i ;

where H is any Hilbert space. In the Kripkean semantics, consider therealizations based on the following frames

hE(H)� f0g ; 6?i ;

where ?= represents the non orthogonality relation between e�ects (E 6? Fi� E 6v F 0). Di�erently from the projection case, here the accessibilityrelation is symmetric but generally non-re exive. For instance, the semi-transparent e�ect 1

21I (representing the prototypical ambiguous property)is a �xed point of the generalized complement 0; hence 1

21I ? 121I and

( 121I)0 ? ( 121I)0. From the physical point of view, possible worlds are hereidenti�ed with possible pieces of information about the physical system un-der investigation. Any information may be either maximal (a pure state)or non maximal (a mixed state); either sharp (a projection) or unsharp(a proper e�ect). Violations of the non contradiction principle are deter-mined by unsharp (ambiguous) pieces of knowledge. Interestingly enough,proper mixed states (which cannot be represented as projections) turn outto coincide with particular e�ects. In other words, within the unsharp ap-proach, it is possible to represent both states and events by a unique kindof mathematical object, an e�ect.

PQL represents a somewhat rough logical abstraction from the classof all e�ect-realizations. An important condition that holds in all e�ectrealizations is represented by the regularity property (which may fail in ageneric PQL-realization).

QUANTUM LOGICS 191

DEFINITION 124. An algebraic PQL realization hB ; v i is called regulari� the involutive bounded lattice B is regular (a u a0 v b t b0).

The regularity property can be naturally formulated also in the frame-work of the Kripkean semantics:

DEFINITION 125. A PQL Kripkean realization hI; R ;� ; �i is regular i�its frame hI ; R i is regular . In other words, 8i; j 2 I : i ? i and j ? j y

i ? j.

One can prove that a symmetric frame hI; Ri is regular i� the involutivebounded lattice of all propositions of hI; Ri is regular. As a consequence,an algebraic realization is regular i� its Kripkean transformation is regularand viceversa (where the Kripkean [algebraic] transformation of an algebraic[Kripkean] realization is de�ned like in OL).

On this basis one can introduce a proper extension of PQL: regular para-consistent quantum logic (RPQL). Semantically RPQL is characterized bythe class of all regular realizations (both in the algebraic and in the Krip-kean semantics). The calculus for RPQL is obtained by adding to thePQL-calculus the following rule:

� ^ :� j�� _ :� (Kleene rule)

A completeness theorem for both PQL and RPQL can be proved, simi-larly to the case of OL. Both logics PQL and RPQL admit a natural modaltranslation (similarly to OL). The suitable modal system which PQL canbe transformed into is the system KB, semantically characterized by theclass of all symmetric frames. A convenient strengthening of KB gives riseto a regular modal system, that is suitable for RPQL.

An interesting question concerns the relation between PQL and the or-thomodular property.

Let B = hA;v ; 0 ; 1 ;0i be an ortholattice. By Lemma 19 the follow-ing three conditions (expressing possible de�nitions of the orthomodularproperty) turn out to be equivalent:

(i) 8a; b 2 B: a v b y b = a t (a0 u b);(ii) 8a; b 2 B: a v b and a0 u b = 0 y a = b;

(iii) 8a; b 2 B: a u (a0 t (a u b)) v b.However, this equivalence breaks down in the case of involutive bounded

lattices. One can prove only:

LEMMA 126. Let B be an involutive bounded lattice. If B satis�es condition(i), then B satis�es conditions (ii) and (iii).

Proof. (i) implies (ii): trivial. Suppose (i); we want to show that (iii) holds.Now, a0 v a0 t b0 = (a u b)0. Therefore, by (i), (a u b)0 = a0 t (a u (a u b)0).By de Morgan law: a u b = (a u (a0 t (a u b)) v b. �


�1

� f 0

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????

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� f??

????

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�0

� cOOOOOOOOOOOOOO � b0

��

��

�a��

��

��

OOOOOOOOOOOOOOO??

????

??? � c0

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� d??

????

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��

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ooooooooooooooo

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? � e0OOOOOOOOOOOOOO

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Figure 6. G14

LEMMA 127. Any involutive bounded lattice B that satis�es condition (iii)is an ortholattice.

Proof. Suppose (iii). It is suÆcient to prove that 8a; b 2 B: a u a0 v b.Now, a u a0 v a; a0. Moreover, a0 v a0 t (a u b). Therefore, by (iii),a u a0 v a u (a0 t (a u b)) v b. Thus, 8a 2 B: a u a0 = 0. �

As a consequence, we can conclude that there exists no proper orthomod-ular paraconsistent quantum logic when orthomodularity is understood inthe sense (i) or (iii). A residual possibility for a proper paraconsistent quan-tum logic to be orthomodular is orthomodularity in the sense (ii). In fact,the lattice G14 (see Figure 6) is an involutive bounded lattice which turnsout be orthomodular (ii) but not orthomodular (i).

Since f u f 0 = f 6= 0, G14 cannot be an ortholattice. Hence, G14 isneither orthomodular (i) nor orthomodular (iii). However, G14 is triviallyorthomodular (ii) since the premiss of condition (ii) is satis�ed only in thetrivial case where both a; b are either 0 or 1.

Hilbert space realizations for orthomodular paraconsistent quantum logiccan be constructed in the algebraic semantics by taking as support thefollowing proper subset of the set of all e�ects:

I(H) := fa1I j a 2 [0; 1]g [ P (H):

QUANTUM LOGICS 193

In other words, a possible meaning of the formula is either a sharp property(projection) or an unsharp property that can be represented as a multipleof the universal property (1I).

The set I(H) determines an orthomodular involutive regular boundedlattice, where the partial order is the partial order of E(H) restricted toI(H), while the fuzzy complement is de�ned like in the class of all e�ects(E0 := 1I�E).

An interesting feature of PQL is represented by the fact that this logicturns out to be a common sublogic in a wide class of important logics.In particular, PQL is a sublogic of Girard's linear logic [Girard, 1987], of Lukasiewicz' in�nite many-valued logic and of some relevant logics.

As we will see in Section 17, PQL represents the most natural quantumlogical extension of a quite weak and general logic, that has been calledbasic logic.

14 THE BROUWER{ZADEH LOGICS

The Brouwer{Zadeh logics (called also fuzzy intuitionistic logics) representnatural abstractions from the class of all BZ-lattices (de�ned in Section 12).As a consequence, a characteristic property of these logics is a splitting ofthe connective \not" into two forms of negation: a fuzzy-like negation, thatgives rise to a paraconsistent behaviour and an intuitionistic-like negation.The fuzzy \not" represents a weak negation, that inverts the two extremetruth-values (truth and falsity), satis�es the double negation principle butgenerally violates the non-contradiction principle. The second \not" is astronger negation, a kind of necessitation of the fuzzy \not".

We will consider two forms of Brouwer{Zadeh logic: BZL (weak Brouwer{Zadeh logic) and BZL3 (strong Brouwer{Zadeh logic). The language of bothBZL and BZL3 is an extension of the language of QL. The primitive con-nectives are: the conjunction (^), the fuzzy negation (:), the intuitionisticnegation (�).

Disjunction is metatheoretically de�ned in terms of conjunction and ofthe fuzzy negation:

� _ � := :(:� ^ :�) :

A necessity operator is de�ned in terms of the intuitionistic and of the fuzzynegation:

L� :=� :� :A possibility operator is de�ned in terms of the necessity operator and ofthe fuzzy negation:

M� := :L:� :Let us �rst consider our weaker logic BZL. Similarly to OL and PQL,

also BZL can be characterized by an algebraic and a Kripkean semantics.


DEFINITION 128 (Algebraic realization for BZL). An algebraic realizationof BZL is a pair hB ; vi, consisting of a BZ-lattice hB ;v ; 0 ;� ;1 ;0i anda valuation-function v that associates to any formula � an element in B,satisfying the following conditions:

(i) v(:�) = v(�)0

(ii) v(� �) = v(�)�

(iii) v(� ^ ) = v(�) u v( ).

The de�nitions of truth, consequence in an algebraic realization for BZL,logical truth and logical consequence are given similarly to the case of OL.

A Kripkean semantics for BZL has been �rst proposed in [Giuntini, 1991].A characteristic of this semantics is the use of frames with two accessibilityrelations.

DEFINITION 129. A Kripkean realization for BZL is a systemK = hI ; 6? ; 6?� ;� ; �i where:

(i) hI ; 6? ; 6?�i is a frame with a non empty set I of possible worldsand two accessibility relations: 6? (the fuzzy accessibility relation)and 6?� (the intuitionistic accessibility relation).

Two worlds i ; j are called fuzzy-accessible i� i 6? j. They arecalled intuitionistically-accessible i� i 6?� j. Instead of not(i 6? j)and not(i 6?� j), we will write i ? j and i ?� j, respectively.

The following conditions are required for the two accessibilityrelations:

(ia) hI; 6?i is a regular symmetric frame;

(ib) any world is fuzzy-accessible to at least one world:

8i 9j : i 6? j :

(ic) hI; 6?�i is an orthoframe;

(id) Fuzzy accessibility implies intuitionistic accessibility:

i 6? j y i 6?� j:

(ie) Any world i has a kind of \twin-world" j such that for anyworld k:

(a) i 6?� k i� j 6?� k(b) i 6?� k y j 6? k.

For any set X of worlds, the fuzzy-orthogonal set X 0 is de-�ned like in OL:

X 0 = fi 2 I j 8j 2 X : i ? jg :

QUANTUM LOGICS 195

Similarly, the intuitionistic orthogonal set X� is de�ned asfollows:

X� = fi 2 I j 8j 2 X : i ?� jg :The notion of proposition is de�ned like in OL. It turns outthat a set of worlds X is a proposition i� X = X 00.One can prove that for any set of worlds X , both X 0 and X�

are propositions. Further, like in OL, X u Y (the greatestproposition included in the propositions X and Y ) is X \Y ,while X t Y (the smallest proposition including X and Y )is (X [ Y )00.

(ii) � is a set of propositions that contains I , and is closed under0 ;� ;u.

(iii) � associates to any formula a proposition in � according to thefollowing conditions:

�(:�) = �(�)0;

�(� �) = �(�)�;

�(� ^ ) = �(�) u �( ).

THEOREM 130. Let hI ; 6? ; 6?� i be a BZ-frame (i.e. a frame satisfying theconditions of De�nition 129(i)) and let �0 be the set of all propositions ofthe frame. Then, the structure

�0 ;� ; 0 ; � ; ; ; I � is a complete BZ-lattice

such that for any set � � �0:

inf (�) := F� =\

� and sup (�) :=G

� =�[

��00:

As a consequence, the proposition-structure h� ;� ; 0 ; � ; ; ; Ii of a BZLrealization, turns out to be a BZ-lattice.

The de�nitions of truth, consequence in a Kripkean realization, logicaltruth and logical consequence, are given similarly to the case of OL.

One can prove, with standard techniques, that the algebraic and theKripkean semantics for BZL characterize the same logic.

We will now introduce a calculus that represents an adequate axiomati-zation for the logic BZL. The most intuitive way to formulate our calculusis to present it as a modal extension of the axiomatic version of regularparaconsistent quantum logic RPQL. (Recall that the modal operators ofBZL are de�ned as follows: L� :=� :�; M� := :L:�).

Rules of BZL.

The BZL-calculus includes, besides the rules of RPQL the following modalrules:


(BZ1) L� j��

(BZ2) L� j�LL�

(BZ3) ML� j�L�

(BZ4)� j��

L� j�L�(BZ5) L� ^ L� j�L(� ^ �)

(BZ6) ; j�:(L� ^ :L�)

The rules (BZ1)-(BZ5) give rise to a S5{like modal behaviour. Therule (BZ6) (the non-contradiction principle for necessitated formulas) is,of course, trivial in any classical modal system.

One can prove a soundness and completeness Theorem with respect to theKripkean semantics (by an appropriate modi�cation of the correspondingproofs for QL).

Characteristic logical properties of BZL are the following:

(a) like in PQL, the distributive principles, Duns Scotus, the non-contradiction and the excluded middle principles (for the fuzzynegation) break down;

(b) like in intuitionistic logic, we have:

j=BZL� (�^ � �); j=

BZL= �_ � � ; � j=

BZL�� ; �� j=

BZL= � ;

�� j=BZL� � ; � j=

BZL� y � � j=

BZL� � ;

(c) moreover, we have:

� � j=BZL:� ; :� j=

BZL= � � ; : � � j=

BZL�� ;

One can prove that BZL has the �nite model property; as a consequenceit is decidable [Giuntini, 1992].

The ortho-pair semanticsOur stronger logic BZL3 has been suggested by a form of fuzzy-intuitionisticsemantics, that has been �rst studied in [Cattaneo and Nistic�o, 1986]. Theintuitive idea, underlying this semantics (which has some features in com-mon with Klaua's partielle Mengen and with Dunn's polarities) can besketched as follows: one supposes that interpreting a language means associ-ating to any sentence two domains of certainty : the domain of the situations

QUANTUM LOGICS 197

where our sentence certainly holds, and the domain of the situations whereour sentence certainly does not hold. Similarly to Kripkean semantics, thesituations we are referring to can be thought of as a kind of possible worlds.However, di�erently from the standard Kripkean behaviour, the positive do-main of a given sentence does not generally determine the negative domainof the same sentence. As a consequence, propositions are here identi�edwith particular pairs of sets of worlds, rather than with particular sets ofworlds.

Let us again assume the BZL language. We will de�ne the notion ofrealization with positive and negative certainty domains (shortly ortho-pairrealization) for a BZL language.

DEFINITION 131. An ortho-pair realization is a system O = hI ; R ; ; vi ;where:

(i) hI ; R i is an orthoframe.

(ii) Let �0 be the set of all propositions of the orthoframe hI ; Ri.As we already know, this set gives rise to an ortholattice withrespect to the operations u;t and 0 (where u is the set-theoreticintersection).

An orthopairproposition of hI ; R i is any pair hA1 ; A0i, whereA1; A0 are propositions in �0 such that A1 � A00. An orthopair-proposition hA1 ; A0i is called exact i� A0 = A01 (in other words,A0 is maximal). The following operations and relations can bede�ned on the set of all orthopairpropositions:

(iia) The fuzzy complement:

hA1 ; A0i 0 := hA0 ; A1i :(iib) The intuitionistic complement:

hA1 ; A0i � := hA0 ; A00i :

(iic) The orthopairpropositional conjunction:

hA1 ; A0i u hB1 ; B0i := hA1 u B1 ; A0 t B0i :(iid) The orthopairpropositional disjunction:

hA1 ; A0i t hB1 ; B0i := hA1 t B1 ; A0 u B0i :(iie) The in�nitary conjunction:

FnfhAn1 ; A

n0 ig :=

*\n

fAn1g ;

Gn

fAn0g+:


(iif) The in�nitary disjunction:

GnfhAn

1 ; An0 ig :=

*Gn

fAn1g ;

\n

fAn0g+:

(iig) The necessity operator:

�(hA1 ; A0i) := hA1 ; A01i :

(iih) The possibility operator:

�(hA1 ; A0i) := (�(hA1 ; A0i 0)) 0 :(iik) The order-relation:

hA1 ; A0i v hB1 ; B0i i� A1 � B1 and B0 � A0:

(iii) is a set of orthopairpropositions, that is closed under 0 ; � ;u ;tand 0 := h; ; Ii :

(iv) v is a valuation-function that maps formulas into orthopairpropo-sitions according to the following conditions:

v(:�) = v(�) 0;

v(� �) = v(�) �;

v(� ^ ) = v(�)uv( ).

The other basic semantic de�nitions are given like in the algebraic se-mantics. One can prove the following Theorem:

THEOREM 132. Let hI ; R i be an orthoframe and let 0 be the set of allorthopairpropositions of hI ; R i. Then, the structure h0 ;� ; 0 ; � ; h;; Ii;hI; ;ii is a complete BZ-lattice with respect to the in�nitary conjunction anddisjunction de�ned above. Further, the following conditions are satis�ed: forany hA1; A0i ; hB1 ; B0i 2 0:

(i) � hA1 ; A0i = hA1 ; A0i 0 �.

(ii) hA1 ; A0i � = �(hA1 ; A0i 0 ).

(iii) � hA1 ; A0i = hA1 ; A0i � 0.(iv) (hA1; A0i u hB1; B0i) � = hA1; A0i � t hB1; B0i � :

(Strong de Morgan law)

(v) (hA1; A0i u hB1; B0i � �) � (hA1; A0i 0 � t hB1; B0i):

Accordingly, in any ortho-pair realization the set of all orthopairproposi-tions 0 gives rise to a BZ-lattice. As a consequence, one can immediately

QUANTUM LOGICS 199

prove a soundness theorem with respect to the ortho-pair semantics. Doesperhaps the ortho-pair semantics characterizes the logic BZL? The answerto this question is negative. As a counterexample, let us consider an in-stance of the fuzzy excluded middle and an instance of the intuitionisticexcluded middle applied to the same formula �:

� _ :� and �_ � �:One can easily check that they are logically equivalent in the ortho-pairsemantics. For, given any ortho-pair realization O, there holds::

� _ :� j=O �_ � � and �_ � � j=O � _ :� :However, generally

� _ :� j=BZL= �_ � � :

For instance, let us consider the following algebraic BZL{realization A =hB ; vi, where the support B of is the real interval [0 ; 1] and the algebraicstructure on B is de�ned as follows:

a v b i� a � b;a0 = 1� a;

a� =

(1 ; if a = 0;

0 ; otherwise:

1 = 1; 0 = 0.

Suppose for a given sentential literal p: 0 < v(p) < 1=2. We will havev(p_ � p) = Max(v(p) ; 0) = v(p) < 1=2. But v(p _ :p) = Max(v(p) ; 1 �v(p)) = 1� v(p) � 1=2. Hence: v(p_ � p) < v(p _ :p).As a consequence, the orthopair-semantics characterizes a logic strongerthan BZL. We will call this logic BZL3. The name is due to the charac-teristic three-valued features of the ortho-pair semantics.

Our logic BZL3 is axiomatizable. A suitable calculus can be obtainedby adding to the BZL-calculus the following rules.

Rules of BZL3.

(BZ31) L(� _ �) j�L� _ L�

(BZ32)L� j��; � j�M�

� j��The following rules turn out to be derivable:

(DR1)L� j�� ;M� j�M�

� j��


(DR2) M� ^M� j�M(� ^ �)

(DR3) � (� ^ �) j� � �_ � �The validity of a strong de Morgan's principle for the connective � (DR3)

shows that this connective represents, in this logic, a kind of strong \super-intuitionistic" negation (di�erently from BZL, where the strong de Morganlaw fails, like in intuitionistic logic).

One can prove a soundness and a completeness theorem of our calculuswith respect to the ortho-pair semantics.

THEOREM 133 (Soundness theorem).

T j��BZL3 � y T j=BZL3

�:

Proof. By routine techniques. �

THEOREM 134. Completeness theorem.

T j=BZL3

� y T j��BZL3 �:

Sketch of the proofInstead of T j=

BZL3� and T j��BZL3�, we will shortly write T j= � and T j��.

It is suÆcient to construct a canonical model O = hI ; R ; ; vi such that:

T j=O � y T j�� :(The other way around follows from the soundness theorem).

De�nition of the canonical model

(i) I is the set of all possible sets i of formulas satisfying the followingconditions:

(ia) i is non contradictory with respect to the fuzzy negation ::for any �, if � 2 i, then :� 62 i;

(ib) i is L-closed : for any �, if � 2 i, then L� 2 i;(ic) i is deductively closed : for any �, if i j��, then � 2 i.

(ii) The accessibility relation R is de�ned as follows:

Rij i� for any formula �: � 2 i y :� 62 j.(In other words, i and j are not contradictory with respectto the fuzzy negation).

Instead of notRij, we will write i ? j.(iii) is the set of all orthopairpropositions of hI; Ri.

QUANTUM LOGICS 201

(iv) For any atomic formula p:

v(p) = hv1(p) ; v0(p)i ;

where:

v1(p) = fi j i j� pg and v0(p) = fi j i j�:pg :

O is well de�ned since one can prove the following Lemmas:

LEMMA 135. R is re exive and symmetric.

LEMMA 136. For any � ; fi j i j��g is a proposition of the orthoframehI ; Ri.LEMMA 137. For any �, fi j i j��g � fi j i j�:�g0.

Further, one can prove

LEMMA 138. For any �, v(�) = hv1(�) ; v0(�)i, where:

v1(�) = fi j i j��gv0(�) = fi j i j�:�g

LEMMA 139. For any formula �:0 := h;; Ii = hfi j i j�L� ^ :L�g ; fi j i j�:(L� ^ :L�)gi.LEMMA 140. Let T = f�1 ; : : : ; �n ; : : : g be a set of formulas and let � beany formula.Tfv1(�n) j �n 2 Tg � v1(�) y L�1 ; : : : ; L�n ; : : : j��.

As a consequence, one can prove:

LEMMA 141. Lemma of the canonical model

T j=O � y T j��:

Suppose T j=O �. Hence (by de�nition of consequence in a given re-alization): for any orthopairproposition hA1 ; A0i 2 , if for all �n 2 T ,hA1 ; A0i v v(�n), then hA1 ; A0i v v(�).

The propositional lattice, consisting of all orthopairpropositions of O iscomplete (see Theorem 132). Hence: Fn fv(�n) j �n 2 Tg v v(�). Inother words, by de�nition of v:

(i)Tfv1(�n) j �n 2 Tg � v1(�);

(ii) v0(�) � Ffv0(�n) j �n 2 Tg.Thus, by (i) and by Lemma 140: L�1 ; : : : ; L�n ; : : : j��. Consequently,there exists a �nite subset f�n1 ; : : : ; �nkg of T such that L�n1 ^ : : : ^L�nk j��. Hence, by the rules for ^ and L: L(�n1 ^ : : : ^ �nk ) j��.


At the same time, we obtain from (ii) and by Lemma 138: v1(:�) vFfv1(:�n) j �n 2 Tg.Whence, by de Morgan,

v1(:�) �h\f(v1(:�n))0 j �n 2 Tg

i0:

Now, one can easily check that in any realization: v1(:�)0 = v1(M�). Asa consequence: v1(:�) � [

Tf(v1(M�n) j �n 2 Tg]0 : Hence, by contraposi-tion: \

fv1(M�n) j �n 2 Tg � (v1(:�))0

and \fv1(M�n) j �n 2 Tg � v1(M�):

Consequently, by Lemma 140 and by the S5-rules:

LM�1 ; : : : ; LM�n ; : : : j�M� ; M�1 ; : : : ;M�n ; : : : j�M� :

By syntactical compactness, there exists a �nite subset f�m1 ; : : : ; �mhg of

T such that M�m1 ; : : : ;M�mhj�M�. Whence, by the rules for ^ and

M : M(�m1 ^ : : : ^ �mh) j�M�. Let us put 1 = �n1 ^ : : : ^ �nk and

2 = �m1^: : :^�mh. We have obtained: L 1 j�� and M 2 j�M�. Whence,

L 1 ^L 2 j��, L( 1 ^ 2) j��, M 1 ^M 2 j�M�, M( 1 ^ 2) j�M�. FromL( 1^ 2) j��, and M( 1^ 2) j�M� we obtain, by the derivable rule (DR1): 1 ^ 2 j��. Consequently: T j��. �

Similarly to other forms of quantum logic, also BZL3 admits an al-gebraic semantic characterization [Giuntini, 1993] based on the notion ofBZ3-lattice.

DEFINITION 142. A BZ3-lattice is a BZ-lattice B = hB ;v ; 0 ;� ;1 ;0i,which satis�es the following conditions:

(i) (a u b)� = a� t b�;

(ii) a u b�� v a0� t b.By Theorem 132, the set of all orthopairpropositions of an orthoframe

determines a complete BZ3-lattice. One can prove the following represen-tation theorem:

THEOREM 143. Every BZ3-lattice is embeddable into the (complete) BZ3-lattice of all orthopairpropositions of an orthoframe.

A slight modi�cation of the proof of Theorem 17 permits us to show thatortho-pair semantics and the algebraic semantics strongly characterize thesame logic.

One can prove that BZL3 can be also characterized by means of a nonstandard version of Kripkean semantics [Giuntini, 1993].

Some problems concerning the Brouwer-Zadeh logics remain still open:

QUANTUM LOGICS 203

1) Is there any Kripkean characterization of the logic that is alge-braically characterized by the class of all de Morgan BZ-lattices(i.e. BZ-lattices satisfying condition (i) of De�nition 142)? Inthis framework, the problem can be reformulated in this way: isthe (strong) de Morgan law elementary?

2) Is it possible to axiomatize a logic based on an in�nite many-valued generalization of the ortho-pair semantics?

3) Find possible conditional connectives in BZL3.

4) Find an appropriate orthomodular extension of BZL3.

Unsharp quantum models for BZL3

The ortho-pair semantics has been suggested by the e�ect-structures inHilbert-space QT. In this framework, natural quantum ortho-pair realiza-tions for BZL3 can be constructed. Let us refer again to the language LQ(whose atoms express possible measurement reports) and let S be a quantumsystem whose associated Hilbert space is H. As usual, E(H) will representthe set of all e�ects of H. Now, an ortho-pair realizationMS = hI ; R ; ; vi(for the system S) can be de�ned as follows:

(i) I is the set of all pure states of S in H.

(ii) Rij i� for any e�ect E 2 E(H) the following condition holds:whenever i assigns to E probability 1, then j assigns to E aprobability di�erent from 0.

In other words, i and j are accessible i� they cannot be stronglydistinguished by any physical property represented by an e�ect.

(iii) The propositions of the orthoframe hI; R i are determined by theset of all closed subspaces of H (sharp properties), like in OQL.

(iv) is the set of all orthopairpropositions of hI; R i. Any e�ect Ecan be transformed into an orthopairproposition f(E) := hXE

1 ;XE0 i of , where:

XE1 := fi j i assigns to E probability 1g ;

XE0 := fi j i assigns to E probability 0g :

In other words, XE1 ; X

E0 represent the positive and the negative

domain of E, respectively. The map f turns out to preserve theorder relation and the two complements:

E v F i� f(E) v f(F ):

f(E0) = f(E) 0 =XE1 ; X

E0

� 0=XE0 ; X

E1

�:

f(E�) = f(E) � =XE1 ; X

E0

� �=XE0 ; (X

E0 )0�:


(v) The valuation-function v follows the intuitive physical meaningof the atomic sentences. Let p express the assertion \the valuefor the observable A lies in the sharp (or fuzzy) Borel set �" andlet Ep be the e�ect that is associated to p in H. We de�ne v asfollows:

v(p) = f(Ep) =DXEp

1 ; XEp

0

E:

It is worthwhile to notice that our map f is not injective: di�erent e�ectswill be transformed into one and the same orthopairproposition. As a con-sequence, moving from e�ects to orthopairpropositions clearly determines aloss of information. In fact, orthopairpropositions are only concerned withthe two extreme probability value (0,1), a situation that corresponds to athree-valued semantics.

15 PARTIAL QUANTUM LOGICS

In Section 12, we have considered examples of partial algebraic structures,where the basic operations are not always de�ned. How to give a seman-tic characterization for di�erent forms of quantum logic, corresponding re-spectively to the class of all e�ect algebras, of all orthoalgebras and of allorthomodular posets? We will call these logics: unsharp partial quantumlogic (UPaQL), weak partial quantum logic (WPaQL) and strong partialquantum logic (SPaQL).

Let us �rst consider the case of UPaQL, that represents the \logic ofe�ect algebras" [Dalla Chiara and Giuntini, 1995].

The language of UPaQL consists of a denumerably in�nite list of atomicsentences and of two primitive connectives: the negation : and the exclusivedisjunction _+ (aut).

The set of sentences is de�ned in the usual way. A conjunction is met-alinguistically de�ned, via de Morgan law:

� : � := :(:�_+ :�):

The intuitive idea underlying our semantics for UPaQL is the follow-ing: disjunctions and conjunctions are always considered \legitimate" froma mere linguistic point of view. However, semantically, a disjunction �_+ �will have the intended meaning only in the \well behaved cases" (wherethe values of � and � are orthogonal in the corresponding e�ect orthoal-gebra). Otherwise, �_+ � will have any meaning whatsoever (generally notconnected with the meanings of � and �). As is well known, a similarsemantic \trick" is used in some classical treatments of the description op-erator � (\the unique individual satisfying a given property"; for instance,\the present king of Italy").

QUANTUM LOGICS 205

DEFINITION 144. A realization for UPaQL is a pair A = hB ; vi, whereB = hB ;� ;1 ;0i is an e�ect algebra (see De�nition 115); v (the valuation-function) associates to any formula � an element of B, satisfying the fol-lowing conditions:

(i) v(:�) = v(�)0, where 0 is the generalized complement (de�ned in B).

(ii)

v(� _+ ) =

(v(�) � v( ); if v(�) � v( ) is de�ned inB;

any element; otherwise.

The other semantic de�nitions (truth, consequence in a given realization,logical truth, logical consequence) are given like in the QL-case.

Weak partial quantum logic (WPaQL) and strong partial quantum logic(SPaQL) (formalized in the same language as UPaQL) will be naturallycharacterized mutatis mutandis . It will be suÆcient to replace, in the de�-nition of realization, the notion of e�ect algebra with the notion of orthoal-gebra and of orthomodular poset (see De�nition 117 and De�nition 110).Of course, UPaQL is weaker than WPaQL, which is, in turn, weaker thanSPaQL.

Partial quantum logics are axiomatizable. We will �rst present a calculusfor UPaQL, which is obtained as a natural transformation of the calculusfor orthologic.

Di�erently from QL, the rules of our calculus will always have the form:

�1 j��1; : : : ; �n j��n� j��

In other words, we will consider only inferences from single formulas.

Rules of UPaQL

(UPa1) � j�� (identity)

(UPa2)� j�� j�

� j� (transitivity)

(UPa3) � j�::� (weak double negation)


(UPa4) ::� j�� (strong double negation)

(UPa5)� j��:� j�:� (contraposition)

(UPa6) � j��_+ :� (excluded middle)

(UPa7)� j�:� �_+ :� j��_+ �

:� j�� (unicity of negation)

(UPa8)� j�:� � j��1 �1 j�� j��1 �1 j��

�_+ � j��1 _+ �1(weak substitutiv-

ity)

(UPa9)� j�:�

�_+ � j�� _+ �(weak commutativity)

(UPa10)� j�: � j�:(� _+ )

� j�:� (weak associativity)

(UPa11)� j�: � j�:(� _+ )

�_+ � j�: (weak associativity)

(UPa12)� j�: � j�:(� _+ )

�_+ (� _+ ) j� (�_+ �)_+ (weak associativity)

(UPa13)� j�: � j�:(� _+ )

(�_+ �)_+ j��_+ (� _+ )(weak associativity)

QUANTUM LOGICS 207

The concepts of derivation and of derivability are de�ned in the expectedway. In order to axiomatize weak partial quantum logic (WPaQL) it issuÆcient to add a rule, which corresponds to a Duns Scotus-principle:

(WPaQL)� j�:�� j�� (Duns Scotus)

Clearly, the Duns Scotus-rule corresponds to the strong consistency con-dition in our de�nition of orthoalgebra (see De�nition 117). In other words,di�erently from UPaQL, the logic WPaQL forbids paraconsistent situa-tions.

Finally, an axiomatization of strong partial quantum logic (SPaQL) canbe obtained, by adding the following rule to (UPa1)-(UPa13), (WPa):

(SPaQL)� j�:� � j� � j�

�_+ � j� In other words, (SPaQL) requires that the disjunction _+ behaves like a

supremum, whenever it has the \right meaning".Let PaQL represent any instance of our three calculi. We will use the

following abbreviations. Instead of � j�PaQL

� we will write � j��. When �and � are logically equivalent (� j�� and � j��) we will write � � �.

Let p represent a particular sentential literal of the language: T will bean abbreviation for p_+ :p; while F will be an abbreviation for : (p_+ :p).

Some important derivable rules of all calculi are the following:

(D1) F j�� ; � j�T (Weak Duns Scotus)

(D2)� j�:�� j��_+ �

(weak sup rule)

(D3)� j��

� � �_+ : (�_+ :�)(orthomodular-like rule)

(D4)� j�: � j�: �_+ � � _+

� � � (cancellation)

As a consequence, the following syntactical lemma holds:

LEMMA 145. For any � ; �: � j�� i� there exists a formula such that

(i) � j�: ;

(ii) � � �_+ .


In other words, the logical implication behaves similarly to the partial orderrelation in the e�ect algebras.

The following derivable rule holds for WPaQL and for SPaQL:

(D5)� j�:� � j� � j� j��_+ �

�_+ � j� Our calculi turn out to be adequate with respect to the corresponding se-mantic characterizations. Soundness proofs are straightforward. Let ussketch the proof of the completeness theorem for our weakest calculus (UP-aQL).

THEOREM 146. Completeness.

� j= � y � j��:

Proof. Following the usual procedure, it is suÆcient to construct a canon-ical model A = hB ; vi such that for any formulas �; �:

� j�� y � j=A �:

De�nition of the canonical model.

(i) The algebra B = hB ;� ;1 ;0i is determined as follows:

(ia) B is the class of all equivalence classes of logically equivalentformulas: B := f[�]� j � is a formulag. (In the following,we will write [�] instead of [�]�).

(ib) [�]� [�] is de�ned i� � j�:�. If de�ned, [�]� [�] := [�_+ �].

(ic) 1 := [T]; 0 := [F].

(ii) The valuation function v is de�ned as follows: v(�) = [�].

One can easily check that A is a \good" model for our logic. The opera-tion � is well de�ned (by the transitivity, contraposition and weak substi-tutivity rules). Further, B is an e�ect algebra: � is weakly commutativeand weakly associative, because of the corresponding rules of our calculus.The strong excluded middle axiom holds by de�nition of � and in virtue ofthe following rules: excluded middle, unicity of negation, double negation.Finally, the weak consistency axiom holds by weak Duns Scotus (D1) andby de�nition of �.

LEMMA 147. Lemma of the canonical model

[�] v [�] i� � j��:

Sketch of the proof. By de�nition of v (in any e�ect algebra) onehas to prove:

� j�� i� for a given such that [�] ? [ ] : [�]� [ ] = [�]:

QUANTUM LOGICS 209

This equivalence holds by Lemma 145 and by de�nition of �.Finally, let us check that v is a \good" valuation function. In other words:

(i) v(:�) = v(�)0

(ii) v(� _+ ) = v(�) � v( ), if v(�) � v( ) is de�ned.

(i) By de�nition of v, we have to show that [:�] is the unique [ ] suchthat [�]� [ ] = 1 := [T]. In other words,

(ia) [T] v [�]� [:�].

(ib) [T] v [�]� [ ] y :� � .

This holds by de�nition of the canonical model, by de�nition of � and bythe following rules: double negation, excluded middle, unicity of negation.(ii) Suppose v(�) � v( ) is de�ned. Then � j�: . Hence, by de�nition of� and of v: v(�)� v( ) = [�]� [ ] = [� _+ ] = v(� _+ ).

As a consequence, we obtain:

� j�� i� [�] v [�] i� v(�) v v(�) i� � j=A �

�

The completeness argument can be easily transformed, mutatis mutandisfor the case of weak and strong partial quantum logic.

16 LUKASIEWICZ QUANTUM LOGIC

As we have seen in Section 12, the class E(H) of all e�ects on a Hilbertspace H determines a quasi-linear QMV-algebra. The theory of QMV-algebras suggests, in a natural way, the semantic characterization of a newform of quantum logic (called Lukasiewicz quantum logic ( LQL)), whichgeneralizes both OQL and L@.

The language of LQL contains the same primitive connectives as WPaQL(_+ ;:). The conjunction ( : ) is de�ned via de Morgan law (like in WPaQL).Further, a new pair of conjunction ( ^ ) and disjunction (__ ) connectivesare de�ned as follows:

� ^ � := (�_+ :�) : �

�__ � := :(:� ^ :�)

DEFINITION 148. A realization for LQL is a pair A = hM ; vi, where

(i) M = hM ;� ;� ;1 ;0i is a QMV-algebra.

(ii) v (the valuation-function) associates to any formula � an elementof M , satisfying the following conditions:


v(:�) = v(�)�.

v(� _+ ) = v(�)� v( ).

The other semantic de�nitions (truth, consequence in a given realization,logical truth, logical consequence) are given like in the QL-case.

LQL can be easily axiomatized by means of a calculus that simply mimicsthe axioms of QMV-algebras.

The quasi-linearity property, which is satis�ed by the QMV-algebras ofe�ects, is highly non equational. Thus, the following question naturallyarises: is LQL characterized by the class of all quasi-linear QMV-algebras(QLQMV)? In the case of L@, Chang has proved that L@ is characterizedby the MV-algebra determined by the real interval [0; 1]. This MV-algebrais clearly quasi-linear, being totally ordered.

The relation between LQL and QMV algebras turns out to be much morecomplicated. In fact one can show that LQL cannot be characterized evenby the class of all weakly linear QMV-algebras (WLQMV). Since WLQMV isstrictly contained in QLQMV, there follows that LQL is not characterizedby QLQMV. To obtain these results, something stronger is proved. Inparticular, we can show that:

� the variety of all QMV-algebras (QMV) strictly includes thevariety generated by the class of all weakly linear QMV-algebras(H SP(WLQMV)).

� H SP(WLQMV) strictly includes the variety generated by the class ofall quasi-linear QMV-algebras (H SP(QLQMV)).

So far, little is known about the axiomatizability of the logic based onHSP(QLQMV). In the case of H SP(WLQMV), instead, one can prove thatthis variety is generated by the QMV-axioms together with the followingaxiom:

a = (a� c� b�) e (a� c� � b):The problem of the axiomatizability of the logic based on H SP(QLQMV)is complicated by the fact that not every (quasi-linear) QMV-algebraM =hM ;� ;� ;1 ;0i admits of a \good polynomial conditional", i.e., a polyno-mial binary operation Æ such that

a Æ b = 1 i� a � b:

Thus, it might happen that the notion of logical truth of the logic basedon H SP(QLQMV) is (�nitely) axiomatizable, while the notion of \logicalentailment" (� j= �) is not.

We will now show that the QMV-algebraM4 (see Figure 7 below) doesnot admit any good polynomial conditional. The operations of M4 are

QUANTUM LOGICS 211

de�ned as follows:

`

�0 0 0

0 a a

0 b b

0 1 1

a 0 a

a a 1

a b 1

a 1 1

b 0 b

b b 1

b a 1

b 1 1

1 0 1

1 a 1

1 b 1

1 1 1

�

0 1

a a

b b

1 0

�1

�a��

��

��

�0

????

????

� b??

????

??

� ��

��

Figure 7. M4

Let us consider the three-valued MV-algebra M3, whose operations are


�1

� 12

�0

Figure 8. M3

de�ned as follows:

�0 0 0

0 12

12

0 1 112 0 1

212

12 1

12 1 1

1 0 1

1 12 1

1 1 1

�

0 112

12

1 0

It is easy to see that the map h : M4 !M3 such that 8x 2M4

h(x) :=

8><>:

0; if x = 0;12 ; if x = a or x = b;

1; otherwise

is a homomorphism of M4 into M3.

Suppose, by contradiction, that M4 admits of a good polynomial condi-tional !M4 . Since a 6� b, we have h(a!M4 b) 6= 1. Thus,

1 6= h(a!M4 b) = h(a)!M3 h(b) =1

2!M3

1

2= 1;

contradiction.

QUANTUM LOGICS 213

17 QUANTUM LOGIC AND THE CUBE OF LOGICS(BY GIULIA BATTILOTTI AND CLAUDIA FAGGIAN)

Di�erent forms of quantum logic can be axiomatized as sequent calculi[Dummett, 1976; Nishimura, 1980; Cutland and Gibbins, 1982; Tamura,1988; Nishimura, 1994]. This permits us to investigate such logics more andmore deeply from the proof-theoretical point of view. A sequent calculus fororthologic can be obtained from a calculus for classical logic, by requiringa special restriction on contexts in the rules that would permit to derivethe distributive laws. The critical rules are the following: the introduc-tion of disjunction on the left side, the introduction of conjunction on theright side, the rules concerning implication and negation. Such a restric-tion, however, determines some serious proof-theoretical diÆculties, becausequantum logic has a suÆciently strong negation that satis�es de Morgan'slaws. The shortcoming becomes apparent when we try to prove the corner-stone result, represented by a cut-elimination theorem (which, essentiallydepends on the formulation of the rules that appear in our proofs).

A simple and compact sequent calculus for orthologic [Faggian and Sam-bin, 1997; Battilotti and Sambin, 1999], which admits cut-elimination bymeans of a neat procedure, can be obtained by a convenient strengtheningof basic logic. This is a new logic that has been introduced in order toinvestigate a general structure for the space of logics [Sambin et al., 1998].

In the framework of basic logic, constraints on contexts are not considereda limitation; on the contrary, they are regarded as a positive feature, whichis called visibility . At the same time, negation is treated by exploiting thesymmetry of the calculus: the main idea is to use Girard's linear negation,which can be interpreted as an orthocomplement in a quite natural way.This approach shows that orthologic (and non-distributive logics, in general)admits a proof-theory, which turns out to be simpler than the proof-theoryfor classical logic. Describing quantum logic in the framework of a uniformand general setting gives many advantages, since it permits us to studyvarious logics and their mutual relations. In particular, we obtain a wholesystem of quantum logics (including linear orthologic); and for each of theselogics we have a proof of the cut-elimination theorem. All this gives rise toa new formulation of classical logic [Faggian, 1997], with respect to whichorthologic and the other quantum-like logics (created by this method) turnout to be characterizable as substructural logics . On this basis it is easy tocompare di�erent logics, and to prove embedding results [Battilotti, 1998].

Basic logic and the cube of logics.As we already know, quantum logic represents a weakening of classical logic,obtained by dropping the distributive laws. There are at least two otherimportant logics that are weaker than classical logic: intuitionistic logicand linear logic [Girard, 1987]. The situation can be sketched as picturedin Figure 9.


C

I��

��

��

��

��

L

QL

????

????

????

????

?

Figure 9. The most important weakenings of classical logic

It is natural to ask whether there exists a logic that represents a kindof common denominator for Q, I and L, in the same way as classical logictheoretically includes all the other logics. A solution to this problem hasbeen found in terms of a suitable sequent calculus B, that represents a basiclogic.

Di�erently from the calculi we have considered in the previous sections,a sequent calculus for a given logic L is based on axioms and rules thatgovern the behaviour of sequents . Any sequent has the form

M ` N

where M;N are (possibly empty) �nite multisets of formulas.14 Axioms areparticular sequents. Any rule has the form

M1 ` N1 : : : Mn ` Nn

M ` N

where M1 ` N1; : : : ;Mn ` Nn are the premisses , while M ` N is theconclusion of the rule. Rules can be structural or operational . Operationalrules introduce a new connective, while structural rules deal only with thestructure of the sequents (orders, repetitions, etc.).

A derivation is a sequence of sequents where any element is either anaxiom or the conclusion of a rule whose premisses are previous elements ofthe sequence.

Basic logic has been introduced in [Battilotti and Sambin, 1999], andsubstantially reformulated in [Sambin et al., 1998]. According to the sec-

14A multiset is a set of pairs such that the �rst element of every pair denotes any object,while the second element denotes the multiplicity of the occurrences of our object. Twomultisets are equal if and only if all their pairs are equal, that is all their objects togetherwith their multiplicities are equal.

QUANTUM LOGICS 215

ond formulation (we will follow here),15 our logic is characterized by threestrictly linked principles: re ection, symmetry , visibility . The re ectionprinciple states the fact that logical constants are the result of importinginto a formal system metalinguistic links between assertions, considered pre-existing. There is a method that leads to the rules of the calculus, startingfrom metalinguistic links between assertions. Such a method analyses thefollowing equivalences, which assert a correspondence between language andmetalanguage:

M ` � � � if and only if M ` � ÆR �

� � � ` N if and only if � ÆL � ` NHere the generic sign \�", corresponding to a metalinguistic link between

assertions, is translated respectively into the connective ÆR, when it appearson the right of the sign `, and into the connective ÆL, when it appears on theleft. In B, the operational rules are completely determined by such equiv-alences. As a consequence, the meaning of a connective turns out to besemantically determined by the correspondence with a metalinguistic link,quite independently of any link with a context. Since every metalinguisticlink is translated into a connective according to two specular ways, the sys-tem of rules, obtained by this method, turns out to be strongly symmetric.In fact B contains, for every axiom and for every (unary or binary) rule R

Mi ` Ni

M ` N R

its symmetric rule Rs, given by

Nsi `Ms

i

Ns `Ms Rs

where the map (�)s is de�ned by induction (on the length of formulas), byputting ÆsR � ÆL and ÆsL � ÆR, given a suitable correspondence betweenpropositional variables.

The third principle that B satis�es is the visibility property. A rulefor a given connective is called visible when the principal formula and thecorresponding secondary formulas appear in the rule without any context.16

15The formulation of the rules of B presented in [Sambin et al., 1998] is based on �nitelists rather than �nite multisets of formulas; hence it contains in addition the structuralrule of exchange. Here we prefer to use multisets, in order to obtain an easy comparisonwith sequent calculi for quantum logics.16In any operational rule, the formula in the conclusion that contains the connective

introduced by the rule itself is called the principal formula; the formulas in the premissesthat are the components of the formula introduced by the rule are called the secondaryformulas.


We shall see below the syntactical consequences of visibility; we stress herethat semantically it corresponds to the fact that basic connectives have aprimitive meaning, in accordance with the re ection principle.

As an example let us refer to a rule that plays an important role in thecase of quantum logic. As is well known, in classical logic, disjunction isintroduced on the left according to the following rule:

M;� ` N M;� ` NM;� _ � ` N

In the case of B, instead, disjunction is introduced according to the fol-lowing visible form:

� ` N � ` N� _ � ` N

where the context M has disappeared.

From the intuitive point of view, one can read the di�erence between thetwo cases as follows: the rule typical of classical logic attaches a meaningto the connective _ in presence of the link \;" with M (such a link is to beinterpreted as a conjunction), whereas the visible rule is intended to explainthe meaning of the connective _ by referring only to the connective itself.In particular, the visible rule does not permit us to prove the equation thatlinks conjunction and disjunction ( the distributive law of ^ with respect to_). As a consequence, any sequent calculus for a quantum logic shall adoptthe visible form for the rule that concerns the introduction of disjunctionon the left. As to the other rules, visibility is not strictly necessary inorder to obtain an adequate sequent calculus for quantum logic. However, amore convenient strategy permits us to axiomatize quantum logic, by addingonly structural rules to basic logic, without any change in the rules for theconnectives. In this way, we can preserve the characteristic properties ofsymmetry and visibility of B, that turn out to be highly convenient fromthe proof-theoretical point of view (as we will see below).

Basic logic B has no structural rules. As a consequence, B can be re-garded as \the logic of connectives" from which various stronger logics canbe obtained by adding suitable structural rules.

Let us now present the sequent calculus for B. Similarly to linear logic,the language of B contains two pairs of conjunctions and disjunctions: theadditive conjunction ^ and the multiplicative conjunction ; the additivedisjunction _ and the multiplicative disjunction

&. Further there are two

conditionals (!, ), and two pairs of propositional literals 1 and >, 0and ?.

QUANTUM LOGICS 217

The basic sequent calculus B

Axioms� ` �

Operational rules

(Formation)�; � ` N� � ` N L M ` �; �

M ` � &�

&R

(Re ection)� ` N1 � ` N2

�&� ` N1; N2

&L

M2 ` � M1 ` �M2;M1 ` � � R

(Formation)` N

1 ` N 1LM `M `? ? R

(Re ection) ?` ? L ` 1 1R

(Formation)� ` N � ` N� _ � ` N _ L M ` � M ` �

M ` � ^ � ^ R

(Re ection)� ` N

� ^ � ` N� ` N

� ^ � ` N ^ L M ` �M ` � _ �

M ` �M ` � _ � _ R

(Formation) 0 ` N 0L M ` > >R

(Formation)� ` �� ` L

� ` �` �! �

! R

(Re ection)` � � ` N�! � ` N ! L

M ` � � `M ` � � R

(Order)� ` � ` Æ� ! ` �! Æ

! U ` Æ � ` � � ` Æ � U

We will distinguish three main kinds of structural rules, labelled by theletters L, R and S. The extensions of B obtained by the addition of anycombination of such rules can be organized in a cube, which is conceived asan architecture whose basis is B (see Figure 9).


BLRS

BLR BR

BL��

��

��

��

��

BRS

BLS��

��

��

��

��

BS

B

��

��

Figure 10. The cube of logics

In the cube, every logic with S satis�es the structural rules of weakeningand contraction:17

M ` NM;O ` P;N weakening

M;O;O ` N;N; PM;O ` N;P contraction

Every logic with \L" allows left contexts in any inference rule; every logicwith R allows right contexts in any inference rule. In particular, the cubesolves our initial problem, sketched in Figure 11. In fact, vertex BLRS,opposed to B represents classical logic, vertex BLR and vertex BLS repre-sent respectively Girard's linear logic and intuitionistic logic; �nally, vertexBS corresponds to paraconsistent quantum logic (see below). Moreover,since logics with R are simply the symmetric copy of logics with L, logicscontaining both L and R (BLRS, BLR) or logics containing neither L nor

17As we have seen, in B (as well as in linear logic) the connectives conjunction and dis-junction are splitted into a multiplicative and an additive connective. Such a distinctiondepends on the fact that there are two ways of formulating contexts in any operationalrule: this leads to a multiplicative and to an additive form for each rule. The multiplica-tive and additive formulation turn out to be equivalent, whenever the structural rules ofweakening and contraction hold. Hence, the distinction plays an essential role in linearlogic and a fortiori in basic logic (where weakening and contraction fail); at the sametime, it vanishes in classical logic and in orthologic.

QUANTUM LOGICS 219

R (BS, B), are symmetric. The study of quantum logics �nds place in thediagonal of symmetric logics, where a �ner distinction of structural rulescan be obtained.

Sequent calculus for Orthologic.The logic BS is non-distributive. Let us consider the fragment of BS re-stricted to the connectives ^ and _. If we want to obtain a quantum logicfrom it, what is still missing is an involutive negation, satisfying de Morgan.

This aim can be reached by extending the language and by adoptingGirard's negation. The key point is to assume as primitive symbols of thelanguage both the propositional variables and their duals. In other words,the propositional literals are assumed to be given in pairs, consisting of apositive element (written p) and of a negative one (written p?). On thisbasis, the negation of a formula is de�ned as follows:

p?? := p (� ^ �)? := �? _ �? (� _ �)? := �? ^ �?

By this choice, we obtain a calculus called basic orthologic and denoted by?BS (where the symbol ? reminds us that our calculus is applied to a duallanguage). Basic orthologic turns out to be equivalent to paraconsistentquantum logic (PQL). As we already know, PQL represents a weakeningof orthologic, that is obtained by dropping the non contradiction and theexcluded middle principles. Hence, in order to have a calculus for orthologic,it will be suÆcient to add such principles to our ?BS. This can be done bymeans of two new structural rules called transfer . The result is a calculusfor orthologic, which will be denoted by ?O.

The rules of ?O are the following (where (i) -(v) are the rules of ?BS 18

while (vi) express the transfer rules).

(i) � ` �

(ii)

� ` N � ` N� _ � ` N _L M ` � M ` �

M ` � ^ � ^R

(iii)� ` N

� ^ � ` N� ` N

� ^ � ` N ^LM ` �

M ` � _ �M ` �

M ` � _ � _R

(iv)M ` N

M;O ` P;N weakening

18Note that, in ?BS, weakening and contraction are redundant. In fact, one can showthat such a calculus admits elimination of contraction. At the same time, weakening onthe right and on the left can be simulated by ^L and _R, respectively. On this basis,PQL turns out to admit a very simple formulation, given by (i), (ii), (iii).


(v)

M;O;O ` N;N; PM;O ` N;P contraction

(vi)

M ` NM;N? ` tr1

M ` N`M?; N

tr2

It is not hard to prove:

THEOREM 149. ?BS is a calculus for paraconsistent quantum logic.

THEOREM 150. ?O is a calculus for orthologic.

As we have seen, our calculus ?O contains both p; q; r::: and p?; q?; r?::: .Moreover, for any rule of the calculus, the calculus shall contain also thesymmetric one. As a consequence, whenever the calculus produces a deriva-tion �, it will also produce the dual derivation �?, obtained substitutingevery axiom � ` � with the axiom �? ` �? and every occurrence of a rulewith an occurrence of its corresponding symmetric rule (e.g. ^R with _L).On this basis there holds:

LEMMA 151. The following rule is derivable for ?O:

M ` NN? `M?

Sketch of the proof One can easily see that M ` N is derivable bya derivation � if and only if N? ` M? is derivable by the symmetricderivation �?. �

The structure of the calculus ?O permits us to prove the following cut-elimination result.

THEOREM 152. ?O admits the elimination of the cuts.

O ` � M;� ` NM;O ` N cutL

O ` �; P � ` NO ` N;P cutR

Sketch of the proof Like in Gentzen, the cut-elimination procedure isobtained by induction on two parameters: the degree and the rank of thecut-formula19.

19Suppose a derivation and a sequent where a formula � occurs. Let us consider thepaths (i.e. the sequences of consecutive sequents) connecting that point with the pointwhere the formula � has been introduced, (by an axiom, or by weakening, or by anoperational rule whose principal formula was �). The rank of this particular occurrenceof � is the maximum among the lengths of all these paths. In other words, the rankrepresents the `maximum length' between the formula- occurrence we are examining andthe point where that occurrence has been introduced.The degree (or length) of a formula � is the number of the connectives occurring in �.

QUANTUM LOGICS 221

The calculus ?O permits us to overcome in a simple way two questionsthat usually make cut elimination for orthologic so complicated: (i) con-straints on contexts and (ii) negation. We give a sketch of the proof, con-sidering the two points. The �rst problem is solved by visibility, while thesecond one is solved by symmetry.

(i) As we have seen, in any calculus for quantum logic the rule thatintroduces _ on the left (here indicated with _L) must have an emptycontext on the left. Now consider, for a generic calculus, the derivation

� ` ^ Æ � ` ^ Æ� _ � ` ^ Æ _L M; ` �

M; ^ Æ ` �

M;� _ � ` �cutL

In this derivation, the cut-formula is principal on the right premiss;hence the right rank is 1. In such a situation, Gentzen's procedure tolower the rank must operate on the left; this would necessarily producethe two derivations

� ` ^ Æ M; ^ Æ ` �

M;� ` �cutL

� ` ^ Æ M; ^ Æ ` �

M;� ` �cutL

Now, one would like to conclude by applying _L, in order to obtainM;� _ � ` �. However, this step is here not allowed, unless M isempty. Such a problem does not arise for the calculus ?O, because,by visibility, every principal formula has an empty context.

(ii) In ?O the only rules about negation are the structural rules of transfer.Let us consider a derivation of the form:

O ` �?

.... �

M ` �M;�? ` tr1

M;O ` cutL

We can reduce the rank in a quick way, by exploiting symmetry. Infact, Girard's negation has the nice property that every formula �and its dual �? have exactly the same degree. The same idea canbe extended to derivations, and hence to the rank of a cut. As wehave seen in Lemma 151, whenever we have a derivation � for thesequent M ` N , we also have the dual derivation �? , which derivesN? ` M?. The two derivations � and �? have exactly the same(symmetrical) structure. Hence in particular, if � is principal, �? isprincipal. If � has rank r, then also �? will have the same rank r. In


C

S

? ?O

S

trBS

S

L? ?OL

tr?B

Figure 11. The diagonal of the cube

such a situation, in order to raise the cut rule, we can substitute �?

by � ( ipping derivation). As a consequence, the initial derivationwill be simply reduced to:

O ` �?.... �?

�? `M?

O `M? cutL

O;M ` tr1

�

Quantum logics and classical logicWe will now consider the symmetric diagonal of the cube in the diagramin Figure 11. In our diagram, the calculus ?O appears as an intermediatepoint between basic orthologic and classical logic. Similarly, we have anotherintermediate point between basic logic and linear logic: this is given by ?B+ tr, which represents the common denominator for orthologic and linearlogic (we will call it \ortholinear logic" ?OL). In the same way, ?B turnsout to be the common denominator of basic orthologic and linear logic. Onthis basis, we obtain a whole system of quantum logics, which are all cut-free. The last of our logics, ?B + tr, seems to be a good candidate in orderto represent a linear quantum logic in the sense of Pratt [1993].

So far we have only dealt with a fragment of basic logic, which has noimplication connective. By means of this linguistic restriction, we haveeasily proved the equivalence between our calculi and the usual formula-tions of paraconsistent quantum logic and of orthologic. However, the samemethods can be naturally applied to the complete versions of our calculi,preserving cut-elimination and ipping of derivations. In this way, we willhave a primitive implication connective! (together with its dual ) in allthe logics we have considered. An interesting question to be investigatedconcerns the possibility of physical interpretations of such new connectives.

QUANTUM LOGICS 223

In the diagram above, we have still a question mark concerning the pathfrom orthologic to classical logic. Our question can be solved as follows:

THEOREM 153. A calculus for classical logic is obtained from a calculusfor orthologic by adding a pair of structural rules, named separation:

(vii)

M;O `M ` O? sep1

` N;PN? ` P sep2

It is easy to see that, in the framework of ?O, separation rules allow usto derive the following full cut

M1 ` �;N1 M2; � ` N2

M1;M2 ` N1; N2cut

The converse is also true: full cut allows us to derive separation (by cuttingwith ` A;A?). In this sense, separation and cut rule are equivalent: addingeither of them to orthologic gives rise to one and the same logic. Theorem153 then expresses, with a more e�ective20 content, the well known factthat adding a full cut rule to orthologic yields classical logic (cf. [Dummett,1976], [Cutland and Gibbins, 1982]).

It is natural to ask what is the meaning of sep. In the same way as thetr rules are equivalent to tertium non datur and non contradiction, the seprules turn out to be equivalent to reductio ad absurdum 21

M;�? `M ` � RAA

Let us consider again our Figure 11, where the question marks have beensubstituted by sep. Given the logic B as a basic calculus, which contains thefundamental rules for the connectives, several structural rules can be added:each rule permits us to reach a \superior" logic. The strongest element isrepresented by classical logic, which can be characterized as ?B + S + tr +sep. With respect to our formulation of classical logic (denoted by ?C) allthe other logics in the diagram can be described as `substructural logics:for, they can be obtained by dropping some structural rules. This situationholds in particular for quantum logics, which turn out to be simpler andmore basic than classical logic, from the proof-theoretical point of view.

As we have seen, the examples of quantum logic (we have considered sofar) are, at the same time, substructural with respect to classical logic andsubstructural one with respect to the other. On this basis, on can prove

20For, in a sequent calculus cut should represent a metarule, that is should be elim-inable.21In [Gibbins, 1985], pag.361, Gibbins shows that dropping the rule RAA has a direct

justi�cation in terms of quantum mechanics, and this is the only case of direct justi�ca-tion, among all the rules which must be restricted in quantum logic.


some embedding theorems, by convenient restriction of our structural rulesto suitable kinds of formulas, by means of special modalities. In the case oflinear logic, exponentials have been introduced in order to express weakeningand contraction. In the case of quantum logics, instead, we should obtainrules of separation and of transfer in a suitable way. How to express theseparation rules in orthologic, in order to obtain an embedding of classicallogic into orthologic? Given ?O, let us �rst assume in the language, besidesthe literals p and p?, two new kinds of literals, #p and #p?. This permits usto obtain a new kind of formulas, that will be named \separable formulas",formulated as follows:

#(p) := #p #(p?) := #p? #(#p) := #p #(#p?) := #p?

for literals;#(� Æ �) := #� Æ #�

for every binary connective Æ.Separable formulas are precisely those formulas that satisfy the separationrules, which are then de�ned as follows:

(vii0)

M; # O `M `# O? # sep1

`# N;P# N? ` P # sep2

where formulas in M , N are any kind of formulas, while formulas in #M ,#N are separable formulas. We can now introduce the system #?O, whichis de�ned by the rules of ?O and by the rules #sep. In this system, thesign # plays the role of a modality (which behaves as an unary monotonicconnective: if M ` N , is a derivable sequent in #?O, then also #M ` #N isa derivable sequent).

Consider now the system ?C for classical logic, and let us describe # asa map from formulas of the language of ?C into formulas of the languageof #?O. It is easy to show, by induction on the depth of the derivation,that:

THEOREM 154. For every M , N , M ` N is derivable in ?C if and onlyif #M ` #N is derivable in #?O.

Sketch of the proof A proof can be obtained by a natural transformationof a similar proof, given in [Battilotti, 1998] for the case of classical logicand basic orthologic. �

As a consequence we obtain an embedding of ?C in #?O. Formulas ofthe kind #� can be interpreted as \the classical part of #?O". Similarly to?, the sign # does not represent here a connective; therefore, there is noneed of introduction rules. One can prove that sequents like #� ` � or like� ` #� are not provable (di�erently from the exponentials in linear logic).

QUANTUM LOGICS 225

In this way, the system #?O is simply a way to represent the coexistence ofclassical and quantum logic: it does not assert that \classical" propositionsare stronger or weaker than \quantum" propositions.

18 CONCLUSION

Some general questions that have been often discussed in connection with(or against) quantum logic are the following:

(a) Why quantum logics?

(b) Are quantum logics helpful to solve the diÆculties of QT?

(c) Are quantum logics \real logics"? And how is their use compatiblewith the mathematical formalism of QT, based on classical logic?

(d) Does quantum logic con�rm the thesis that \logic is empirical"?

Our answers to these questions are, in a sense, trivial, and close to aposition that Gibbins (1991) has de�ned a \quietist view of quantum logic".It seems to us that quantum logics are not to be regarded as a kind of \clue",capable of solving the main physical and epistemological diÆculties of QT.This was perhaps an illusion of some pioneering workers in quantum logic.Let us think of the attempts to recover a realistic interpretation of QT basedon the properties of the quantum logical connectives22.

Why quantum logics? Simply because \quantum logics are there!" Theyseem to be deeply incorporated in the abstract structures generated by QT.Quantum logics are, without any doubt, logics . As we have seen, theysatisfy all the canonical conditions that the present community of logiciansrequire in order to call a given abstract object a logic. A question that hasbeen often discussed concerns the compatibility between quantum logic andthe mathematical formalism of quantum theory, based on classical logic.Is the quantum physicist bound to a kind of \logical schizophrenia"? At�rst sight, the compresence of di�erent logics in one and the same theorymay give a sense of uneasiness. However, the splitting of the basic logicaloperations (negation, conjunction, disjunction,...) into di�erent connectiveswith di�erent meanings and uses is now a well accepted logical phenomenon,that admits consistent descriptions. Classical and quantum logic turn outto apply to di�erent sublanguages of quantum theory, that must be sharplydistinguished.

Finally, does quantum logic con�rm the thesis that \logic is empirical"?At the very beginning of the contemporary discussion about the nature oflogic, the claim that the \right logic" to be used in a given theoreticalsituation may depend also on experimental data appeared to be a kind of

22See for instance [Putnam, 1969]


extremistic view, in contrast with a leading philosophical tradition accordingto which a characteristic feature of logic should be its absolute independencefrom any content.

These days, an empirical position in logic is generally no longer regardedas a \daring heresy" . At the same time, as we have seen, we are facingnot only a variety of logics, but even a variety of quantum logics . As aconsequence, the original question seems to have turned to the new one :to what extent is it reasonable to look for the \right logic" of QT?

ACKNOWLEDGEMENTS

The authors are grateful to Giulia Battilotti and Claudia Faggian who wrotea section of this chapter.

M. L. Dalla ChiaraUniversit�a Firenze, Italy.

R. GiuntiniUniversit�a Cagliari, Italy.

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[Ludwig, 1983] G. Ludwig. Foundations of Quantum Mechanics, Vol. 1, Springer, Berlin,1983.

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MARTIN BUNDER

COMBINATORS, PROOFS AND

IMPLICATIONAL LOGICS

1 INTRODUCTION

In this chapter we �rst look at operators called combinators. These arevery simple but extremely powerful. They provide a means of doing logicand mathematics without using variables, are powerful enough to allow thede�nition of all recursive functions and have more recently been used as abasis for certain \functional" computer languages.

We are interested in another use here which involves the functionalcharacter or type possessed by many combinators. Each type can beinterpreted as a theorem of the intuitionistic implicational logic H! andcombinators possessing that type can be interpreted as Hilbert-style proofsof that theorem. Weaker sets of combinators can be used to representproofs in sublogics of H! , these include the substructural logics, suchas the relevance logics R! and T!. There is a further interpretation ofcombinators and types as programs and speci�cations which we will notdiscuss here.

Next we look at lambda calculus. This also allows the de�nition of allrecursive functions and has also been used in foundations of mathematicsand computer language development. Many lambda terms also have typesand these again are the theorems of H!. The lambda terms representnatural deduction style proofs of these theorems.

In the third section of this chapter we look at translations from combi-nators to lambda terms and vice versa. For the combinators and lambdaterms that represent proofs in H! these translations are well known, forthose corresponding to proofs in weaker logics they are quite new.

In a fourth section we develop a new algorithm which, given an implica-tional formula, allows us to �nd lambda terms representing natural deduc-tion style proofs of the formula or demonstrates that the formula has noproof.

Most implicational substructural logics are speci�ed by substructuralrules or by axioms and not by rules in the natural deduction form. Ourtranslation procedure, together with the algorithm, provides us with a sim-ple constructive means of �nding Hilbert-style proofs in many of these logics.As the translation procedure tells us which lambda terms are translatableinto which sets of combinators, the algorithm can be directed to look onlyfor the lambda terms of the appropriate kind. The algorithm is inherently�nite; for any given formula, and for many logics, bounds for the proofsearches can be written down.


230 MARTIN BUNDER

The H! algorithm has been implemented by Anthony Dekker as Brouwer7.9.0 (see [Dekker, 1996]) and, for any implicational formula, produces a�-term proof (or even 50 alternative proofs) or a guarantee that there isno proof, virtually instantly. The implementation has more recently beenextended by Martijn Oostdijk as LambdaCal2, (see [Oostdijk, 1996]) tocover the other implicational systems in this chapter as well as certain sys-tems with other connectives. This implementation supplies combinator andlambda calculus proofs.

2 COMBINATORY LOGIC

Combinators are operators which manipulate arbitrary expressions by can-cellation, duplication, bracketing and permutation. Combinators were �rstde�ned by Sch�on�nkel in his 1924 paper and rediscovered by Curry in [1930].

To illustrate their use we consider the following examples:let Axy (rather than A(x; y)) represent x+ y. The commutative law for

addition can then be written as

Axy = Ayx:

Given a combinator C with the property:

Cxyz = xzy

this becomesAxy = CAxy

which could be simply written, without variables, as

A = CA:

Given an identity combinator I, i.e. one such that

Ix = x

we can write0 + x = x

asA0x = x

orA0x = Ix

and so, without variables, asA0 = I:

x+ 0 = x

COMBINATORS, PROOFS AND IMPLICATIONAL LOGICS 231

would beCA0 = I:

Sch�on�nkel found that only two combinators, K and S, were enough tode�ne all others.

We will now introduce these, other combinators, and our method of writ-ing functional expressions (such as Axy rather than A(x; y)) more formally.

2.1 Combinators and Application

DEFINITION 1 (Combinator).

1. K and S are combinators.

2. If X and Y are combinators so is (XY ).

(The operation in (2) is called application.)

Though it is possible, it is often not convenient to work without vari-ables; we therefore introduce terms which are made up of combinators andvariables using application. Other constants could also be included in (1)below.

DEFINITION 2 (Term).

1. K, S, x; y; z; : : : ; x1; x2; : : : are terms

2. If X and Y are terms so is (XY ).

Notation We use association to the left for terms. This means that ourAxy is short for ((Ax)y). A binary function over the real numbers such asA is therefore interpreted as a unary function from real numbers into theset of unary functions from real numbers to real numbers.

The process of going from CXY Z to XZY or from IX to X is calledreduction. This is de�ned as follows:

DEFINITION 3 (Reduction). The relation X . Y (X reduces weakly toY ) is de�ned as follows:

(K) KXY . X

(S) SXY Z . XZ(Y Z)

(�) X . X

(�) X . Y ) UX . UY

(�) X . Y ) XU . Y U

(�) X . Y and Y . Z ) X . Z

232 MARTIN BUNDER

(K) and (S) are called the reduction axioms for K and S.

DEFINITION 4 (Weak equality). X = Y if this can be derived from theaxioms and rules of De�nition 3 with \=" instead of \ .", together with

(�) X = Y ) Y = X .

The formal system consisting of at least De�nition 1 and the postulates inDe�nitions 3 we call combinatory logic. Other axioms and rules may beadded.

We now show how the combinators we met earlier, and others, can bede�ned. We use \�" for \equals by de�nition".

DEFINITION 5.I � SKK

B � S(KS)KC � S(BBS)(KK)B0 � CBW � SS(KI)S0 � B(BW)(BBB0)

Each of these de�ned combinators has a characteristic reduction theorem:

THEOREM 6.

1. IX . X

2. BXYZ . X(Y Z)

3. CXYZ . XZY

4. B0XY Z . Y (XZ)

5. WXY . XY Y

6. S0XY Z . Y Z(XZ)

Proof.

1. SKKX . KX(KX) by (S)so SKKX . X by (K) and (�)2. S(KS)KXY Z . KSX(KX)Y Z by (�) and (�)

. S(KX)Y Z by (K) and (�)

. KXZ(Y Z) by (S)

. X(Y Z) by (K) and (�).so S(KS)KXY Z . X(Y Z) by (�) �

If M(X1; : : : ; Xn) is a term made up by application using zero or moreoccurrences of each of X1; : : : ; Xn; we can �nd a combinator Z such thatZX1X2 : : :Xn . M(X1; : : : ; Xn).


This property is called the \combinatory completeness" of the com-binatory logic based on K and S.

The combinator Z above is represented by

Z � [x1]([x2](: : : ([xn]M(x1; : : : ; xn)) : : :))

where each [xi](: : :) is called a bracket abstraction.Bracket abstractions can be de�ned in various ways, a simple de�nition,

involving S and K, is as follows:

DEFINITION 7 (Bracket abstraction [xi]).

(i) [xi] xi � I

(k) [xi] Y � KY if xi 62 Y(�) [xi]Y xi � Y if xi 62 Y(s) [xi]Y Z � S([xi]Y )([xi]Z),

where xi 62 Y stands for xi does not appear in Y .

The above clauses must be used in the order given i.e. (ik�s). In the order(iks�), we would always obtain S([xi]Y )I for [xi]Y xi, if xi 62 Y , instead ofthe simpler Y .

Repeated bracket abstraction as in [x1]([x2](: : : ([xn]M) : : :)) we will writeas [x1; x2; : : : ; xn]M .

EXAMPLE 8.

[x1; x2; x3] x3(x1x3) � [x1; x2]S([x3]x3)([x3](x1x3)) by (s)� [x1; x2]SIx1 by (i) and (�)� [x1]K(SIx1) by (k)� S(KK)(SI) by (k) and (�):

S(KK)(SI)x1x2x3 . KKx1(SIx1)x2x3. K(SIx1)x2x3. SIx1x3. Ix3(x1x3). x3(x1x3):

Bracket abstraction has the following property which we call (�) forlambda abstraction in Section 3.

THEOREM 9. ([x]X)Y . [Y=x]X where [Y=x]X is the result of substitutingY for all occurrences of x in X.

Proof. By a simple induction on the length of X . �

234 MARTIN BUNDER

From this theorem we get:

([x]X)x . X

and if x1; : : : ; xn 62 X1X2 : : :Xn,

([x1; : : : ; xn]X)X1 : : : Xn . [X1=x1; : : : ; Xn=xn]X ;

where [X1=x1; : : : ; Xn=xn]X is the result of substituting simultaneously X1

for all occurrences of x1 in X; : : : ;Xn for all occurrences of xn in X .

2.2 Combinators, Types, Proofs and Theorems

If a term X is an element of a set � (which we write as X 2 � or X : �below) we have as

KXY = X;KXY 2 �:

If Y 2 � we have that KX is a function from the set � into the set �,i.e. in the usual mathematical notation:

KX : � ! � ;

where � ! � represents the set of all functions from � into �.From this it follows that K is a function from � into � ! �, so we can

write:K : �! (� ! �) :

Sets such as the �; � and � ! (� ! �) above will be denoted byexpressions called types. Types are de�ned as follows:

DEFINITION 10 (Types).

1. a; b; c; : : : are (atomic) types.

2. If � and � are types so is (�! �).

For types we use association to the right, �! (� ! �) can therefore bewritten as �! � ! �.

The type variables or atomic types can be interpreted as arbitrary sets,the compound types then represent sets of functions.

Above we arrived at K : � ! � ! �; such a derivation we call a typeassignment, we call � ! � ! � the type of K and K an inhabitant of�! � ! �.

Type assignments can be more formally derived from:

DEFINITION 11 (Type Assignment).

1. Variables can be assigned arbitrary types


2. If X : �! � and Y : � then (XY ) : �.

3. If Xx : �; x 62 X and x : � then X : � ! �.

We will illustrate this by assigning a type to S.

If we let x : �! � ! ; y : �! � and z : � we have by (2) xz : � ! and yz : � and so xz(yz) : which is, by (S), Sxyz : .

Now by (3) Sxy : �! ,

Sx : (�! �)! �!

and S : (�! � ! ) ! (�! �) ! �! :

In particular we have

S : (a! b! c) ! (a! b) ! a! c ;

and it can be seen, from the work above, that every type of S has to be asubstitution instance of this. A type with this property we call the princi-pal type scheme (PTS).

The PTS of K is given by

K : a! b! a :

Notice that the types of K and S are exactly the axioms of H!, intu-itionistic implicational logic, when ! is read as implication and �; �; : : :are read as well formed formulas or propositions.

De�nition 11.2 and 3 can be rewritten as:

!e

X : �! � Y : �

XY : �

and

!i

[x : �]....

Xx : �

X : �! � (x 62 X)

We have, considering only the right hand sides of the :s, the rules ofinference of a natural deduction formulation of H!.

If we take the types of K and S as axiom schemes and use only the partsof !e to the right of the :s, we have a Hilbert-style formulation of H!.What appears on the left hand side of the �nal step in such a proof givesus a unique representation of a proof of the theorem expressed on the rightof the :.

236 MARTIN BUNDER

EXAMPLE 12. If we abbreviate (� ! ! Æ) ! (� ! ) ! � ! Æ by� we have:

K :�! ( ! Æ)! �S :�

KS : ( ! Æ)! �S : (( ! Æ)! �)! (( ! Æ)! � ! ! Æ)! ( ! Æ)! (� ! )! � ! Æ

S(KS) : (( ! Æ)! � ! ! Æ)! ( ! Æ)! (� ! )! � ! ÆK : ( ! Æ)! � ! ! Æ

S(KS)K : ( ! Æ)! (� ! )! � ! Æ

We note that each S or K in S(KS)K represents the use of an axiomand each application a use of !e. We also note that the above representsa type for the combinator B of De�nition 5.

Some combinators do not have types, for example if we want to �nd atype for SSS we would proceed as follows:

let S : (� ! ! Æ) ! (� ! ) ! � ! Æand S : (� ! �! �) ! (� ! �) ! � ! �

then we have putting � ! �! � = �; � ! � = and � ! � = Æ:

SS : ((� ! �! �) ! � ! �) ! (� ! �! �) ! � ! � :

Now with S : (� ! � ! �) ! (� ! �) ! � ! �, we would need, toassign a type to SSS :

� ! �! � = �! � ! � ;� = �! � ;

and � = �! � :

However this requires both � = � and � = �! � which is impossible.

Also there are types that have no inhabitant, for example a ! b and((a! b)! a)! a. In fact:

THEOREM 13.

1. If a type � has a combinator inhabitant, � is a theorem of the intu-itionistic implicational logic H!.

2. If � is a theorem of H!, then it is the type of a combinator.


Proof.

1. It can be proved by an easy induction on the length of X that

x1 : �1; : : : ; xn : �n ` X : �

implies

�1; : : : ; �n ` �

2. It can be proved by an induction on the length of the deduction leadingto

�1; : : : ; �n ` �that there are variables x11; : : : x1m1 ; : : : ; xnmn

and a term X suchthat

FV (X) � fx11; : : : ; xnmng

and

x11 : �1; : : : ; x1m1 : �1; x21 : �2; : : : ; xnmn: �n ` X : � :

For more details on this see Hindley [1997, Section 6B2 and Section 6B5].�

The isomorphism between inhabitants and types and proofs and theoremsof H!, which can be extended to �t programs and speci�cations, is calledthe Curry-Howard or Formulas-as types isomorphism.

Curry was the �rst to recognise the relation between types and theoremsof H! (see [Curry and Feys, 1958]). The idea was taken up and extendedto other connectives and quanti�ers in Lauchli [1965; 1970], Howard [1980]

(but written in 1969), de Bruin [1970; 1980] and Scott [1970]. Recently itwas extended to include a large amount of mathematics in Crossley andShepherdson [1993].

2.3 Types and Weaker Logics

If we consider an arbitrary set of combinators Q, we can de�ne the set ofQ-combinators and Q-terms as follows:

DEFINITION 14 (Q-terms).

1. Elements of Q and x; y; z; : : : ; x1; x2; : : : are Q-terms

2. If X and Y are Q-terms so is (XY ).

238 MARTIN BUNDER

DEFINITION 15 (Q-combinators). A Q-term containing no variables is aQ-combinator.

The formal system consisting of at least De�nition 15 and the postulatesof De�nition 3, with the reduction axioms (K) and (S) replaced by onesappropriate to Q, is called Q-combinatory logic. Q-logic is the implica-tional logic whose axioms are the types of the combinators in Q and whoserule is ! e.

Thus combinatory logic, as de�ned before, is fS;Kg-(or simply SK-)combinatory logic, terms are SK-terms and combinators are SK-combinators.

DEFINITION 16 (Weaker sets of combinators). A set Q1 of combinatorsis said to be weaker than a set Q2, if for every Q1-combinator X thereis a Q2-combinator Y with the same reduction theorem. (In that case wesay X is Q2-de�nable.) Also there must be a Q2-combinator which is notQ1-de�nable.

DEFINITION 17 (Weaker combinatory logics). Q1-combinatory logic isweaker than Q2-combinatory logic if Q1 is weaker than Q2.

BCKW-combinatory logic is just as strong as SK-combinatory logic asB,C,K and W are (by De�nition 5) SK-de�nable and S is also BCKW-de�nable (S � B(BW)(BC(BB))).

BCW and BCIW-combinatory logics are both weaker than SK-combin-atory logic as S is not de�nable using B,C,I and W.

It is clear that BCK- and BCIW-combinatory logics are not combina-torially complete.

The set of types of the BCK-combinators can easily be seen to be theset of theorems generated by !e using the types of B,C and K. This wecall BCK-(implicational) logic, which is a subsystem of H!.

In general, if all the combinators of Q1 [Q2 have types and Q1 is weakerthan Q2 then Q1-logic is weaker than Q2-logic.

As before, given a Q-combinator, a Q-theorem and its proof can be reado�.

EXAMPLE 18. Determine the type of BC(BK) and the BCK-proof ofthis as a BCK-theorem.

B : (�! �) ! ( ! �) ! ! �C : (Æ ! � ! �) ! � ! Æ ! �

BC : ( ! Æ ! � ! �) ! ! � ! Æ ! �(Here � = Æ ! � ! �, � = � ! Æ ! � :)

K :�! � ! �BK : (� ! �) ! � ! � ! �

(Here � = �; � = � ! � and = �.)BC(BK) : (� ! �) ! � ! � ! �

(Here = � ! �, Æ = �; � = � and � = �)


2.4 Combinator Reduction and Proof Reduction

We illustrate here what happens to (part of) a proof represented by a com-binator KXY or SXY Z when this combinator is reduced by (K) or (S).

The original proof involving KXY must look like:

K : �! � ! �D1

X : �

KX : � ! �D2

Y : �

KXY : �D3

with D1; D2 and D3 representing other proof steps.With the reduction of KXY to X the proof reduces (or normalises) to:

D1

X : �D3

If the proof involving SXY Z is:

D2

Y : �! �

S : (�! � ! )! (�! �)! �! D1

X : �! � !

SX : (�! �)! �!

SXY : �! D3

Z : �

SXY Z : D4

With the reduction of SXY Z to XZ(Y Z) the proof reduces (or nor-malises) to:

D1

X : �! � ! D3

Z : �

XZ : � !

D2

Y : �! �D3

Z : �

Y Z : �

XZ(Y Z) : D4

It may be, by the way, if D3 is a particularly long part of the proof, thatthe \reduced proof" is actually longer than the original. In the same way,if Z is long, XZ(Y Z) may be longer than SXY Z.

DEFINITION 19. If a term has no subterms of the form KXY or SXY Z,the term is said to be in normal form.

Not every combinator has a normal form, for example WI(WI) andWW(WW) do not, but every combinator that has a type also has a normalform.

240 MARTIN BUNDER

A combinator in normal form will represent a normalised proof, thesehowever are not unique. For example:

K : a! a! a

and alsoKI : a! a! a:

3 LAMBDA CALCULUS

Lambda calculus, like combinatory logic, provides a means of representingall recursive functions. It is, these days, much used as the basis for functionalcomputer languages. Extensions of the typed lambda calculus we introducein Section 3.2 below, also have applications in program veri�cation.

Lambda calculus was �rst developed by Church in the early 30s (see[Church, 1932; Church, 1933]) as part of a foundation of logic and mathe-matics. This was also the aim of Curry's \illative combinatory logic", butboth Church's and Curry's extended systems proved to be inconsistent.

The use of the lambda calculus notation is best seen through an examplesuch as the following:

If the value of the sin function at x is sinx and the value of the logfunction at x is logx, what is the function whose value at x is x2? Usuallythis function is also called x2. The lambda calculus allows us to eliminatethis ambiguity by using �x:x2 for the name of the function.

3.1 Lambda terms and lambda reductions

We will now set up the system more formally:

DEFINITION 20 (Lambda terms or �-terms).

1. Variables are �-terms.

2. If X is a �-term and x a variable then, the abstraction of X withrespect to x, (�x:X) is a �-term.

3. If X and Y are �-terms then (XY ), the application of X to Y , is a�-term.

(�x:X) is interpreted as the function whose value at x is X .

Given this we would expect the following to hold:

EXAMPLE 21.(�x: sin x) = sin

((�x: sin x)x) = sinx((�x: sin x)�) = sin�

((�x:x2)2) = 4((�x:2)x) = 2


(�x:2) is the constant function whose value is 2.

For terms formed by application we use association to the left as for termsof combinatory logic. Repeated �-abstraction as in (�x1:(�x2 : : : (�xn:X) : : :))we abbreviate to �x1:�x2 : : : �xn:X or to �x1x2 : : : xn:X .

Note that while �x1x2 : : : xn:X represents a function of n variables, it isalso a function of one variable whose value, �x2x3 : : : xn:X at x1, is also afunction (if n � 2).

The process of simplifying (�x: sin x)x to sinx or (�x:x2)2 to 22 is called�-reduction . To explain this we need to de�ne free and bound variablesand, using these, a substitution operator.

As in combinatory logic we use � for equality by de�nition or identity.

DEFINITION 22 (Free and bound variables, closed terms).

1. x is a free variable in x.

2. If x is free in Y or Z then x is free in (Y Z).

3. If x is free in Y and y 6� x, x is free in �y:Y .

4. Every x that appears in �x:Y is bound in �x:Y .

We write FV (X) for the set of free variables of X .A closed term is one without free variables.

EXAMPLE 23.

1. If X � �xy:xyx; X is a closed term and x and y are bound in X .

2. If X � �xy:xyzx(�u:zx)w; x; y and u are bound in X and FV (X) =fz; wg

3. If X � x(�x:xy)x, the �rst and last occurrences of x are free occur-rences of x. Both the occurrences of x in �x:xy are bound.

DEFINITION 24. ([Y=x]X - the result of substituting Y for all free oc-currences of x in X)

1. [Y=x]x � Y2. [Y=x]y � y if y is an atom, x 6� y3. [Y=x](WZ) � ([Y=x]W )([Y=x]Z)

4. [Y=x](�x:Z) � �x:Z5. [Y=x](�y:Z) � �y:[Y=x]Z if y 6� x

and, y 62 FV (Y ) or x 62 FV (Z).

242 MARTIN BUNDER

6. [Y=x]�y:Z � �z:[Y=x][z=y]Zif y 6� x; y 2 FV (Y ); x 2 FV (Z) and z 62 FV (Y Z).

As we saw in Example 21, a �-term of the form (�x:X)Y can be simpli�edor \reduced". This kind of reduction is speci�ed by the following axiom:

(�) (�x:X)Y . [Y=x]X :

A �-term of the form (�x:X)Y is called a �-redex.The following axiom allows a change of bound variables and is called an

�-reduction.

(�) �x:X . �y:[y=x]X if y 62 FV (X):

Rules (�), (�) and (�) of De�nition 3 together with:

(�) X . Y ) �x:X . �x:Y ;

allow us to perform �- and �-reductions within a term.

DEFINITION 25 (�- and ��-reduction).The reduction . speci�ed by (�), (�), (�), (�), (�), (�) and (�) is called

�-reduction.This becomes ��- (or just �-) reduction if the following axiom is added:

(�) �x:Xx . X if x 62 FV (X):

We denote the two forms of reduction, when we wish to distinguish them,by .� and .��. If in a ��-reduction (�) is not used we sometimes write .�instead of .��.

We will call �-, �- and �- reductions �-reductions to distinguish themfrom combinator reductions.

EXAMPLE 26.

1. �x1:x1((�x2:x3x2x2)x1) .� �x1:x1(x3x1x1)

2. �x3:(�x1:x1x2(�x2:x3x1x2))(�x1:x1).� �x3:(�x1:x1x2(x3x1))(�x1:x1).� �x3:(�x1:x1)x2(x3(�x1:x1)).� �x3:x2(x3(�x1:x1))

so �x3:(�x1:x1x2(�x2:x3x1x2))(�x1:x1).�� x3:x2(x3(�x1:x1))

A term of the form �x:Xx where x 62 FV (X), is called an �-redex.A term with no �-redexes is said to be in �-normal form. One without

�- and �- redexes is in ��-normal form. One without �-redexes is said to


be in �-normal form. �-, �- and ��-normal forms are unique (see [Hindleyand Seldin, 1986]).

DEFINITION 27 (�-,�- and ��-equality).The relation =� is speci�ed by (�); (�); (�); (�); (�) and (�), all with

=� for . and

(�) X =� Y ) Y =� X .

The relation =� is speci�ed by all the above postulates, as well as (�),with =� for . and =�.

The relation =�� is speci�ed by all the =� postulates, as well as (�), with=�� replacing =� and ..

Note that the weak equality of combinatory logic obeys all the abovepostulates (with [..] for �..) except (�) and (�). It is possible to extendweak equality by means of some extra equations involving combinators tomake (�) and/or (�) admissible (see [Curry and Feys, 1958, Section 6C4]).We will call the corresponding equalities for combinatory logic =� and =��

respectively.

3.2 Lambda Terms, Types, Proofs and Theorems

If a term X is an element of a set � and the variable y is in a set �; �y:Xwill represent a function from � into � i.e. �y:X : � ! �. We will denotesets such as these by the types introduced in De�nition 10. We assign typesto �-terms in a similar way to the assignment to combinators.

DEFINITION 28 (Type Assignment).

1. Variables can be assigned arbitrary types.

2. If X : �! � and Y : � then (XY ) : �

3. If X : � and y : � then �y:X : � ! �.

EXAMPLE 29.

1. If x2 : a! a! b; x3 : a then x2x3x3 : b and �x3:x2x3x3 : a! b.

If x1 : (a! b)! c then x1(�x3:x2x3x3) : c, so �x1x2:x1(�x3:x2x3x3) :((a! b)! c)! (a! a! b)! c.

2. If x1 : � ! � then for x1x1 to have a type we must have � = � ! �which is impossible. Hence x1x1 and so �x1:x1x1 have no types.

DEFINITION 30. If X is a closed lambda term and ` X : � is derived bythe type assignment rules then X is said to be an inhabitant of � and � atype of X .

244 MARTIN BUNDER

Note that, as with combinatory logic, if we ignore the terms on the leftof the :, we have in De�nition 20.2 and 3 the elimination and introductionrules for ! of the natural deduction version of H!. For any Y : �, thetypes of the variables free in Y constitute the uncancelled hypothesis whichyield �. When Y has no free variables, � is a theorem of H! and Yrepresents a natural deduction proof of �. This can be best seen when thetype assignment rules are written in tree form as in the example below.

EXAMPLE 31.

(6 2)x2 : a

(6 1)x1 : a! b! c

(6 2)x2 : a

x1x2 : b! c(6 3)

x3 : (b! c) ! a! d

x3(x1x2) : a! d

x3(x1x2)x2 : d

�x2:x3(x1x2)x2 : a! d�(2) (6 4)

x4 : (a! d) ! e

x4(�x2:x3(x1x2)x2) : e

�x3:x4(�x2:x3(x1x2)x2) : ((b! c) ! a! d) ! e�(3)

�x1x3:x4(�x2:x3(x1x2)x2) : (a! b! c) !((b! c) ! a! d) ! e

�(1)

�x4x1x3:x4(�x2:x3(x1x2)x2) : ((a! d) ! e) !(a! b! c) ! ((b! c) ! a! d) ! e

�(4)!i

!i

!i

!e

!i

!e

!e

!e

At a !i step the assumption cancelled is indicated by -(n). At this timethe assumption is relabelled with (6 n).

In the lambda term proof each application represents a use of !e andeach �-abstraction a use of !i. It follows that:

THEOREM 32.

1. Every type of a �-term is a theorem of H!.

2. Every natural deduction style proof of a theorem of H! can be repre-sented by a closed �-term with that theorem as its type.

Proof.

1. We will prove the following more general result by induction on X :

If FV (X) � �x1; : : : ; xn and if

x1 : �1; : : : ; xn : �n ` X : � �(a)

then

�1; : : : ; �n ` � �(b)

in H!.


If X is an atom, X = xi for some i; 1 � i � n and � = �i, so (b)holds.

If (a) is obtained from

x1 : �1; : : : ; xn : �n; xn+1 : �n+1 ` Y : � ;

where � = �n+1 ! � and X = �xn+1:Y then by the inductionhypothesis

�1; : : : ; �n; �n+1 ` �is valid in H! and so also (b).

If X � UV wherexi1 : �i1 ; : : : ; xik : �ik ` U : �! �;xj1 : �j1 ; : : : ; xim : �jm ` V : �

and�xi1 : �i1 ; : : : ; xin : �ik

[ �xj1 : �j1 ; : : : ; �jm : �jm

=�x1 : �1; : : : ; xn : �n

we have by the induction hypothesis in H!:

�i1 ; : : : ; �ik ` �! �

and

�j1 ; : : : ; �jm ` �and so (b).

2. We show by induction on the length of a proof in H! of

�1; : : : ; �n ` � (c)

that there is a term X with FV (X) � �x1; : : : ; xn such that

x1 : �1; : : : ; xn : �n ` X : � (d)

If � is one of the �is this is obvious with X = xi.

If (c) is obtained by the !i rule from

�1; : : : ; �n; ` Æ

where � = ! Æ, then by the induction hypothesis we have

x1 : �1; : : : ; xn : �n; xn+1 : ` Y : Æ

246 MARTIN BUNDER

where FV (Y ) � �x1; : : : ; xn; xn+1:Then (d) follows with X = �xn+1:Y .

If (c) is obtained from

�i1 : : : ; �ik ` �! �and �j1 : : : ; �jm ` �where

��i1 ; : : : ; �ik

[ ��j1 ; : : : ; �jm =��1; : : : ; �n

We have, by the induction hypothesis:

xi1 : �i1 ; : : : ; xik : �ik ` Y : �! �xj1 : �j1 ; : : : ; xjm : �jm ` Z : �

where FV (Y ) � �xik ; : : : ; xik and FV (Z) � �xj1 ; : : : ; xjm.

(d) then follows with X = Y Z.

�

3.3 Long Normal Forms

In Section 5 we will develop an algorithm, which, when given a type, willproduce an inhabitant of this type if it has one. The inhabitant that isproduced is in long normal form. This is de�ned below.

DEFINITION 33 (Long normal form). A typed �-term �x1 : : : xn:xiX1::Xk

(n � 0; k � 0) is said to be in long normal form (lnf) if X1; : : : ; Xk arein lnf and have types �1; : : : ; �k and xi has type �1 ! : : : ! �k ! a wherea is an atom.

THEOREM 34. If X is a �-term such that ` X : �, then there is a termY in lnf such that ` Y : � and Y .� X.

Proof. By induction on the number of parts xiX1 : : :Xk of X that are notin lnf and are not the initial part of a term xiX1 : : : Xm, with m > k.

Consider the shortest of these. X1; : : : ; Xk must then be in lnf and ifeach Xj has type �j , xi must have type �1 ! : : :! �m ! a where m > k.

Let xp+1; xp+2; : : : ; xp+m�k be variables not free in xiX1 : : : Xk withtypes �k+1; �k+2; : : : ; �m respectively.

Then xiX1 : : : Xkxp+1 : : : xp+m�k has type a and �xp+1 : : : xp+m�k:xiX1 : : : Xkxp+1 : : : xp+m�k has type �k+1 : : : ! �m ! a, the same asxiX1 : : : Xk.

This new term with the same type as xiX1; : : : ; Xk is in lnf, so when itreplaces xiX1 : : : Xk in X , there is one fewer part not in lnf.

Hence X can be expanded to a term Y in lnf such that Y .� X . Thetypes of the parts of X being changed are not a�ected so the type of Y willbe the same as the type of X . �


3.4 Lambda Reductions and Proof Reductions

We illustrate here what happens to (a part of) a proof represented by a�-term (�x:X)Y or �y:Xy with y 62 FV (X) when this is reduced by (�) or(�).

(6 1)x : �D1

X : � �(1)�x:X : �! �

D2

Y : �

(�x:X)Y : �D3

reduces to:D2

Y : �[Y=x]D1

[Y=x]X : �[[Y=x]X=(�x:X)Y ]D3

and

!e

D1

X : �! �(6 1)x : �

!i

Xx : �

�x:Xx : �! �D2

�(1)(x 62 FV (X))

reduces toD1

X : �! �

[X=�x:Xx]D2

An expansion of a typed �-term to lnf reverses the second reduction.The e�ect of these reductions on a type assignment shown by the above

is stated in the theorem below.

THEOREM 35 (Subject Reduction Theorem). If

x1 : �1; : : : ; xn : �n ` X : �

and

X .�� Y ;

then

x1 : �1; : : : ; xn : �n ` Y : �

248 MARTIN BUNDER

Proof. A full proof of this is lengthy. A good outline appears in [Hindleyand Seldin, 1986, Chapter 15]. �

The converse of this is not true, for example (�uv:v)(�x:xx) has no typebut it �-reduces to �v:v which has type a! a.�xyz:(�u:y)(xz) has type (c ! d) ! b ! c ! b, but not type a ! b !

c! b; it reduces to �xyz:y which has type a! b! c! b.The �-expansion used in forming a lnf in the proof of Theorem 34, illus-

trates the following limited form of the Subject Expansion Theorem.

THEOREM 36. If

x1 : �1; : : : ; xn : �n ` X : �! �

thenx1 : �1; : : : ; xn : �n ` �x:Xx : �! �

if x 62 FV (X).

Proof. See Hindley and Seldin [1986, Chapter 15]. �

References Much more detail on the work in this section can be foundin [Hindley, 1997, Chapter 2]. The system TA� discussed there is e�ectivelythe system we have introduced. See also [Barendregt, 1984, Appendix A].

4 TRANSLATIONS

As �-terms and combinators describe the same set of (recursive) functionsit is not surprising that for each �-term there is a combinator representingthe same function and, in a simple case, with the same reduction theorems(as in (K), (S) or Theorem 6) and the same types.

For every combinator there is also a similar �-term given by:

DEFINITION 37 ((X�)).

S� � �xyz:xz(yz)K� � �xy:x

(XY )� � X�Y�:

Any �-term X can be translated into a combinator by taking parts ofX of the form �xi : : : xj :Y where Y contains no �xks and changing theseto [xi; : : : ; xj ]Y as de�ned in De�nition 7, then further parts of the form�xi : : : xj :Y can be changed in the same way until there are no �xks left. Ifwe call this translation � we might hope to have

X�� X


andY�� Y

for any combinator X and �-term Y .The former identity holds, but the latter one does not, in general.

EXAMPLE 38.

1. (SKK)�� (�xyz:xz(yz))�(�uv:u)�(�ts:t)�� ([x; y; z]:xz(yz))([u:v]u)([t; s]t)� ([x; y]:Sxy)([u]:Ku)([t]:Kt) � SKK

2. (z(�xyz:xyyz))�� z([x; y; z]:xyyz)�� z([x; y]:xyy)� � z([x]:SxI)� � z(SS(KI))�� z(�xyz:xz(yz))(�uvw:uw(vw))((�st:s)(�r:r)). z(�yzw:zw(yzw))(�tr:r). z(�zw:zw((�tr:r)zw)). z(�zw:zww)

In the second example above we have only

Y�� =�� Y :

This turns out to be about as much as we can hope to have.Weaker sets of combinators can be translated into �-calculus in the same

way as above with the translations of S and K in De�nition 37 replaced byappropriate translations such as:

B� � �xyz:x(yz) ;

for each element of the given basis set Q.The reverse process however is not so simple. It is not clear which �-

terms can be translated, say, into BCIW combinators, nor how to performthis translation.

We will resolve this problem later for several basis sets Q.The process is important as, as was mentioned in the introduction, it

provides a decision procedure and a constructive proof �nding algorithmfor axiomatic logics.

4.1 Q-Translation Algorithms

Trigg, Bunder and Hindley in [Trigg et al., 1994] have extended the notion ofabstractibility from that of De�nition 7 for SK-combinators to abstractibil-ity for Q-combinators for various basis sets Q. These de�nitions will formpart of our translation procedures.

First we will give the de�nition, from [Bunder, 1996], of a translationfrom �-terms into Q-combinators.

DEFINITION 39. A mapping � from �-terms to Q-terms is de�ned by:

250 MARTIN BUNDER

1. X� � X (X an atom)

2. (XY )� � X�Y�3. (�x:X)� � ��x:X�;

��x varies with Q but is, in simple cases, like the bracket abstraction [x] ofDe�nition 7, a sequence of some of the following clauses:

(i) ��x:x � I

(k) ��x:X � KX if x 62 FV (X)

(�) ��x:Xx � X if x 62 FV (X)

(s) ��x:XY � S(��x:X)(��x:Y )

(b) ��x:XY � BX(��x:Y ) if x 62 FV (X)

(c) ��x:XY � C(��x:X) if x 62 FV (Y )

Note that, as before, the clauses in an algorithm � are used strictly inthe order in which they appear.

Note also that the mapping starts by assigning a � to each � in theterm starting from the outermost ones and working inwards. Then, to theinnermost terms ��x:Z i.e. those for which Z contains no ��s, we applythe appropriate abstraction clause. Later steps now evaluate terms ��x:Z1where Z1 is a Q-term. The Q-combinators in Z1 arise from the evaluationof previous ��x:Zs.

EXAMPLE 40.

�(ik�s)xy:xzy � �(ik�s)x:xz � SI(Kz)�(isk�)xy:xzy � �(isk�)x:S(S(Kx)(Kz))I� S(S(KS)S(S(KS)(S(KK)I))(S(KK)(Kz))))(KI)

DEFINITION 41. A mapping � from �-terms to Q-terms is said to be aQ-translation algorithm if:

(A) For every Q-combinator X , X�� is de�ned and

X�� X :

(B) If for a �-term Y; Y� is de�ned and is a Q-term, then there is a �-termY1 such that:

Y�� .Q Y1 /� Y

where .Q means that only full or partial reductions involving the �-versions of Q-combinators are used and .� involves only � and �-reductions.


EXAMPLE 38.2(again) In this example we had,

with � � (ik�s);Y � z(�xyz:xyyz);

and Y�� z(�zw:zww);

so (B) holds with Y1 =� Y��.

EXAMPLE 42. (BCC)�� (�xyz:x(yz))�(�uvw:uwv)�(�rst:rts)�If � is (i�bc) we have

(BCC)�� BCC

as required by (A).If however � is (ikc) (�xyz:x(yz))� is not de�nable so (ikc) is not a

BCI-translation algorithm.If � is (ibc) we have

B�� (�xyz:x(yz))�� C(BB(BBI))(C(BBI)I) 6� B;

so (ibc) is also not a BCI-translation algorithm.(note that we do have B�� =�� B.)

EXAMPLE 43. When � is (i�bc)

(�yz:z(�x:yx))�� (��yz:z(��x:yx))�� (��yz:zy)� � (��z:CIz)�� (CI)�� (�uvw:uwv)(�x:x).BCI �vw:(�x:x)wv.BCI �vw:wv:

while also �yz:z(�x:yx) .� �vw:wv

The following theorem lists some properties that are preserved under theoperations � and �.THEOREM 44.

1. If X and Y are Q-terms then

X .Q Y ) X� .Q Y� :

If � is a Q-translation algorithm then:

2. for every Q-term X, X�� is de�ned and

X�� X

252 MARTIN BUNDER

3. For every pair of �-terms U and V for which U� and V� are de�nedand are Q-terms,

U =�� V , U� =�� V�

4. If Y� is de�ned and is a Q-term then

(�x:Y )�x =�� Y�

Proof.

1. It suÆces to prove the result for X � PX1 : : : Xn where P is anyQ-combinator and Y � f(X1; : : : ; Xn) which results by a single P -reduction step. Then

X� � P�X1� : : : Xn�

� (�x1 : : : xn:f(x1; : : : ; xn))X1� : : : Xn� .Q f(X1�; : : : ; Xn�)� Y�:

2. By an easy induction using (A) and De�nition 39.2.

3. By Curry and Feys [1958, Section 6C4, Theorem 1]:

U� =�� V� () U�� =�� V��

so the result follows by (B).

4. By (3) ((�x:Y )x)� =�� Y� so the result holds by De�nition 39.2. �

4.2 Q-de�nability

We now come to the important question as to which �-terms can be trans-lated into Q-combinators, for a given Q. We �rst de�ne Q-de�nability.

DEFINITION 45. A �-term Y is Q-de�nable if there is a Q-translationalgorithm � for which Y� is de�ned and is a Q-term.

The following theorem relates the ( )� operation to Q-de�nability.

THEOREM 46.

1. If Z is Q-de�nable there is a Q-term X such that Z =�� X�.

2. If X is a Q-term and there is a Q-translation algorithm then X� isQ-de�nable.


Proof.

1. If Z is Q-de�nable there is a Q-translation algorithm � so that, by (B)

Z�� =�� Z:

thus Z� is the required X .

2. If X is a Q-term and � is a Q-translation algorithm, by Theorem 44.2X�� X; so clearly X� is Q-de�nable. �

The importance of Q-de�nability, for implicational logics is that a typed�-term Y is Q-de�nable if and only if its type is a theorem of Q-logic. Inall the cases we deal with below we �nd that, for a given Q, we can �nd aQ-translation algorithm which translates all Q-de�nable �-terms.

We now give translations algorithms for a number of sets of combinators.First we need a lemma.

LEMMA 47. If U is a �-term for which U� is de�ned then

1. if � is the (i�ks) algorithm (��x:U�)x .KS U�:

2. if � is the (i�kbc) algorithm (��x:U�)x .BCK U�:

3. if � is the (i�bc) algorithm (��x:U�)x .BCI U�:

4. if � is the (i�bcs) algorithm (��x:U�)x .BCIW U�:

Proof.

1. By induction on the length of U�. If U� � x

(��x:U�)x � Ix � SKKx . KS x � U�:

If U� � U1�x where x 2=FV (U1�);

(��x:U�)x � U1�x � U�:

If x 2=FV (U�);

(��x:U�)x �KU�x . KS U�

If U� � U1�U2�, where x 2 FV (U1�U2�) and either x 2 FV (U1�)or U2� 6� x, (��x:U�)x = S(��x:U1�)(�

�x:U2�)x . KS (��x:U1�)x((��x:U2�)x) . KS U1�U2� � U� by the induction hypothesis.

Cases 2. to 4. are similar. �

254 MARTIN BUNDER

THEOREM 48.

1. (i�ks) is an SK-translation algorithm.

2. (i�kbc) is a BCK-translation algorithm.

3. (i�bc) is a BCI-translation algorithm.

4. (i�bcs) is a BCIW-translation algorithm.

Proof.

1. It is easy to check that (A) holds for the (i�ks) abstraction algorithm.

We prove (B) by induction on the length of the �-term Y .

If Y is an atom Y�� Y:If Y � UV; Y�� U��V�� and by the inductive hypothesis we havea U1 and V1 such that Y�� U��V�� .KS U1V1 /� UV � Y:If Y � �x:Xx, where x 62 FV (X),

Y�� (�x:Xx)�� X��:

By the induction hypothesis there is an X1 such that

Y�� X�� .KS X1 /� X /� Y:

If Y = �x:UV where x 2 FV (U) or x 6� V; but x 2 FV (UV ):

Y�� S�(�x:U)��(�x:V )�� (�uvx:ux(vx))

(�x:U)��(�x:V )��.SK�x:((�x:U)��x)((�x:V )��x) � �x:((��x:U�)x)�((��x:V�)x)�

.SK�x:U��V��

by Lemma 47 and Theorem 44.1.

Now by the induction hypothesis there exist U1 and V1 such that

�x:U��V�� .SK �x:U1V1 /� �x:UV � Y :

Note that if x had not been chosen as the third bound variable inS�, an extra �-reduction would have been required from �x:U1V1 toreach the term obtained by the SK-reduction. We will use similarsimpli�cations below.

If Y � �x:x;Y�� I� =� Y :


If Y � �x:X; where x 62 FV (X),

Y�� K�X�� (�yx:y)X�� .SK �x:X�� :

By the induction hypothesis there exists an X1 such that

X�� .SK X1 /� X

soY�� .SK �x:X1 /� Y

2., 3. and 4. are similar. �

EXAMPLE 49.

1. �i�kbcx1x2:x2��i�kbcx3:x1x4

� � �i�kbcx1x2:x2 (K(x1x4))

� �i�kbcx1:CI(K(x1x4))

� B(CI)(BK(CIx4))

2. �i�bcx1x2:x2��i�bcx3:x3x1

�� i�bcx1x2:x2(CIx1)

� �i�bcx1:CI(CIx1)

� B(CI)(CI):

3. �i�kbcsx1x2:x2��i�kbcsx3:x2(x1x3)

�� i�kbcsx1x2:x2(Bx2x1)

� �i�kbcsx1:SI(CBx1)

� B(SI)(CB):

4.3 SK, BCI, BCK and BKIW De�nable Terms

For many sets Q we can delineate the Q-de�nable terms. First we needsome notation.

DEFINITION 50.

1. � is the set of all �-terms.

2. A �-term is in Once (i1; : : : ; in) if each xi1 ; : : : ; xin appears free exactlyonce in the term and if in every subterm �xj1 : : : xjk :Y of the term, Yis in Once (j1; : : : ; jk).

3. A �-term is in Once�(i1; : : : ; in) if each xi1 ; : : : ; xin appears free atmost once in the term and if in every subterm �xj1 : : : xjk :Y of theterm, Y is in Once�(j1; : : : ; jk).

256 MARTIN BUNDER

4. A �-term is in Once+(i1; : : : ; in) if each of xi1 ; : : : ; xin appears at leastonce in the term and if in every subterm �xj1 : : : xjk :Y of the term, Yis in Once+(j1; : : : ; jk).

THEOREM 51.

1. The set of SK-de�nable terms is �.

2. The set of BCI-de�nable terms is Once( ).

3. The set of BCK-de�nable terms is Once�( ).

4. The set of BCIW-de�nable terms is Once+( ).

Proof.

1. is trivial.

2. If Y 2 Once(i1; : : : ; in) for some i1; : : : ; in we show by induction on Ythat Y is BCI-de�nable using the (i�bc) algorithm.

Case 1 Y is an atom Y(i�bc) � Y .

Case 2 Y � UV; U 2 Once(j1; : : : ; jr) and V 2 Once(m1; : : : ;ms),where (j1; : : : ; jr) and (m1; : : : ;ms) are disjoint subsequences of (i1;: : : ; in) and r + s = n:

Then by the induction hypothesis U and V are BCI-de�nable usingthe (i�bc) algorithm and

Y(i�bc) = U(i�bc) V(i�bc):

Case 3 Y � �xp:Zxp where xp 2=FV (Z).

Zxp 2 Once(i1 : : : ; in; p)

and by the induction hypothesis Zxp is BCI-de�nable as Z(i�bc)xpand

Y(i�bc) = Z(i�bc):

Case 4 Y � �xp:UV; UV 2 Once(i1 : : : ; in; p) and so xp 2 FV (V )�FV (U) or xp 2 FV (U)� FV (V ):

Hence, for similar disjoint sequences to the above we have U 2 Once(j1; : : : ; jr) and V 2 Once(m1; : : : ;ms; p) or U 2 Once(j1; : : : ; jr; p)and V 2 Once(m1; : : : ;ms):

In the former case U; V and �xp:V are BCI de�nable and

Y(i�bc) � BU(i�bc)(�(i�bc)xp:V(i�bc)):


In the latter case U; V and �xp:U are BCI-de�nable and

Y(i�bc) � C(�(i�bc)xp:U(i�bc))V(i�bc):

3. and 4. are similar, except that in 4. (m1; : : : ;ms) and (j1; : : : ; jr)need not be disjoint. �

Note that in each of the four cases above the set of de�nable �-termsleads us to a natural deduction style system for the logic.

In the case of BCIW-logic (the relevance logic R!), the only �-termsallowable are those in Once+( ). These can be generated by allowing �x:Xto be de�ned only when x 2 FV (X). This restriction when translated totyped terms becomes:

(6 1)

x : �

D1

X : �

�x:X : �! ��(1)

only if x : � is used in the proof D1. This therefore gives the appropriaterestriction on an R! ! introduction rule.

4.4 Bases Without C

We now look at some basis sets that do not include (as de�ned or primitive)the combinator C. Its lack causes a problem.

Previously the algorithms we used in evaluating ��x2:X did not a�ectthe de�nability of ��x1:�

�x2:X . When we are dealing with algorithms thatdo not include (c) (or (s)), this de�nability may fail depending on a choiceof �.

If, for example, we de�ne ��x3:x4x2(x1x3) to be B(x4x2)x1, usingclause (b), we cannot easily de�ne ��x2:B(x4x2)x1. If however we de-�ned ��x3:x4x2(x1x3) as B0x1(x4x2), using a clause (b0), we can de�ne��x2:�

�x3:x4x2(x1x3) as B(BB0)x1x4 using (b). Clearly if (b) and (b0) (andso B and B0) are both available in our translation algorithm, the choice ofwhich to use would depend on the variables to be abstracted later.

If a subterm �xin+1 :Y of a �-term X is to be translated by � and is inthe scope of (from left to right in X): �xi1 ; : : : ; �xin , we will write thistranslation as �

xi1 ;:::;xinxin+1

:Y�, so that the variables with respect to which weneed to abstract later are agged as: xin to be done next, then xin�1 etc.These abstractions are of course tied to some set Q and to algorithm clauseswhich we still denote by �.

To ensure that the agged xij s are distinct we will assume that any �-term X being translated has, if necessary, �rst been altered so that no �xkappears more than once in X .

258 MARTIN BUNDER

For Q-translation algorithms �, where C is not in Q, such as BB0I andBB0IW, we replace De�nition 39 by:

DEFINITION 52. (xi1 ; : : : ; xin ;Y )� and in particular ( ;Y )� � Y� are givenby:

(xi1 ; : : : ; xin ;P )� � P if P is an atom(xi1 ; : : : ; xin ;PQ)� � (xi1 ; : : : ; xin ;P )�(xi1 ; : : : ; xin ;Q)�

(xi1 ; : : : ; xin ;�xin+1:R)� � �

xi1 ;:::;xinxin+1

:(xi1 ; : : : ; xin ; xin+1 ;R)�

( ;�xi1 :R)� � ��xi1(xi1 ;R)�:

Note that Theorem 44 still holds under this de�nition.

EXAMPLE 53.

( ;�x4x1:x3(�x5x6:x1(�x7:x1x5))(�x2:x2(�x9:x2)))� ��x4�

x4x1:x3(�x4x1x5

:(�x4x1x5x6:x1(�

x4x1x5x6x7

:x1x5))(�x4x1x2:x2(�x4x1x2x9

:x2))

Before we can write down the algorithm clauses used for logics without C,we need the notion of the index of a term with respect to a set of variables.

DEFINITION 54. (idx(M; i1; : : : ; in))

idx(M; i1; : : : ; in) = maxnp j 1 � p � n ^ xip 2 FV (M)

oDEFINITION 55 ((�x

xi1 :::xinin+1

:P )). This is de�ned using some or all of thefollowing clauses, depending on �.

(i) �xi1 :::xinxin+1

:xin+1 � I

(�) �xi1 :::xinxin+1

:Uxin+1 � U if xin+1 62 FV (U)

(b) �xi1 :::xinxin+1

:PQ � BP (�xi1 :::xinxin+1

:Q)

if idx(P; i1; : : : ; in) � idx(Q; i1; : : : ; in) or xi1 : : : xin is replaced by �;and xin+1 =2 FV (P ).

(b0) �xi1 :::xinxin+1

:PQ � B0(�xi1 :::xinxin+1

:Q)P

if idx(P; i1; : : : ; in) > idx(Q; i1; : : : ; in) and xin+1 =2 FV (P ).

(s) �xi1 :::xinxin+1

:PQ � S(�xi1 :::xinxin+1

:P )(�xi1 :::xinxin+1

:Q)

if idx(P; i1; : : : ; in) � idx(Q; i1; : : : ; in) or if xi1 : : : xin is replaced by�

(s0) �xi1 :::xinxin+1

:PQ � S0(�xi1 :::xinxin+1

:Q)(�xi1 :::xinxin+1

:P )

if idx(Q; i1; : : : ; in) < idx(P; i1; : : : ; in)


EXAMPLE 56.

1. ( ;�x2x4:x2(�x3:(�x5x6:x5(x4x6))x3)(i�bb0)

� ��x2�x2x4 :x2(�x2x4x3 :(�x2x4x3x5 �x2x4x3x5x6 :x5(x4x6))x3)� ��x2�x2x4 :x2(�x2x4x3 :(�x2x4x3x5 :B0x4x5)x3)� ��x2�x2x4x2(�x2x4x3 :B0x4x3)� ��x2�x2x4 :x2(B0x4)� B0B0 where � is (i�bb0)

2. (;�x2x4:x2(�x3:x2x4(�x1:x3(x4x1))))(�ikbb0ss0)

� ��x2:�x2x4 :x2(�x2x4x3 :x2x4(�x2x4x3x1 :x3(x4x1)))� ��x2:�x2x4 :x2(�x2x4x3 :x2x4(B0x4x3))� ��x2:�x2x4 :x2(B(x2x4)(B

0x4))� ��x2:Bx2(S0B0(BBx2))� SB(B(S0B0)(BB)) where � is (�ikbb0ss0) :

4.5 Translation Algorithms and De�nable Terms for BB0I andBB0IW

Before we give the algorithms we need a lemma

LEMMA 57. If Q is (i) BB0I or (ii) BB0IW, X is a Q-term and �xi1 :::xinxin+1

:Xis de�ned then �

�xi1 :::xinxin+1

:X�xin+1 .Q X :

Proof. By induction on the length of X .

Cases 1 to 4 apply where Q is BBI0 or BB0IW, Cases 5 and 6 only forBB0IW.

Case 1 X � xin+1 .

��xi1 :::xinxin+1

:X�xin+1 � Ixin+1 .Q xin+1 � X

Case 2 X � Uxin+1 where xin+1 62 FV (U)


:X�xin+1 � Uxin+1 � X :

Case 3 X � UV where xin+1 62 FV (U), xin+1 6� V , and idx(U; i1; : : : ; in) �idx(V; i1; : : : ; in).

260 MARTIN BUNDER


:X�xin+1 � BU

��xi;:::xinxin+1

:V�xin+1

.QU��xi1 :::xinxin+1

:V�xin+1

�.QUV � X ;

by the induction hypothesis.

Case 4 X � UV where xin+1 62 FV (U), xin+1 6� V and idx(V; i1; : : : ; in) <idx(U; i1; : : : ; in). Similar to Case 3.

Case 5 X � UV where xin+1 2 FV (U) \ FV (V ) and idx(U; i1; : : : ; in) �idx(V; i1; : : : ; in). Similar to Case 3.

Case 6 X � UV where xin+1 2 FV (U) \ FV (V ) and idx(V; i1; : : : ; in) <idx(U; i1; : : : ; in). Similar to Case 3. �

THEOREM 58. The following are translation algorithms:

1. ( ; )(i�bb0), for BB0I

2. ( ; )(i�bb0ss0), for BB0IW

Proof. In each case (A) is obvious.We will prove, for each algorithm � and each basis Q, if ((xi1 ; : : : ; xin ;Y )�

is de�ned, that there is a Y1 such that:��xi1 ; : : : ; xin ; Y

��.Q Y1 /� Y :

This we do by induction on the number k of clauses of � that are neededto evaluate (xi1 ; : : : ; xin ;Y )�.

If k = 0 there are no �s in Y and so

((xi1 ; : : : ; xin ;Y )�)� � Y � Y1If k > 0 it is suÆcient to consider a subterm (xi1 ; : : : ; xim ;�xim+1 :Z)�

(� �xi1 :::ximxm+1 :Z) of (xi1 ; : : : ; xin ;Y )� where Z contains no �s and to show

that there is a Z1 such that:��xi1 :::ximxm+1 :Z

��.Q Z1 /� �xim+1 :Z

We then have, by Theorem 44.1,

((xi1 ; : : : ; xin ;Y )�)� .Q ((xi1 ; : : : ; xin ;Y 0)�)� /� Y0 /� Y

where Y 0 is Y with Z1 for �xim+1 :Z.Now Y 0 needs fewer than k clauses of � for its evaluation, so by the

induction hypothesis we have a Y1 such that

((xi1 ; : : : ; xin ;Y 0)�)� .Q Y1 /� Y0 ;


which provides the result.To prove the above result for �xin+1 :Z we consider 6 cases for BB0IW.

The �rst 4 also apply to BB0I.Note that in each case if U is Z or a subterm of Z, as this contains no

�s, we have U� � U .

(a) If Z � xim+1 , then��xi1 ;:::;ximxim+1

:Z�� I� � �xim+1 :Z

(b) If Z � Uxim+1 where xim+1 =2 FV (U)then �

�xi1 ;:::;ximxim+1

:Z�� U�� :

By the induction hypothesis there is a U1, such that

(�xi1 :::ximxim+1

:Z)� .Q U�� .Q U1 /� U /� �xim+1 :Z

(c) If Z � UV , where xim+1 =2 FV (U); V 6� xim+1 and idx(U; i1; : : : ;im) � idx(V; i1; : : : ; im)�

�xi1 ;:::;ximxim+1

:Z��BU

��x

xi1 :::ximim+1

:V��

�

� B�U��(�xi1 ;:::;ximxim+1

:V )�

.Q�xim+1 :U��

��xi1 ;:::;ximxim+1

:V�xim+1

��

so by Theorem 44.1 and Lemma 57,��xi1 ;:::;ximxim+1

:Z��.Q �xim+1 :U��V� � �xin+1 :U��V��

Now by the induction hypothesis we have a U1 and V1 suchthat:��xi1 ;:::;ximxim+1

:Z��.Q �xim+1 :U1V1 /� �xim+1 :UV � �xim+1 :Z

(d), (e), (f) The cases Z � UV where xim+1 =2 FV (U) and idx(U; i1; : : : ;in) > idx(V; i1; : : : ; im) or where xim+1 2 FV (U)\ FV (U) aresimilar. �

Our classi�cation of the BB0I and BB0IW-de�nable �-terms involves aclass HRM(i1; : : : ; in) of heriditary right maximal terms with respectto xi1 : : : xin ; which we now de�ne.

DEFINITION 59 ((HRM(i1; : : : ; in))).

1. Every variable and every basis combinator is in HRM(i1; : : : ; in).

262 MARTIN BUNDER

2. If M;N 2 HRM(i1; : : : ; in)and idx(M; i1; : : : ; in) � idx(N; i1; : : : ; in)then MN 2 HRM(i1; : : : ; in).

3. If M 2 HRM(i1; : : : ; in+1)then �xin+1:M 2 HRM(i1; : : : ; in).

Strictly we should write HRMQ(i1; : : : ; in), but in each case below thebasis Q will be clear from the context.

HRMBB0I(1; : : : ; n) isHRM(x1; : : : ; xn) of Hirokawa 1996. OurHRM(1;: : : ; n) is also HRMn of Trigg et al. [1994] where the basis is also takenfrom the context.

Before obtaining the classi�cations we need a lemma.

LEMMA 60. If X is a BB0IW-term or X � Y� where Y is a BB0IW-termand X .BB0IW Z or Z .� X then, if X is in HRM(i1; : : : ; in), so is Z.

Proof. If X is a BB0IW-term this is easy to prove by induction on thelength of the BB0IW-reduction or of the �-expansion.

If X � Y�, where Y is a BB0IW-term more single BB0IW-reductionsare possible. For example instead of

B0UVW .B0 V (UW )

we can have (�uvw:v(uw)) U�V�W� .B0 (�vw:v(U�w))V�W�

.B0(�w:V�(U�w))W�

.B0V�(U�W�);

however in each case the membership of HRM(i1; : : : ; in) is preserved. Sim-ilarly for �-expansions. �

THEOREM 61.

1. The set of BB0I-de�nable terms is HRM( ) \ Once( ).

2. The set of BB0IW-de�nable terms is HRM( ) \ Once+( ).

Proof. Theorem 58 gives translation algorithms.

1. If Y 2 HRM(i1; : : : in)\Once(i1; : : : ; in), it is easy to show by induc-tion on the length of Y that Y is BB0I de�nable by (xi1 ; : : : ; xin ;Y )i�bb

0

.This holds in particular when n = 0.

If Y is BB0I-de�nable then there is a BB0I-translation algorithm �and a �-term Y1 such that

Y�� .BB0I Y1 /� Y

Now Y� is a BB0I-term and is therefore in HRM( ) \ Once( ). Itfollows by Lemma 60 that Y1 and Y are also in HRM( ) \ Once( ).


2. is similar. �

The (i�bb0)-algorithm for BB0I was �rst used (for abstraction only) in[Helman, 1977]. It was also used by Hirokawa in his proof of the ) half of(B) in [Hirokawa, 1996]. Hirokawa proved the result of Theorem 61.1 there.

5 THE BASES BB0IK; BB0; BB0W AND BB0K

The BB0IK(i1; : : : ; in) abstractable terms of Trigg et al 1994 were termsobtainable from terms of HRM(i1; : : : ; in) \ Once(i1; : : : ; in) by deletingcertain variables. The BB0IK(i1; : : : ; in)-translation algorithm that wedevelop here has as its �rst stage, a \full ordering algorithm" whichreverses the deletion process by building up elements of a subclass ofOnce�(i1; : : : ; in) to elements of HRM(i1; : : : ; in)\ Once(i1; : : : ; in). Suchelements can then be translated by ( ; )(bb

0i�). A partial ordering algorithm,which builds up to elements of HRM(i1; : : : ; in)\ Once�(i1; : : : ; in) couldalso be used and requires only simple alterations to 1. and 2. below.

THE FULL ORDERING ALGORITHM

Aim To extend, if possible, a ��BB0IK-term Y 2 Once�(i1; : : : ; in) to a�-BB0IK-term Y o 2 HRM(i1; : : : ; in)\ Once(i1; : : : ; in) so that Y o .KI Y .

1. If Y � a, an atom not in fxi1 ; : : : ; xingY o � K�a (xi1xi2 : : : xin)

2. If Y � xim , and 1 � m < n thenY o � K�xim(xi1 : : : xim�1xim+1 : : : xin)

3. If Y � xin ,Y o � K�I�(xi1 : : : xin�1)xin

4. If Y � �xin+1 :Z, �nd, if possible, Zo such that

Zo 2 HRM(i1; : : : ; in+1) \ Once (i1; : : : ; in+1)

and Zo .KI Zthen Y o � �xin+1 :Z

o.

5. If Y � Zxin , �nd, if possible, Zo such that

Zo 2 HRM(i1; : : : ; in�1) \ Once (i1; : : : ; in�1)

and Zo .KI Zthen Y o � Zoxin .

264 MARTIN BUNDER

6. If Y � UV , where V 6� xin �nd, by going back to (1), a term Uo anda subsequence (xj1 ; : : : ; xjr ) of (xi1 ; : : : ; xin) such that:

(a) FV (U) \ fxi1 ; : : : ; xing � fxj1 ; : : : ; xjrg andFV (V ) \ fxj1; : : : ; xj rg = ;.

(b) Uo 2 HRM(j1; : : : ; jr) \ Once(j1; : : : ; jr)

(c) Uo .KI U

(d) jr 6= in

(e) maxfpjxjp 2 fxj1 ; : : : ; xjrg � FV (U)g is minimal.

(f) Given (e), the number of variables in fxj1 ; : : : ; xjrg � FV (U) isminimal.

Now let (xs1 ; : : : ; xst) be the sequence obtained from (xi1 ; : : : ; xin)by removing (xj1 ; : : : ; xjr ).

Now if possible (i.e. by going back to (1)) �nd V o such that

(g) V o 2 HRM(s1; : : : ; st) \ Once(s1; : : : ; st)

and

(h) V o .KI V

then Y o � UoV o.

Choosing the maximal p in xjp 2 FV (Uo)\ fxi1 ; : : : ; xing to be minimalin (e) and then using as few as possible variables new to U in (f), gives usmaximal exibility for expanding V to V o using the remaining, especiallythe higher subscripted, variables. These clauses also ensure that a uniqueY o is produced by the algorithm. Other Y os satisfying the above aim mayexist as well.

The algorithm is applied in two examples below.

EXAMPLE 62. Y � x3x2x1 cannot be ordered relative to (1; 2; 3) or even(1; 2; 3; 4), but relative to (1; 2; 3; 4; 5)

Y o � x3(K�x2x4)(K�x1x5)

EXAMPLE 63.Y � x7x0(�x9:x5(x4(x3x2))x1x9)

Relative to (0; 1; 2; : : : 7; 8; 10; 11) we have

Y o � x7(K�x0x8)(�x9:x5(x4(x3(K�x2x6))(K�x1(x10x11))x9)2 HRM(0; 1; 2; : : : ; 8; 10; 11)\ Once(0; 1; 2; : : : ; 8; 10; 11)

The �-terms that are BB0IK-translatable will be represented in terms ofa generalisation of the classHRM(i1; : : : ; in). If it is not possible to extend a�-BB0IK-term Y to a Y o 2 HRM(i1; : : : ; in)\Once(i1; : : : ; in), it is always


possible to choose variables xin+1 ; : : : ; xim and a Y o 2 HRM(i1; : : : ; im) \Once(i1; : : : ; im) so that Y o .KI Y .

If (xj1 ; : : : ; xjr ) is (xi1 ; : : : ; xin) with the free variables of Y deletedand if we named the atom occurrences in Y from the leftmost to therightmost a1; : : : ; ap then Y o could be de�ned as: Y with a1 replaced byKa1(xj1 : : : xjrxin+1) and ai (1 < i � p) replaced by Kaixin+i . Repeatedlyusing the full reordering algorithm, with n increased by one each time, willproduce a minimal set of extra variables that need to be added to form aY o.

DEFINITION 64 (Potentially Right Maximal (i1; : : : ; in)-�-terms).(PRM(i1; : : : ; in)-�-terms)

1. If X is an atom X 2 PRM( ).

2. xe 2 PRM(e).

3. If X 2 PRM(i1; : : : ; in�1) and xj =2 FV (X) thenX 2 PRM(i1; : : : ; ik; ij ; ik+1; : : : ; in) for 0 � k � n.

4. If X 2 PRM(i1; : : : ; in+1) then �xin+1:X 2 PRM(i1; : : : ; in).

5. If X 2 PRM(j1; : : : ; jp)and Y 2 PRM(r1; : : : ; rq)where p = q = n = 0 or rq = in,

fj1; : : : ; jpg \ fr1; : : : ; rqg = ;;FV (X) \ fxr1 ; : : : ; xrqg = ;;FV (Y ) \ fxj1; : : : ; xjpg = ;;

and (i1; : : : ; in) is a merge of (j1; : : : ; jp) and (r1; : : : ; rq) (i.e. n = p+qand (i1; : : : ; in) has the elements of the two sequences with the orderspreserved) then XY 2 PRM(i1; : : : ; in).

EXAMPLE 65.x1 2 PRM(1)

sox1 2 PRM(1; 5)

Similarlyx2 2 PRM(2; 4)

andx3 2 PRM(3)

sox3x2 2 PRM(2; 3; 4)

266 MARTIN BUNDER

andx3x2x1 2 PRM(1; 2; 3; 4; 5)

Note:x3x2x1 =2 PRM(1; 2; 3)[ PRM(1; 2; 3; 4)

Note that the variables in fxi1; : : : ; xing�FV (X) are used in the orderingof a term Y relative to (i1; : : : ; in), in the same way these extra variablesubscripts are needed to show X 2 PRM(i1; : : : ; in). The connection isgiven by the following lemmas.

LEMMA 66. If Y , ordered relative to (i1; : : : ; in) by the full ordering algo-rithm, becomes Y o, then

Y o .KI Y

where each single K-reduction eliminates one or more of xi1 ; : : : ; xin .

Proof. Obvious from the algorithm. �

LEMMA 67. Y 2 PRM(i1; : : : ; in) \ Once�(i1; : : : ; in)() there is a

Y o 2 HRM(i1; : : : ; in) \ Once(i1; : : : ; in) de�ned by the full ordering algo-rithm.

Proof. ) By induction on Y .The case where Y is an atom is obvious.If Y � �xin+1:Z, then as each bounded variable of Y appears at most

once in Y , we have Z 2 PRM(i1; : : : ; in+1) \ Once�(i1; : : : ; in+1).By the induction hypothesis we have an appropriate

Zo 2 HRM(i1; : : : ; in+1) \ Once(i1; : : : ; in+1)

and so a Y o � �xin+1:Zo 2 HRM(i1; : : : ; in) \ Once(i1; : : : ; in).If Y � Zxin, then as each bounded variable of Y appears at most once

in Y , we have Z 2 PRM(i1; : : : ; in�1) \ Once�(i1; : : : ; in�1).By the induction hypothesis we have an appropriate

Zo 2 HRM(i1; : : : ; in�1) \ Once(i1; : : : ; in�1)

and so a Y o � Zoxin 2 HRM(i1; : : : ; in) \ Once(i1; : : : ; in).If Y � UV (V 6� xin) then we have (j1; : : : ; jp) and (r1; : : : ; rq) such that

fj1; : : : ; jpg \ fr1; : : : ; rqg = ;;FV (U) \ fxr1 ; : : : ; xrqg = ;; rq = in; or n = p = q = 0;

FV (V ) \ fxj1; : : : ; xjpg = ;U 2 PRM(j1; : : : ; jp) \ Once�(j1; : : : ; jp)

andV 2 PRM(r1; : : : ; rq) \ Once�(r1; : : : ; rq)


Also the order of (j1; : : : ; jp) and (r1; : : : ; rq) is preserved in (i1; : : : ; in),where fi1; : : : ; ing = fj1; : : : ; jpg [ fr1; : : : ; rqg.

Of the sequences (j1; : : : ; jp); (r1; : : : ; rq) that satisfy these properties(and so (a), (b), (c), (d), (e) and (h) of (4) of the full ordering algorithm),choose those that also satisfy (e) and (f).

Then, as by the induction hypothesis we have:

Uo 2 HRM(j1; : : : ; jp) \ Once(j1; : : : ; jp)

andV;o 2 HRM(r1; : : : ; rq) \ Once(r1; : : : ; rq);

we have

Y o � UoV o 2 HRM(i1; : : : ; in) \ Once(i1; : : : ; in):

( To prove this we only need to show, by the previous lemma, thatK� or I�-reductions in �-terms eliminating some of xi1; : : : ; xin, preservemembership of PRM(i1; : : : ; in) in a reduction T .BB0IK R.

We prove this by induction on T .If T is an atom or contains an I� redex this is obvious.If

T � K�Wxis 2 PRM(i1; : : : ; in)

thenK�W 2 PRM(j1; : : : ; jp)

andxis 2 PRM(r1; : : : ; rq)

where in = rq and the other conditions apply.From there it follows that xr1 ; : : : ; xrp =2 FV (W ).Also W 2 PRM(j1; : : : ; jp)

and as (j1; : : : ; jp) is a subsequence of (i1; : : : ; in), by 3,

W 2 PRM(i1; : : : ; in):

If T � �xin+1 :W; R � �xin+1 :S and W . S,

W 2 PRM(i1; : : : ; in+1)

and by the induction hypothesis S 2 PRM(i1; : : : ; in+1) and soR 2 PRM(i1; : : : ; in).

If T � UV , where R �WS; U .BB0IKW and V .BB0IK S, we have: forsome (j1; : : : ; jp) and (r1; : : : ; rq)

U 2 PRM(j1; : : : ; jp)V 2 PRM(r1; : : : ; rq)

268 MARTIN BUNDER

with the usual conditions.By the induction hypothesis

W 2 PRM(j1; : : : ; jp)

andS 2 PRM(r1; : : : ; rq)

and soR �WS 2 PRM(i1; : : : ; in)

�

LEMMA 68. If A.�B and A.KIC then there is a term D such that B.KIDand C .� D.

Proof. By induction on the number of reduction steps in A .� B and asecondary induction on the number of steps in A .KI C (i.e. a standardChurch-Rosser theorem proof.) �

THEOREM 69. ( ;Y o)(i�bb0) is a BB0IK-translation algorithm.

Proof. (A) holds as before. By Theorem 58.1 there is a term Y1 such that(( ;Y o)(i�bb

0))� .BB0I Y1 /� Yo.

Now also by Lemma 66Y o .KI Y;

so by Lemma 68 there is a Y2 such that

Y1 .KI Y2 /� Y

So (( ;Y o)(i�bb0))� .BB0IK Y2 /� Y , i.e. (B) holds. �

Thus ( ;Y o)(i�bb0) is a BB0IK-translation algorithm.

THEOREM 70. The set of BB0IK-translatable terms is PRM( )\ Once�( ).

Proof. We have a BB0IK-translation algorithm by Theorem 69.If Y 2 PRM( )\ Once�( ), then by Lemma 67 Y o 2 HRM( )\ Once( )

and so ( ;Y o)(i�bb0) is a BB0IK-term, so Y is BB0IK-de�nable.

If Y is BB0IK-de�nable, the proof of Y 2 PRM( )\ Once�( ) proceedsas for Theorem 61. �

We now consider some bases without I.We de�ne �� as the set of all �-terms whose �-normal forms do not have

subterms of the form �xi1 : : : xin :xijxij+1 : : : xin or �xi1 : : : xin :xi1xi1xi2xi3: : : xin .

THEOREM 71.


1. HRM( ) \ �� \ Once is the set of BB0-de�nable terms.

2. HRM( ) \ �� \ Once+ is the set of BB0W-de�nable terms.

3. PRM( ) \ �� \ Once� is the set of BB0K-de�nable terms.

Proof. 3. Any BB0K-translation algorithm � must contain (�), otherwise,for example. B�� would not be de�nable. If a Q-translation algorithm �contains (�) it is easy to show, by induction on the length of a �Q-term Y ,with �-normal form Z that Y� � Z�.

The full ordering algorithm is such that the combinator I is used onlywhen it is essential (i.e. just K, B0 and B won't do) and it is clear that theonly terms in �-normal form in PRM( )\�� \ Once� that can have an Iin their translation are of the form:�xi1 : : : xin :xijxij+1 : : : xin .

Hence our result follows from Theorems 69 and 70.

1. and 2. are similar but simpler. �

Note that in a BB0IW abstraction of a term only terms of the form�xi1 : : : xin :xi1xi2 : : : xin and �xi1 : : : xin :xi1xi1xi2 : : : xin can contain an Iand in BB0I abstraction only the former.

6 THE � PROOF FINDING ALGORITHMS

We have shown that for each theorem of a wide range of implicational logicsthere is a �-term of a particular kind, depending on the logic, that has thattheorem as a type. We have also shown that we can expand any such �-terminto long normal form.

If a theorem, or type, � takes the form �1 ! : : : ! �n ! a, where a isan atom, we know that any lnf inhabitant must take the form �x1 : : : xn:X ,where X has type a and has FV (X) � fx1; : : : ; xng. This is the basis for theBen-Yelles algorithm (see [Ben-Yelles, 1979; Hindley, 1997; Bunder, 1995]),which constructs potential inhabitants of a given type from the outside in.As there may be several potential Xs with type a the process can branchseveral times. There is, at least for SK and sets of combinators without Sand W a simple bound to this inhabitant, or proof, �nding procedure. Theversion of the algorithm given in [Bunder, 1995], which is very eÆcient, hasbeen implemented in [Dekker, 1996].

In this section we will be looking at an alternative inhabitant buildingalgorithm, generally even more eÆcient, which builds the inhabitants of atype from the inside outwards. The given type provides the building blockswe use for this. This algorithm has been implemented in [Oostdijk, 1996].

270 MARTIN BUNDER

6.1 The Variables and Subterms of an Inhabitant of a Type

In order to describe the way in which a �-term inhabitant of a type is builtup we need some notation.

DEFINITION 72.

1. � is a positive subtype of a type �.

2. If � is a positive subtype of � or a negative subtype of then � is anegative subtype of �! .

3. If � is a negative subtype of � or a positive subtype of then � is apositive subtype of �! .

DEFINITION 73. An occurrence of a positive (negative) subtype � of atype � is said to be long if the occurrence of � is not the right hand partof a positive (negative) subtype �! � of � .

All types other than atomic types are said to be composite.

DEFINITION 74. The rightmost atomic subtype of a type is known as itstail.

EXAMPLE 75. � = (a ! b) ! (b ! c) ! ((a ! b) ! c) ! c hasb ! c; (a ! b) ! c, the second occurrence of a and the �rst occurrence ofa ! b as long negative subtypes and � , the �rst occurrence of a and thesecond occurrences of b and a! b as long positive subtypes of � .

DEFINITION 76.

dn(�) = the number of distinct long negative subtypes of �:do(�) = the number of occurrences of long negative subtypes of �:dcp(�) = the number of distinct long positive composite subtypes of �:

dapn(�) = the number of distinct atoms that are tails of both longpositive and long negative subtypes of �:

F (�) = 2dn(�)(dapn(�) + dcp(�)) + dn(�):G(�) = 2do(�)(dapn(�) + dcp(�)) + do(�):j� j= the total number of subtypes of �:

DEFINITION 77.

1. X is of �-depth 0 in X .

2. If an occurrence of a term Y is of �-depth d in X then Y is of �-depthd in UX and XU , provided these are in �-normal form.

3. If an occurrence of a term Y is of �-depth d in �xi:U , it is of �-depthd in �xjxi:U .


4. If an occurrence of a term Y is of �-depth d in V 6� �xi:U for any xior U , then Y is of �-depth d+ 1 in �xj :V .

The two lemmas that now follow will help to tell us with what a longnormal form inhabitant of a given type � can be constructed and also howto restrict the search for components.

LEMMA 78. If X is a normal form inhabitant of � , U is a subterm of Xof type � and V is a term of type � with FV (V ) � FV (U), then the resultof replacing U by V in X is another inhabitant of � .

Proof. By a simple induction on the length of X . �

EXAMPLE 79.

�x1x2:x2(�x3:x2(�x4:x1x3x4)) : (a! a! b)! ((a! b)! b)! b

Here x1 : a! a! b, x2 : (a! b)! b, x3 : a and x4 : a, so

�x3:x2(�x4:x1x3x4) : a! b

and�x3:x1x3x3 : a! b:

So we have by Lemma 77:

�x1x2:x2(�x3:x1x3x3) : (a! a! b)! ((a! b)! b)! b:

LEMMA 80. If ` Z : � , then there is a term X in long normal form, inwhich no two distinct variables have the same type, such that:

1. ` X : �

2. For every long subterm Y of X with FV (Y ) = fxi1 ; : : : ; xikg we have:

xi1 : �i1 ; : : : ; xik : �ik ` Y : �

where � is either a long occurrence of a composite positive subtype of �or an atom which is the tail of both a long positive and a long negativesubtype of � . �i1 ; : : : ; �ik are distinct long negative subtypes of � .

Proof.

1. By Theorem 34 there is an X 0 in lnf such that ` X 0 : � .

If X 0 now contains two variables xk and xe with the same type � wecan change any part �xe:B(xk; xe) of X 0 to �xk:B(xk ; xk) and anypart �xe:B(xe) to �xk :B(xk).

Let X be the term obtained when all possible changes of this kindhave been made to X 0. In X no two distinct variables will have thesame type.

272 MARTIN BUNDER

2. We prove this by induction on the �-depth d of Y in X .

If X is formed by application, as it is in long normal form, it takesthe form:

X � xiX1 : : : Xm

and the type of X must be dependent on that of xi, which does notagree with 1.

Hence X is not formed by application and if � = �1 ! : : : ! �n ! atakes the form

X � �x1 : : : xn:xiX1 : : : Xm:

(Note: some of x1; : : : ; xn could be identical.)

If d = 0 and Y is long, we must have Y � X and the result holds withk = 0 and � = � . If d = 1 and X = �x1 : : : xn:xiX1 : : : Xm; Y willappear in xiX1 : : : Xm, not in the scope of any �xjs. Thus it appearsin a part xtZ1 : : : ZrY Zr+2 : : : Zq of X for some t(1 � t � n).

If �t = �1 ! : : :! �q ! b we have, x1 : �1; : : : xn : �n ` Y : �r+1.

�r+1 is a long negative subtype of �t and so a long positive subtypeof � . �1; : : : ; �n are occurrences of long negative subtypes of � .

If �r+1 is an atom, then �r+1 must also be the tail of a long negativesubtype of � (namely one of �1; : : : ; �n).

Leaving out the variables not free in Y and variables identical to othersgives the result.

If d > 1,X � �x1 : : : xn:xiX1 : : :Xj�1(�xn+1 : : : xk:xsZ1 : : : Zp)Xj+1 : : : Xm,where Y is a long subterm of xsZ1 : : : Zp.

As xsZ1 : : : Zp is long and only in the scope of �x1 : : : xn : : : xk, wehave, if �i = 1 ! : : :! m ! c and j = �n+1 ! : : :! �k ! b:

x1 : �1; : : : ; xn : �n; : : : ; xk : �k ` xsZ1 : : : Zp : b:

Thus ` �x1 : : : xk:xsZ1 : : : Zp : �1 ! : : : ! �k ! b where Y is a longsubterm of �x1 : : : xk :xsZ1 : : : Zp of �-depth d� 1.

Thus 2. holds by the induction hypothesis with � a long positivesubtype of �1 ! : : : ! �k ! b and so of � or an atom which isthe tail of a long positive as well as of a long negative subtype of�1 ! : : :! �k ! b and so of � .

�

Note The �rst inhabitant given in Example 79 is a counterexample, due toRyo Kashima,to an earlier version of Lemma 80. This claimed property 2.


for any inhabitant X of the given type � in long normal form, rather thanjust some X .

It follows from Lemmas 78 and 80 that an X in long normal form suchthat

` X : �;

can be built up from subterms of the form

�xr : : : xs:xiX1 : : : Xn;

where xi : �i; �i is a long negative subtype of � , and �i has tail a, where ais an atom which is also the tail of a positive subtype of � .

The compound types of these subterms in long normal form must beamong the long positive subtypes of � .

With this in mind we arrive at the following algorithm for �nding inhab-itants of types, i.e. proofs in intuitionistic implicational logic.

6.2 SK-logic (H !)

The H ! Decision Procedure or the SK Long Inhabitant Search Algorithm

AimGiven a type � , to �nd a closed X in long normal form (if any) such that

` X : �

Step 1To each distinct long negative subtype �i of � assign a variable xi giving a�nite list:

x1 : �1; : : : ; xm : �m

Step 2For each atomic type b that is the tail of both a long negative and a longpositive subtype of � , form by application, (if possible) a lnf inhabitant Yof b if we don't already have a Y 0 : b with FV (Y 0) � FV (Y ) in Step 1 orthis or earlier Steps 2.

274 MARTIN BUNDER

Step 3For each long positive composite subtype � of � , form, by abstraction, withrespect to some of x1; : : : ; xm, a term Y in lnf such that Y : � if we don'talready have a Y 0 : � with FV (Y 0) � FV (Y ) in Step 1 or earlier Steps 3.

If one of these terms Y is closed and has type � we stop.If there is no such term we continue with Steps 2 and 3 until we obtain

no more terms with a \new" set of free variables or a new (atomic tail orlong positive) type. If there are no new terms there is no solution X .Note

In the work below and in all examples we will select our variables inStep 1 in the following order: x1 is assigned to the leftmost shallowest longnegative subtype of �; x2 to the next to leftmost shallowest long negativesubtype etc. until the shallowest long negative subtypes are used up. Thenext variable xn+1 is assigned to the leftmost next shallowest long negativesubtype and eventually xm to the rightmost deepest long negative subtype.

In Example 81 below, x1 to x4 are assigned to the shallowest subtypesand x5 to the next shallowest (the deepest) subtype.

Because of this ordering of the variables of X any subterm Y formedby the algorithm will be in the scope of �x1x2 : : : xnxp1xp1+1 : : : xp2xp3 : : :xp4 : : : xpq where n < p1 < p2 < p3 : : : < pq and where each �xp2i+1 : : : �xp2i+2

represents a single set of abstractions.

EXAMPLE 81.

� = [((a! b)! d)! d]! [(a! b)! d! e]! [a! a! b]! a! b

Step 1 x1 : ((a! b)! d)! d; x2 : (a! b)! d! e;x3 : a! a! b; x4 : a; x5 : a! b,

Step 2 x3x4x4 : b; x5x4 : b;

Step 3 �x4:x3x4x4 : a! b; �x4:x5x4 : a! b; �x1x2x3x4:x3x4x4 : �:

EXAMPLE 82.� = ((a! b)! a)! a:

Step 1 x1 : (a! b)! a; x2 : a:

Step 2 No new terms can be formed by application.

Step 3 �x1:x2 : �:No new terms can be formed. �x1:x2 is not closed so � has noinhabitants and no proof in H !.

EXAMPLE 83.

� = [((a! a)! a)! a]! [(a! a)! a! a]! [a! a! a]! a! a:


Step 1 x1 : ((a! a)! a)! a; x2 : (a! a)! a! a,x3 : a! a! a; x4 : a; x5 : a! a;

Step 2 x3x4x4 : a; x5x4 : a;

Step 3 �x4:x4 : a! a, �x5:x5x4 : (a! a)! a, �x1x2x3x4:x3x4x4 : � .

THEOREM 84. Given a type � , the SK long inhabitant search algorithmwill, in �nite time, produce an inhabitant of � or will demonstrate that �has no inhabitants. The algorithm will produce at most F (�) terms beforeterminating.

Proof. It follows from the Weak Normalisation Theorem (see [Turing, 1942;Hindley, 1997]) that if � has an inhabitant, this inhabitant has a normalform and this will also have type � .

By Lemma 80, � will have an inhabitant X of the form prescribed there.We show that our SK-algorithm provides such an inhabitant X of � .

Step 1 of the SK-algorithm provides us with the largest set of variablesx1; : : : ; xm that, by Lemma 80, need appear in a solution for X .

Step 2 of the algorithm considers terms U = xiX1 : : : Xn with an atomictype having a particular subset of x1; : : : ; xm as free variables. By Lemma78 other terms xjY1 : : : Yk with the same atomic type and a superset ofthese free variables can at most produce alternative inhabitants and so donot need to be considered.

The total number of variables we can have is dn(�). These are the termsgenerated by Step 1. The number of subsets of these is at most 2dn(�), thenumber of atomic types we can have is at most dapn(�) so the number ofterms generated by Steps 2 is at most 2dn(�):dapn(�).

Step 3 forms terms in long normal form which have composite long pos-itive subtypes of � as types. There are dcp(�) of these and we can form atmost one of these terms for each set of variables. Hence the most terms wecan form using Steps 3 is 2dn(�):dcp(�).

The maximal number of terms formed using the algorithm is thereforedn(�) + 2dn(�):(dapn(�) + dcp(�)) = F (�). �

NoteAs dapn(�) + dcp(�) � dp(�) where dp(�) is the number of occurrences ofdistinct positive subtypes in � , F (�) < dn(�)+2dn(�):dp(�) < 2dn(�)+dp(�) �2d(�) where d(�) is the number of long subtypes of � so F (�) < 2j� j.

In Example 1, F (�) = 197 while 2d(�) = 212 and 2j� j = 225. The actualnumber of terms formed by the algorithm was 10.

276 MARTIN BUNDER

6.3 BCK Logic

We now adapt our long Inhabitant Search Algorithm to search for BCK-�-terms in long normal form. By Theorem 51(3), these need to be elementsof Once�( ).

The BCK Logic Decision Procedure or the BCK-Long Inhabitant SearchAlgorithm

AimGiven a type � to �nd a closed BCK-de�nable �-term in long normal form(if any) such that

` X : �:

Step 1 To each occurrence of a long negative subtype �i of � assign avariable xi giving a �nite list:

x1 : �1; : : : ; xm : �m:

Step 2 For each atomic type b that is the tail of both a long positive and along negative subtype of � form (if possible), by application, from the termswe have so far, an inhabitant Y of b such that no free variables appear morethan once in Y , if we don't already have a Y 0 : b with FV (Y 0) � FV (Y ).

Step 3 For each long positive subtype � of � , form, by abstraction, withrespect to some of x1; : : : ; xm, a term Y such that Y : �, if we don't alreadyhave a Y 0 : � where FV (Y 0) � FV (Y ).

EXAMPLE 85.

� = [((a! b)! d)! d]! [(a! b)! d! e]! [a! a! b]! a! b

Step 1 x1 : ((a! b)! d)! d; x2 : (a! b)! d! e;x3 : a! a! b; x4 : a; x5 : a! b; x6 : a;

Step 2 x5x4 : b; x5x6 : b; x3x4x6 : b;

Step 3 �x4:x5x4 : a! b;�x4:x3x4x6 : a! b; �x6:x3x4x6 : a! b; �x1x2x3x4:x5x4 : �;�x1x2x3x6:x3x4x6 : �; �x1x2x3x4:x3x4x6 : �:

No new terms are generated by further uses of steps 2 and 3 and as theterms with type � are not closed, there is no BCK inhabitant of � . Notethat � does have an SK inhabitant �x1x2x3x4:x3x4x4, but this is not BCKbecause x4 is used twice.

EXAMPLE 86.

� = ((a! a)! a! a! b)! a! a! b


Step 1 x1 : (a! a)! a! a! b; x2 : a; x3 : a; x4 : a;

Step 2 No new terms are formed.

Step 3 �x2:x2 : a! a;

Step 2 x1(�x2:x2)x2x3 : b;x1(�x2:x2)x2x4 : b; x1(�x2:x2)x3x4 : b;

Step 3 �x1x2x3:x1(�x2:x2)x2x3 : �:

Notes: 1. The SK algorithm would have produced only �x1x2x3:x1(�x2:x2)x2x2 : � which is not a BCK-�-term.2. It was essential here to have a variable for each distinct long negativeoccurrence of a in � .

LEMMA 87. If X is a BCK term which is a normal form inhabitant of�; U a subterm of X of type � and V a BCK term of type �, in whichno free variable appears more than once, with FV (V ) � FV (U), then theresult of replacing U by V in X is another BCK inhabitant of � .

Proof. As for Lemma 78. �

LEMMA 88. If `BCK Z : � , then there is a term X is long normal formsuch that:

1. `BCK X : �

2. For every long subterm Y of X with FV (Y ) = fxi1 ; : : : ; xing we have

xi1 : �i1 ; : : : ; xin : �in `BCK Y : �;

where � is either a long occurrence of a composite positive subtype of� or an atom which is the tail of both a long positive and a long nega-tive subtype of � . �i1 ; : : : ; �in are distinct occurrences of long negativesubtypes of � .

Proof.

1. The formation of an X in long normal form is as in the proof of Lemma80(1) except that the extra variables xm+1; : : : ; xn that are choosenmust not be free in xiX1; : : : ; Xm, (otherwise �xm+1 : : : xn:xiX1; : : : ;Xmxm+1; : : : ; xn would not be a BCK-term). Also for the same rea-son, we don't identify distinct variables with the same type. We letX be a term obtained by the expansion of Z to long normal form.

278 MARTIN BUNDER

2. As for Lemma 80(2) except that we have to show that we have at mostone variable for each distinct occurrence of a long negative subtype of� . We extend the induction proof to include this.

When d = 1, we clearly have one variable for every long negativesubtype �1 : : : ; �n of � = �1 ! : : :! �n ! a and no others.

When d > 1 the subterm Y appears as Zj in a term

xtU1 : : : Uswhere Ue = �xu : : : xv :xrZ1 : : : Zp

xt : 1 ! : : :! s ! cand w = Æu ! : : :! Æv ! dand 1 � j � p; 1 � e � s and 1 � w � s:

The extra typed variables added at this stage are xu : Æu; : : : ; xv : Æv.

By the inductive hypothesis we have that 1 ! : : : ! s ! c is anoccurrence of a long negative subtype of � and it therefore followsthat the same holds for Æu; : : : ; Æv. Note that as in BCK (and BCI)logic there can be only one free occurrence of the variable xt in a termbefore we abstract with respect to xt, so there can be no other use ofxt that might generate another set of variables with types Æu; : : : ; Ævi.e. one occurrence of 1 ! : : : ! q ! c in � generates at most oneoccurrence of a variable for each occurrence of Æu; : : : ; Æv which arelong negative subtypes of depth one lower than 1 ! : : :! q ! c in� . �

THEOREM 89. Given a type � , the BCK long inhabitant search algorithmwill in �nite time produce an inhabitant or will demonstrate that � has noBCK-inhabitants. The algorithm will produce at most G(�) terms beforeterminating.

Proof. As for Theorem 84, except that Lemmas 87 and 88 replace Lemmas78 and 80. �

It might be thought that the BCK algorithm might require fewer thanthe maximal F (�) terms required for SK, in fact it requires more becausethere may be several variables with the same type. Even when this is notthe case, both algorithms require at most one term for each given type andeach set of variables. For SK some variables may appear several times, forBCK they may not. For BCI in addition all abstracted variables will haveto appear in the term being abstracted.

The bounds F (�) and G(�) are not directly related to standard complex-ity measures. The algorithm will in fact generate some other terms but notrecord them because they have the same type and set of free variables toanother already recorded.


6.4 BCI Logic

Again we can adapt the Long Inhabitant Search Algorithm. This time theinhabitants found must, by Theorem 51(2) be in Once( ).

The BCI-Logic Decision Procedure or the BCI Long Inhabitant Search Al-gorithm

AimGiven a type � to �nd a closed BCI-�-term X in long normal form (if any)such that:

` X : �:

MethodAs for the BCK-algorithm except that Step 2 and 3 end in \FV (Y 0) =FV (Y )" and in Step 3 we may only form �xi : : : xj :Z if xi; : : : ; xj occur freein Z exactly once each.

EXAMPLE 90.

� = ((a! b)! c)! b! d! (c! e)! e

Step 1 x1 : (a! b)! c; x2 : b; x3 : d; x4 : c! e; x5 : a

Step 2 No new terms are formed.

Step 3 �x5:x2 is not a BCI-�-term, so no new terms are generated and sothere is no BCI-proof of � .

EXAMPLE 91.

((a! b! c)! d)! (b! a! c)! d:

Step 1 x1 : (a! b! c)! d; x2 : b! a! c; x3 : a; x4 : b;

Step 2 x2x4x3 : c;

Step 3 �x3x4:x2x4x3 : a! b! c;

Step 2 x1(�x3x4:x2x4x3) : d;

Step 3 �x1x2:x1(�x3x4:x2x4x3) : �:

Example 86 also produced a BCI-term.

THEOREM 92. Given a type � , the BCI long inhabitant search algorithmwill, in �nite time, produce an inhabitant or will demonstrate that � has noinhabitants. The algorithm will produce at most G(�) terms before termi-nating.

280 MARTIN BUNDER

Proof. Lemma 87 holds provided that V is a BCI term such that FV (V ) =FV (U) and each free variable of V appears exactly once in V . If Z in Lemma87 is a BCI term so is X .

The proof of the theorem now proceeds as for Theorem 89 except thatin any substitution [V=U ]X the above restrictions apply. Hence FV (Y 0) =FV (Y ) was needed in Step 2 of the BCI-algorithm. In Steps 2 and 3we must, for each subset of fx1; : : : ; xmg, consider a term with those freevariables with a particular atomic negative or long positive subtype of T .

�

6.5 BCIW Logic (R!)

The BCI-search algorithm can easily be extended to BCIW-logic. ByTheorem 51(4), the �-terms required are from Once+().

The BCIW Long Inhabitant Search Algorithm

AimGiven a type � to �nd a closed BCIW-�-term X in long normal form (ifany) such that

` X : �

Method

As for the BCI algorithm except that in Steps 2 and 3 each variable mustappear in Y and Z at least once.

If the algorithm that we have to this stage fails, additional variables withthe same types as the ones �rst given in Step 1 are added and the previousalgorithm is repeated.

Note that, as it is not clear as to how many times new variables mightneed to be added, this method, while, as shown in Theorem 96, it leadsto �nding an inhabitant if there is one, does not constitute a decision pro-cedure. The need for extra variables is illustrated in Example 95 below.This logic does have a decision procedure (see [Urquhart, 1990]), but itsmaximum complexity is related to Ackermann's function.

EXAMPLE 93.

� = [((a! b)! d)! d]! [(a! b)! d! e]! [a! a! b]! a! b

Step 1 x1 : ((a! b)! d)! d; x2 : (a! b)! d! e;x3 : a! a! b; x4 : a; x5 : a! b; x6 : a

Step 2 x3x4x4 : b; x3x4x6 : b; x3x6x6 : b; x5x4 : b; x5x6 : b

Step 3 �x4:x3x4x4 : a! b; �x4:x3x4x6 : a! b; �x6:x3x4x6 : a! b;�x4:x5x4 : a! b


We cannot form �x1x2x3x4:x3x4x4 as this is not a BCIW-�-term.We can form no more terms by Step 2 as we have no terms of type d

or (a ! b) ! d and no new terms of type a. Hence � has no BCIWinhabitant.

EXAMPLE 94.

� = (c! a! a! a)! c! a! a

Step 1 x1 : c! a! a! a; x2 : c; x3 : a:

Step 2 x1x2x3x3 : a:

Step 3 �x1x2x3:x1x2x3x3 : �:

EXAMPLE 95.

� = c! c! (a! a! b)! (c! (a! b)! b)! b

Step 1 x1 : c; x2 : c; x3 : a! a! b; x4 : c! (a! b)! b; x5 : a

Step 2 x3x5x5 : b

Step 3 �x5:x3x5x5 : a! b

Step 2 x4x1(�x5:x3x5x5) : b;x4x2(�x5:x3x5x5) : b;

Step 3 No new terms can be formed.

(Add to) Step 1 x6 : a

Step 2 x3x5x6 : b, x3x6x6 : b

Step 3 �x6:x3x5x6 : a! b; �x5:x3x5x6 : a! b

Step 2 x4x1(�x6:x3x5x6) : b; x4x2(�x6:x3x5x6) : b;x4x1(�x5:x3x5x6) : b; x4x2(�x5:x3x5x6) : b;

Step 3 �x5:x4x1(�x6:x3x5x6) : a! b; �x5:x4x2(�x6:x3x5x6) : a! b;

Step 2 x4x1(�x5:x4x1(�x6:x3x5x6)) : b;x4x2(�x5:x4x1(�x6:x3x5x6)) : b;x4x2(�x5:x4x2(�x6:x3x5x6)) : b;

Step 3 �x1x2x3x4:x4x2(�x5:x4x1(�x6:x3x5x6)) : �:

Note that in Example 95 x4 is used twice and so the one occurrence of a inc ! (a ! b) ! b, requires two variables of type a. If these were identi�edthe resultant �-term would no longer be BCIW-de�nable.

THEOREM 96. Given a type � , the BCIW long inhabitant search algo-rithm will, in �nite time, produce an inhabitant.

282 MARTIN BUNDER

Proof. Lemma 87 holds for BCIW-terms provided that we have FV (U) =FV (V ).

If Z in Lemma 88 is a BCIW-term so is X . In the counterpart to Lemma88 the word \distinct" must also be dropped for the reasons illustrated inExample 95 above.

The proof of the theorem now proceeds as that of Theorem 91 exceptthat multiple copies of variables may appear in substitutions and in termsformed by the algorithm. �

6.6 BB0IW Logic (T !)

BB0IW search algorithm �nds �-terms of a given type that are in HRM( )\Once+( ), as required by Theorem 61.

The BB0IW Long Inhabitant Search Algorithm

AimGiven a type � to �nd a closed BB0IW-�-term X in long normal form (ifany) such that

` X : �

Step 1 As for BCIW.

Step 2 For each atom b and for each subsequence (j1; : : : ; jr) of (1; : : : ;m)�nd one BB0IW-�-term Y � xj iX1 : : :Xk (k � 0; 1 � i � r),such that Y 2 HRM(j1; : : : ; jr); Y : b and FV (Y ) = fxj1 ; : : : ;xjrg, if there is not already such a Y .

Step 3 For each subsequence (j1; : : : ; jr) of (1; : : : ;m) and for each longpositive subtype � of � , form a BB0IW-�-term Y by abstractionso that Y : � and FV (Y ) = fxj1 ; : : : ; xj rg, if we don't alreadyhave such a Y .

Now repeat steps 2 and 3 and if needs be add extra variables as for BCIW.As with the BCIW-algorithm this does not, in general, provide a decision

procedure.

EXAMPLE 97.

� = (a! b! c! d)! a! c! b! d

Step 1 x1 : a! b! c! d; x2 : a; x3 : c; x4 : b

Step 2 x1x2x4x3 : d and x1x2x4x3 2 HRM(1; 2; 4; 3)

Step 3 The only term with a positive subtype of � that can be formed is�x1x2x3x4:x1x2x4x3 : �; but this is not a BB0IW-�-term. Addingextra variables with the same types only allows us to generate thissame (modulo-� conversion) inhabitant of � .


EXAMPLE 98.

� = (c! a! a! a)! c! a! a

The only BCIW de�nable-�-term inhabitant of � was �x1x2x3:x1x2x3x3;this is also a BB0IW de�nable �-term.

THEOREM 99. Given a type � , the BB0IW long inhabitant search algo-rithm will, in �nite time, produce an inhabitant.

Proof. As for Theorem 96, except that we can replace subterms only bysubterms belonging to the same class HRM(j1; : : : ; jr) (1 � j1 < : : : <jr � m). �

6.7 BB0I Logic (T-W, P-W)

The search algorithm for this logic �nds elements of the appropriate typein HRM( ) \Once( ), as requried by Theorem 61(1).

The BB0I Logic or T-W(P-W) Decision Procedure or the BB0I Long In-habitant Search Algorithm

AimGiven a type � to �nd a closed BB0I-�-term X in long normal form (if any)such that

` X : �

MethodAs for BB0IW logic except that in the terms formed in Step 2 no freevariable may appear twice and that no extra variables need be added.

EXAMPLE 100.

� = (c! a! a! a)! c! a! a

The only BB0IW inhabitant of � is �x1x2x3:x1x2x3x3 and this is not aBB0I-de�nable �-term. Thus � has no BB0I inhabitants.

EXAMPLE 101.

� = [(a! a)! a]! (a! a)! a

Step 1 x1 : (a! a)! a; x2 : a! a; x3 : a

Step 2 x2x3 : a

Step 3 �x3:x3 : a! a �x3:x2x3 : a! a

Step 2 x1(�x3:x2x3) : a; x1(�x3:x3) : a

284 MARTIN BUNDER

Step 3 �x1x2:x1(�x3:x2x3) : �

THEOREM 102. Given a type � the BB0I long inhabitant search algorithmwill, in �nite time, produce an inhabitant or will demonstrate that � has noinhabitants. The algorithm will produce at most G(�) terms before termi-nating.

Proof. As for Theorem 99, except that each variable xi must appear exactlyonce in Y in any �xi:Y as with BCI logic. Also as in Theorem 89 and 92the procedure can be bounded. Note that the number of subsequences of asequence is the same as the number of subsets of the corresponding set. �

6.8 BB0IK Logic

The search algorithm for this logic �nds �-terms of the appropriate type inPRM( ) \ Once�( ).

The BB0IK Logic Decision Procedure or the BB0IK Long Inhabitant SearchAlgorithm

AimGiven a type � to �nd a closed BB0IK-� term X in long normal form (ifany) such that

` X : �:

MethodAs for BB0I logic except that in Step 2 HRM is replaced by PRM .

EXAMPLE 103.

� = b! (b! c)! a! c

Step 1 x1 : b; x2 : b! c; x3 : a

Step 2 x2x1 : c (x2x1 2 PRM (1; 2; 3))

Step 3 �x1x2x3:x2x1 2 �:

THEOREM 104. Given a type � the BB0IK long inhabitant search algo-rithm will, in �nite time, produce an inhabitant or will demonstrate that �has no inhabitant. The algorithm will produce at most G(�) terms beforeterminating.

Proof. As for Theorem 99. �


6.9 Some Other Logics

Bunder [1996] also gives the �-terms de�nable in terms of the combina-tors BB0, BT, BB0K, BIT, BITK, BITW, BTK, and BTW. Those forBCW and BB0W can easily be found.

Inhabitant �nding algorithms for these logics can easily be obtained asabove. Decision procedures can be obtained for those without W.

University of Wollongong, Australia

BIBLIOGRAPHY

[Barendregt, 1984] H. P. Barendregt. The Lambda Calculus, North Holland, Amsterdam,1984.

[Ben-Yelles, 1979] C.-B. Ben Yelles. Type Assignments in the Lambda Calculus, Ph.D.thesis, University College, Swansea, Wales, 1979.

[Bunder, 1996] M. W. Bunder. Lambda terms de�nable as combinators. TheoreticalComputer Science, 169, 3{21, 1996.

[Bunder, 1995] M. W. Bunder. Ben-Yelles type algorithms and the generation ofproofs in implicational logics. University of Wollongong, Department of Mathematics,Preprint Series no 3/95, 1995.

[Church, 1932] A. Church. A set of postulates for the foundation of logic. Annals ofMathematics, 33, 346{366, 1932.

[Church, 1933] A. Church. A set of postulates for the foundation of logic. (second paper).Annals of Mathematics, 34, 839{864, 1933.

[Crossley and Shepherdson, 1993] J. N. Crossley and J. C. Shepherdson. Extracting pro-grams from proofs by an extension of the Curry-Howard process. In Logical Methods,J. N. Crossley, J. B. Remmel, R. A. Shore and M. E. Sweedler, eds. pp. 222{288.Birkh�auser, Boston, 1993.

[Curry, 1930] H. B. Curry. Grundlagen der Kombinatorischen Logik. American Journalof Mathematics, 52, 509{536, 789{834, 1930.

[Curry and Feys, 1958] H. B. Curry and R. Feys. Combinatory Logic, Vol 1, North Hol-land, Amsterdam, 1958.

[de Bruin, 1970] N. G. de Bruin. The Mathematical Language AUTOMATH, Its Usage,and Some of Its Extensions. Vol. 125 of Lecture Notes in Mathematics, SpringerVerlag, Berlin, 1970.

[de Bruin, 1980] N. G. de Bruin. A survey of the project AUTOMATH. In To H. B.Curry, Essays on Combinatory Logic, Lambda Calculus and Formalism, J. R. Hindleyand J. P. Seldin eds. pp. 576{606. Academic Press, London, 1980.

[Dekker, 1996] A. H. Dekker. Brouwer 0.7.9- a proof �nding program for intuitionistic,BCI, BCK and classical logic, 1996.

[Helman, 1977] G. H. Helman. Restricted lambda abstraction and the interpretation ofsome nonclassical logics, Ph.D. thesis, Philosophy Department, University of Pitts-burgh, 1977

[Hindley, 1997] J. R. Hindley. Basic Simple Type Theory. Cambridge University Press,Cambridge, 1997.

[Hindley and Seldin, 1986] J. R. Hindley and J. P. Seldin. Introduction to Combinatorsand �-Calculus. Cambridge University Press, Cambridge, 1986.

[Hirokawa, 1996] S. Hirokawa. The proofs of � ! � in P � W . Journal of SymbolicLogic, 61, 195{211, 1996.

[Howard, 1980] W. A. Howard. The formulae-as-types notion of construction. In To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. R.Hindley and J. P. Seldin eds. pp 479{490. Academics Press, London, 1980.

286 MARTIN BUNDER

[Lauchli, 1965] H. Lauchli. Intuitionistic propositional calculus and de�nably non-emptyterms (abstract). Journal of Symbolic Logic, 30, 263, 1965.

[Lauchli, 1970] H. Lauchli. An abstract notion of realizability for which intuitionisticpredicate calculus is complete. In Intuitionism and Proof Theory, A. Kino et al., eds.pp. 227{234. North Holland, Amsterdam, 1970.

[Oostdijk, 1996] M. Oostdijk. LambdaCal. 2|proof �nding algorithms for intuitionisticand various (sub)classical propositional logics.Available on: http://www.win.tue.nl/�martijno/lambdacal/html, 1996.

[Sch�on�nkel, 1924] M. Sch�on�nkel. �Uber die Bausteine der mathematische Logik. Math-ematische Annale, 92, 305{316, 1924. English translation in From Frege to G�odel, J.van Heijenoort, ed. pp. 355{366. Harvard University Press,1967.

[Scott, 1970] D. S. Scott. Constructive validity, In Vol 125 of Lecture notes in Mathe-matics, pp. 237{275. Springer Verlag, Berlin, 1970.

[Trigg et al., 1994] P. Trigg, J. R. Hindley and M. W. Bunder. Combinatory abstractionusing B,B0 and friends. Theoretical Computer Science, 135, 405{422, 1994.

[Turing, 1942] A. M. Turing. R. O. Gandy An early proof of normalization by A. M.Turing. In To H. B. Curry: Essays... , pp. pp 453-455, 1980.

[Urquhart, 1990] A. Urquhart. The complexity of decision procedures in relevant logic.In Truth or Consequences: Essays in honour of Nuel Belnap. J. M. Dunn and A.Gupta, eds. pp. 61{75, Reidel, Dordrecht, 1990.

GRAHAM PRIEST

PARACONSISTENT LOGIC

Indeed, even at this stage, I predict a time when there will bemathematical investigations of calculi containing contradictions,and people will actually be proud of having emancipated them-selves from `consistency'. Ludwig Wittgenstein, 1930.1

1 INTRODUCTION

Paraconsistent logics are those which permit inference from inconsistent in-formation in a non-trivial fashion. Their articulation and investigation is arelatively recent phenomenon, even by the standards of modern logic. (Forexample, there was no article on them in the �rst edition of the Handbook.)The area has grown so rapidly, though, that a comprehensive survey is al-ready impossible. The aim of this article is to spell out the basic ideas andsome applications. Paraconsist logic has interest for philosophers, mathe-maticians and computer scientists. As be�ts the Handbook, I will concen-trate on those aspects of the subject that are likely to be of more interest tophilosopher-logicians. The subject also raises many important philosophicalissues. However, here I shall tread over these very lightly|except in thelast section, where I shall tread over them lightly.

I will start in part 2 by explaining the nature of, and motivation for, thesubject. Part 3 gives a brief history of it. The next three parts explain thestandard systems of paraconsistent logic; part 4 explains the basic ideas, andhow, in particular, negation is treated; parts 5 and 6 discuss how this basicapparatus is extended to handle conditionals and quanti�ers, respectively.In part 7 we look at how a paraconsistent logic may handle various othersorts of machinery, including modal operators and probability. The nexttwo parts discuss the applications of paraconsistent logic to some impor-tant theories; part 8 concerns set theory and semantics; part 9, arithmetic.The �nal part of the essay, 10, provides a brief discussion of some centralphilosophical aspects of paraconsistency.

In writing an essay of this nature, there is a decision to be made as to howmuch detail to include concerning proofs. It is certainly necessary to includemany proofs, since an understanding of them is essential for anything otherthan a relatively modest grasp of the subject. On the other hand, to proveeverything in full would not only make the essay extremely long, but distractfrom more important issues. I hope that I have struck a happy via media.

1Wittgenstein [1975], p. 332.


288 GRAHAM PRIEST

Where proofs are given, the basic de�nitions and constructions are spelledout, and the harder parts of the proof worked. Routine details are usuallyleft to the reader to check, even where this leaves a considerable amount ofwork to be done. In many places, particularly where the material is a deadend for the purposes of this essay, and is easily available elsewhere, I havenot given proofs at all, but simply references. Those for whom a modestgrasp of the subject is suÆcient may, I think, skip all proofs entirely.

Paraconsistent logic is strongly connected with many other branches oflogic. I have tried, in this essay, not to duplicate material to be found inother chapters of this Handbook, and especially, the chapter on RelevantLogic. At several points I therefore defer to these. There is no section ofthis essay entitled `Further Reading'. I have preferred to indicate in the textwhere further reading appropriate to any particular topic may be found.2

2 DEFINITION AND MOTIVATION

2.1 De�nition

The major motivation behind paraconsistent logic has always been thethought that in certain circumstances we may be in a situation where ourinformation or theory is inconsistent, and yet where we are required to drawinferences in a sensible fashion. Let ` be any relationship of logical conse-quence. Call it explosive if it satis�es the condition that for all � and �,f�;:�g ` �, ex contradictione quodlibet (ECQ). (In future I will omit setbraces in this context.) Both classical and intuitionist logics are explosive.Clearly, if ` is explosive it is not a sensible inference relation in an incon-sistent context, for applying it gives rise to triviality: everything. Thus, aminimal condition for a suitable inference relation in this context is thatit not be explosive. Such inference relationships (and the logics that havethem) have come to be called paraconsistent.3

Paraconsistency, so de�ned, is something of a minimal condition for alogic to be used as envisaged; and there are logics that are paraconsistentbut not really appropriate for the use. For example, Johansson's minimallogic is paraconsistent, but satis�es �;:� ` :�. One might therefore at-tempt a stronger constraint on the de�nition of `paraconsistent', such as: forno syntactically de�nable class of sentences (e.g., negated sentences), �, do

2The most useful general reference is Priest et al. [1989] (though this is already alittle dated). That book also contains a bibliography of paraconsistency up to about themid-1980s.

3The word was coined by Mir�o Quesada at the Third Latin American Symposium onMathematical Logic, in 1976. Note that a paraconsistent logic need not itself have aninconsistent set of logical truths: most do not. But there are some that do, e.g., any logicproduced by adding the connexivist principle :(� ! :�) to a relevant logic at least asstrong as B. See Mortensen [1984].

PARACONSISTENT LOGIC 289

we have �;:� ` �, for all � 2 �. This seems too strong, however. In manylogics, �;:� ` �, for every logical truth, �. If the logic is decidable, thenthere is a clear sense in which the set of logical truths is syntactically charac-terisable. Yet such logics would still be acceptable for many paraconsistentpurposes. Hence, this de�nition would seem to be too strong.4

In his [1974], da Costa suggests another couple of natural constraints ona paraconsistent logic, of a rather di�erent nature. One is to the e�ect thatthe logic should not contain :(�^ :�) as a logical truth. The rationale forthis is not spelled out. However, I take it that the idea is that if one hasinformation that contains � and :� one does not want to have a logicaltruth that contradicts this. Why not though? Since one is not ruling outinconsistency a priori, there would seem to be nothing a priori against this(though maybe for particular applications one would not want the situationto arise). As a general condition, then, it seems too strong. And certainly anumber of the logics that we will consider have :(�^:�) as a logical truth.

Another of the constraints that da Costa suggests is to the e�ect that thelogic should contain as much of classical|or at least intuitionist|logic, asdoes not interfere with its paraconsistent nature. The condition is somewhatvague, though its intent is clear enough; and again, it is too strong. Itassumes that a paraconsistent logician must have no objection to otheraspects of classical or intuitionist logic, and this is clearly not true. Forexample, a relevant logician might well object to paradoxes of implication,such as �! (� ! �).5

As an aside, let me clarify the relationship between relevant logics andparaconsistent logics. The motivating concern of relevant logic is somewhatdi�erent from that of paraconsistency, namely to avoid paradoxes of theconditional. Thus, one may take a relevant (propositional) logic to be onesuch that if �! � is a logical truth then � and � share a propositional pa-rameter. The interests of relevant and paraconsistent logics clearly convergeat many points. Relevant logics and paraconsistent logics are not coexten-sive, however. There are many paraconsistent logics that are not relevant,as we shall see. The relationship the other way is more complex, since thereare di�erent ways of using a relevant logic to de�ne a consequence relation.A natural way is to say that � ` � i� � ! � is a logical truth. Such aconsequence relation is clearly paraconsistent. Another is to de�ne logicalconsequence as deducibility, de�ned in the standard way, using some set ofaxioms and rules for the relevant logic. Such a consequence relation may,but need not, be relevant. For example, Ackermann's original formulationof E contained the rule : if ` � and ` :� _ � then ` �. This gives explo-

4Further attempts to tighten up the de�nition of paraconsistency along these lines canbe found in Batens [1980] (in the de�nition of `A-destructive', p. 201, clause (i) shouldread 6`L A), and Urbas [1990].

5Indeed, it is just this principle that ruins minimal logic for serious paraconsistentpurposes. For � and �!? (i.e., :�) give ?, and the principle then gives � !?.

290 GRAHAM PRIEST

sion by an argument often called the `Lewis Independent Argument', thatwe will meet in a moment.

Anyway, and to return from the digression: the de�nition of paraconsis-tency given here is weaker than suÆcient to guarantee sensible applicationin inconsistent contexts; but an elegant stronger de�nition is not at hand,and since the one in question has become standard, I will use it to de�nethe contents of this essay.

2.2 Inconsistency and Dialetheism

Numerous examples of inconsistent information/theories from which onemight want to draw inferences in a controlled way have been o�ered byparaconsistent logicians. For example:

1. information in a computer data base;

2. various scienti�c theories;

3. constitutions and other legal documents;

4. descriptions of �ctional (and other non-existent) objects;

5. descriptions of counterfactual situations.

The �rst of these is fairly obvious. As an example of the second, consider,e.g., Bohr's theory of the atom, which required bound electrons both toradiate energy (by Maxwell's equations) and not to (since they do not spiralinwards towards the nucleus). As an example of the third, just consider aconstitution that gives persons of kind A the right to do something, x, andforbids persons of kind B from doing x. Suppose, then, that a person inboth categories turns up. (We may assume that it had never occurred tothe legislators that there might be such a person.) In the fourth case, theinformation (in, say, a novel or a myth) characterises an object, and turnsout|deliberately or otherwise|to be inconsistent. To illustrate the �fth,suppose, for example, that we need to compute the truth of the conditional:if you were to square the circle, I would give you all my money. Applyingthe Ramsey-test, we see what follows from the antecedent (which is logicallyimpossible), together with appropriate background assumptions. (And Iwould not give you all my money!)6

There is no suggestion here that in every case one must remain contentwith the inconsistent information in question. One might well like to remove

6Many of these examples are discussed further in Priest et al. [1989], ch. 18. TheBohr case is discussed in Brown [1993]. Another kind of example that is sometimes citedis the information provided by witnesses at a trial. I �nd this less persuasive. It seems tome that the relevant information here is all of the form: witness x says so and so. (Thata witness is lying, or making an honest mistake, is always a possibility to be taken intoaccount.) And any collection of statements of this form is quite consistent.


some of the inconsistent information in the data base; reject or revise thescienti�c theory; change the law to eliminate the inconsistency. But thisis not possible in all of the cases given, e.g., for counterfactual condition-als with impossible antecedents. And even where it is, this not only maytake time; it is often not clear how to do so satisfactorily. (The matter iscertainly not algorithmic.) While we �gure out how to do it, we may stillbe in a situation where inference is necessary, perhaps for practical ends,e.g., so that we can act on the information in the data base; or manipulatesome piece of scienti�c technology; or make decisions of law (on other thanan obviously inconsistent case). Moreover, since there is no decision pro-cedure for consistency, there is no guarantee that any revision will achieveconsistency. We cannot, therefore, be sure that we have succeeded. (This isparticularly important in the case of the data base, where the deductions goon \behind our back", and the need to revise may never become apparent.)

In cases of this kind, then, even though we may not, ideally, be satis�edwith the inconsistent information, it may be desirable|indeed, practicallynecessary|to use a paraconsistent logic. Moreover, we know that manyscienti�c theories are false; they may still be important because they makecorrect predications in most, or even all, cases; they may be good approxi-mations to the truth, and so on. These points remain in force, even if thetheories in question contain contradictions, and so are (thought to be) falsefor logical reasons. Of course, this is not so if the theories are trivial; butthat's the whole point of using a paraconsistent logic.

One can thus subscribe to the use of paraconsistent logics for some pur-poses without believing that inconsistent information or theories may betrue. The view that some are true has come to be called dialetheism, adialetheia being a true contradiction.7 If the truth about some subjectis dialetheic then, clearly, a paraconsistent logic needs to be employed inreasoning about that subject. (I take it to be uncontentious that the setof truths is not trivial. Why this is so, especially once one has accepteddialetheism is, however, a substantial question.)

Examples of situations that may give rise to dialetheias, and that havebeen proposed, are of several kinds, including:

1. certain kinds of moral and legal dilemmas;

2. borderline cases of vague predicates;

3. states of change.

Thus, one may suppose, in the legal example mentioned before, that aperson who is A and B both has and has not the right to do x; or that in

7The term was coined by Priest and Routley in 1981. See Priest et al. [1989], p. xx.Note that some writers prefer `dialethism'.

292 GRAHAM PRIEST

a case of light drizzle it both is and is not raining; or that at the instant amoving object comes to rest, it both is and is not in motion.8

The most frequent and, arguably, most persuasive examples of dialetheiasthat have been given are the paradoxes of self-reference, such as the LiarParadox and Russell's Paradox. What we have in such cases, are apparentlysound arguments resulting in contradictions. There are many suggestions asto what is wrong with such arguments, but none of them is entirely happy.Indeed, in the case of the semantic paradoxes there is not (even after 2,000years) any consensus concerning the most plausible way to go. This givesthe thought that the arguments are, after all, sound, its appeal.9

Naturally, all the examples cited in this section are contestable. I willreturn to the issue of possible objections in the last part of this essay.

3 A BRIEF HISTORY OF PARACONSISTENT LOGIC

3.1 The Law of Non-contradiction and Paraconsistency

During the history of Western Philosophy, there have been a number of �g-ures who deliberately endorsed inconsistent views.10 The earliest were somePresocratics, including Heraklitus. In the middle ages, some Neo-Platonists,such as Nicholas of Cusa, endorsed contradictory views. In the modern pe-riod, the most notable advocate of inconsistent views was Hegel.11 These�gures are relatively isolated, however. It is something of an understate-ment to say that the dominant orthodoxy in Western Philosophy has beenstrongly hostile to inconsistency.12 Consistency has been taken to be piv-otal to a number of fundamental notions, such as truth and rational belief.This antipathy to contradiction is, historically, due in large part to Aristo-

8Many of these examples are discussed further in Priest et al. [1989], ch. 18, 2.2. Adiscussion of transition states and legal dialetheias can be found in chs. 11 and 13 ofPriest [1987]. Moral dilemmas are also discussed in Routley and Routley [1989]. Thedialetheic nature of vagueness is advocated in Pe~na [1989]. It has also been suggestedthat some contradictions in the Hegel/Marx tradition are dialetheic. For a discussion ofthis, see Priest [1989a].

9For further discussion, see Routley [1979] and Priest [1987], chs. 1-3.10And nearly every great philosopher has unwittingly endorsed inconsistent views.11In each case, there is, of course, some|though, I would argue, misguided|possibility

for exegetical attempts to render the views consistent. Other modern philosophers whosethought also appears to endorse inconsistency are Meinong and the later Wittgenstein.In their cases there is more scope for exegetical evasion. For further discussion on allthese matters, see Priest et al [1989], chs. 1, 2.12Eastern philosophy has been notably less so|though there is, again, room for ex-

egetical debate. The most natural interpretation of Jaina philosophy has them endorsinginconsistent positions. And major Buddhist logicians of the stature of Nargarjuna heldthat it was quite possible for statements to be both true and false. Signi�cant elemementsof inconsistency can also be found in Chinese philosophy. For further discussion of allthis, see Priest et al [1989], ch. 1, sect. 2.


tle's defense of the Law of Non-contradiction in the Metaphysics.13 Giventhis situation, it may therefore be surprising that the orthodoxy againstparaconsistency is a relatively recent phenomenon.

3.2 Paraconsistency Before the Twentieth Century

The major account of validity until this century was, of course, AristotelianSyllogistic. Now, consider any sentences of the Syllogistic E and I forms;for example, `No women are white' and `Some women are white'. Theseare contradictories. But the inference from them to, e.g., `All cows areblack', is not a valid syllogism. Syllogistic is not, therefore, explosive: it isparaconsistent.

It might be suggested that it is more appropriate to look for explosion inaccounts of propositional inference. Here the story is more complex, but theconclusion is similar. Aristotle had no elaborated account of propositionalinference. However, there are comments that bear on the matter scatteredthrough the Organon, and they have a distinctly paraconsistent avour. Forexample, in the Prior Analytics (57b3), Aristotle states that contradictoriescannot both entail the same thing. It would seem to follow that Aristotle didnot endorse at least one of (in modern notation) �^:� ` � and �^:� ` :�.For contraposing (a move that Aristotle endorses immediately before), weobtain � ` :(� ^ :�) and :� ` :(� ^ :�). Hence, not everything canfollow from a contradiction. In fact, there are reasons to suppose thatAristotle held a view of negation according to which the negation of anyclaim cancels that claim out. A contradiction has, therefore, no content,and entails nothing. This view of negation (which would now be called`connexivist') was endorsed by a number of subsequent logicians (notablyAbelard) well into the late middle ages.14

A theory of propositional inference was worked out much more thoroughlyby Stoic logicians, and the explosive nature of their theories is more plausiblefor the following reason. There is a famous argument for ECQ, often calledthe Lewis (independent) argument, after C. I. Lewis. This goes (in naturaldeduction form) as follows:

:�� :� _ �

�

13Book �, ch 4. The historical success of this defence is, however, out of all proportionto its intellectual weight. See Priest [1998e].14Much of this and the rest of the material in this subsection is documented in Sylvan

[2000], ch. 4. The discussion there is carried out in terms of the conditional, though it isequally applicable to the consequence relation.

294 GRAHAM PRIEST

The argument uses just two principles (three if you include the transitivityof deducibility): Addition (� ` �_�) and the Disjunctive Syllogism (�;:�_� ` �). As we shall see in due course, the Disjunctive Syllogism (DS) has,unsurprisingly, been rejected by most paraconsistent logicians. Now Stoiclogicians endorsed just this principle. The explosive nature of their logicwould therefore seem a good bet. Despite this, it probably was not: there isreason to suppose that their disjunction was an intensional one that requiredsome kind of connection between � and � for the truth of � _ �. If this isthe case, Addition fails in general, as does the Lewis argument.

It is not known who discovered the Lewis argument. Martin [1985] conjec-tures that it was William of Soissons in the 12th Century. (It was certainlyknown to, and endorsed by, some later logicians, such as Scotus and Buri-dan.) At any rate, William was a member of a group of logicians called theParvipontanians, who were known not only for living by a small bridge, butfor defending ECQ. This group may therefore herald the arrival of explosionon the philosophical stage. Whether or not this is so, after this time, somelogicians endorsed explosion, some rejected it, di�erent orthodoxies ruling atdi�erent times and di�erent places (though, possibly, the explosive view wasmore common). One group of logicians who rejected it is notable, since theyvery much pre�gure modern paraconsistent logicians. This is the CologneSchool of the late 15th Century, who argued against the DS on the groundthat if you start by assuming that � and :�, then you cannot appeal to �to rule out :� as the DS manifestly does.

Notoriously, logic made little progress between the end of the MiddleAges and the start of the third great period in logic, towards the end of the19th Century. With the work of logicians such as Boole and Frege, we seethe mathematical articulation of an explosive logical theory that has cometo be know, entirely inappropriately, as `classical logic'. Though, in its earlyyears, many objected to its explosive features, it has achieved a hegemony(though never a universality) in the logical community, in a (historically)very brief space of time. Whether this is because the truth was de�nitivelyand transparently revealed, or because at the time it was the only game intown, history will tell.

3.3 The Twentieth Century

A feature of paraconsistent logic this century is that the idea appears tohave occurred independently to many di�erent people, at di�erent timesand places, working in ignorance of each other, and often motivated bysomewhat di�erent considerations. Some, notably, for example, da Costa,have been motivated by the idea that inconsistent theories might be ofintrinsic importance. Others, notably the early relevant logicians, weremotivated simply by the idea that explosion, as a property of entailment, is


just too counter-intuitive.15

The earliest paraconsistent logics (that I am aware of) were given by twoRussians. The �rst of these was Vasil'�ev. Starting about 1910, Vasil'�evproposed a modi�ed Aristotelian syllogistic, according to which there isa new form: S is both P and not P . How, exactly, this form was tobe interpreted is contentious, though, a problem exacerbated by the factthat he was not in a position to employ the techniques of modern logic.This is not true of the second logician, Orlov, who, in 1929, gave the �rstaxiomatisation of the relevant logic R. Sadly, the work of neither Vasil'�evnor Orlov made any impact at the time.16

An important �gure who did have a good deal of in uence was the Polishlogician and philosopher Lukasiewicz. Partly in uenced by Meinong's ac-count of impossible objects, Lukasiewicz clearly envisaged the constructionof paraconsistent logics in his seminal 1910 critique of Aristotle on the Lawof Non-contradiction.17 And it was his erstwhile student, Ja�skowski, who,in 1948, produced the �rst non-adjunctive paraconsistent logic.18

Paraconsistent logics were again, independently, proposed in South Amer-ica in doctoral dissertations by Asenjo (1954, Argentina) and da Costa(1963, Brazil). Asenjo proposed the �rst many-valued paraconsistent logic.Da Costa gave axiom systems for a certain family of paraconsistent log-ics (the C systems), and produced the �rst quanti�ed paraconsistent logic.Many co-workers, such as Arruda and Lopari�c, joined da Costa in the next20 years, to produce an active school of paraconsistent logicians at Camp-inas (and later S~ao Paulo). They developed non-truth-functional semanticsfor the C systems, and articulated the subject in various other ways; thisincluded \rediscovering" Vasil'�ev, taking up the work of Ja�skowski, andformulating various other paraconsistent systems.19

Guided by considerations of relevance, an entirely di�erent approach toparaconsistency was proposed in England by Smiley in [1959], who artic-ulated the �rst �lter logic. Starting at about the same time, and drawingon the earlier work of Ackermann and Church, Anderson and Belnap inthe USA proposed a number of relevant paraconsistent logics of a di�erentkind. A research school quickly grew up around them in Pittsburgh, whichincluded co-workers such as Meyer and Dunn.20 The algebraic semantics

15The later Wittgenstein was also sympathetic to paraconsistency for various reasons,though he never articulated a paraconsistent logic. See, e.g., Marconi [1984].16On Vasil'�ev see Priest et al. [1989], ch. 3, 2.2 and Arruda [1977]. On Orlov, see

Anderson et al. [1992], p. xvii.17A synopsis of this is published in English in Lukasiewicz [1971].18For a discussion of Lukasiewicz and Ja�skowski, see Priest et al. [1989], ch. 3, 2.1,

2.3. Ja�skowski's work is translated into English in his [1969].19Discussion and bibliography can be found in Priest et al. [1989], 5.6. The most

accessible introduction to Asenjo's work is his [1966], and to da Costa's is his [1974]. DaCosta and Marconi [1989] reports much of the work of da Costa and his co-workers.20The work of this school is recorded in Anderson and Belnap [1975], and Anderson et

296 GRAHAM PRIEST

for relevant logics, in particular, was inaugurated largely by Dunn's 1966doctoral thesis.21

Investigation of things paraconsistent in Australia took o� in the early1970s with the discovery of world (intensional) semantics for negation byR. Routley (now Sylvan) and V. Routley (now Plumwood). This was de-veloped into an intensional semantics for the Anderson/Belnap logics|andmany others|by Routley,22 Meyer (now in Australia), and a school thatdeveloped around them in Canberra, which included workers such as Bradyand Mortensen. These semantics made the paraconsistent aspects of rele-vant logics plain.23 Later in the 1970s the cudgel for dialetheism was takenup by Priest (now Priest) and Routley.24

By the mid-1970s the paraconsistent movement was a fully internationalone, with workers in all countries cooperating (though not necessarily agree-ing!), and with logicians working in numerous countries other than the onesalready mentioned, including Belgium, Bulgaria, Canada and Italy. Somefeel for the state of the subject at the end of the 70s can be obtained fromPriest et al. [1989].25 The rest, as they say, is not history.

4 BASIC TECHNIQUES OF PARACONSISTENT LOGICS

An understanding of most paraconsistent logics can be obtained by look-ing at the strategies employed in virtue of which ECQ fails. There aremany techniques for achieving this end. In this part, I will describe themost fundamental. In the process, we will meet dozens of di�erent sys-tems of paraconsistent logic, often constructed along very di�erent lines.It is therefore necessary to have some common medium for comparison. Ihave chosen to make this semantics, and will specify systems in terms ofthese. (I would warn straight away though, that many of the systems wewill meet appeared �rst in proof theoretic terms. Indeed, some of the au-thors of these systems|e.g. Tennant|would privilege proof theory oversemantics.) When I give details of corresponding proof theories, I will usethe sort of proof theory (natural deduction, sequent calculus, or axiomatic)that seems most natural for the logic.

Because paraconsistency concerns only negation essentially, we can seethe essentials of paraconsistent logics in languages with very little logical

al. [1992].21See Anderson and Belnap [1975], and also the article on Relevant Logic in this volume

of the Handbook.22Whenever the name `Routley' is used without initial in this essay, it refers to Sylvan.23The work of this group is most accessible in Routley et al. [1982].24See, e.g., Routley [1979]. Priest's early work on the area is most accessible in Priest

[1987].25Despite the date, all the work in the collection was �nished by 1980. A number of

papers produced at the same time, that were not included in this, were published in aspecial issue of Studia Logica on paraconsistent logics (43 (1984), nos. 1 & 2).


apparatus. In this part, we will be concerned with a propositional languagewhose only connectives are negation, :, conjunction, ^, and disjunction, _.I will use lower case Roman letters, starting with p, for propositional param-eters, lower case Greeks, starting with �, for arbitrary formulas, and uppercase Greeks for sets of formulas. I will use j=C , j=I , and j=S5 for the con-sequence relations of classical logic, intuitionist logic and S5, respectively,and j= for the semantic consequence relation of whichever paraconsistentlogic happens to be the topic of discussion. If a proof theory is involved, Iwill use ` for the corresponding notion of deducibility.

4.1 Filtration

One of the simplest ways to prevent explosion is to �lter it, and any otherundesirables, out. Consider, for simplicity, the one-premise case. (Finitesets of premises can always be reduced to this by conjoining.) Let F (�; �)be any relationship between formulas. De�ne an inference from � to � to beprevalid i� � j=C � and F (�; �). The thought here is that for an inferenceto be correct, something more than classical truth-preservation is required,e.g., some connection between premise and conclusion. This is expressed byF . Usually, prevalidity is too weak as a notion of validity, since, in general,it is not closed under uniform substitution, and this is normally taken to bea desideratum for any notion of validity. However, closure can be ensuredif we de�ne an inference to be valid i� it is a uniform substitution instanceof a prevalid inference.

What inferences are valid depends, of course, entirely on the �lter, F .One that naturally and obviously gives rise to a paraconsistent logic is:F (�; �) i� � and � share a propositional parameter. (This collapses thenotions of validity and prevalidity, since if � and � share a propositionalparameter, so do uniform substitution instances thereof.) This logic is nota very interesting paraconsistent one, however, since, as is clear, p^:p j= �where � is any formula containing the parameter p.26

A di�erent �lter, proposed by Smiley [1959] is: F (�; �) i� � is not a(classical) contradiction and � is not a (classical) tautology.27 (Note that,according to this de�nition, � ^ :� = � is not prevalid, but it is valid,since it is an instance of p ^ q = p.) It is easy to see that on this accountp ^ :p does not entail q. The major notable feature of �lter logics is that,in general, transitivity of deducibility breaks down.28 For example, using

26A stronger �lter is one to the e�ect that all the variables of the premise occur inthe conclusion. This gives rise to a logic in the family of analytic implications. On thisfamily, see Anderson and Belnap [1975], sect. 29.6.27Filters of a related kind were also suggested by Geach and von Wright. See Anderson

and Belnap [1975], sect. 20.1.28Though it need not. First Degree Entailment, where transitivity holds, can be seen

as a �lter logic. See Dunn [1980].

298 GRAHAM PRIEST

Smiley's �lter, it is easy to see that p^:p j= p^ (:p_ q), p^ (:p_ q) j= q,but p ^ :p 6j= q.

One of the most interesting �lter logics, given by Tennant [1984], is ob-tained by generalising Smiley's approach. Let � and � be sets of sentences,and let `� j=C �' be understood in the natural way (every classical evalu-ation that makes every member of � true makes some member of � true).De�ne the inference from � to � to be prevalid i�: � j=C � and for noproper subsets of � and �, �0 and �0, respectively, do we have �0 j=C �0.Validity is then de�ned by closing under substitution as before. In this ac-count, a valid inference is one which is classically valid, and minimally so:there is no \noise" amongst premise and conclusion set.29

Tennant's j= is obviously non-monotonic (that is, adding extra premisesmay invalidate an inference). It also has the following property: if � j=C �,then there are subsets of � and �, �0 and �0, respectively, such that �0 j=�0. For if � j=C �, we can simply throw out premises and/or conclusionsuntil this is no longer true; the result is a prevalid, and so valid, inference.In particular, if � j=C � then for some �0 � �, �0 j= � or �0 j= �. In the�rst case, � follows validly from part of �; in the second, part of � can beshown to be inconsistent by valid reasoning.

Filtration can also be applied proof theoretically: we start with classicalproofs and throw out those that do not satisfy some speci�c criteria. Ten-nant's logic can be characterised proof-theoretically in just this way. For�nite premises and conclusions, the valid inferences are exactly those thatare provable in the Gentzen sequent calculus for classical logic, but whichdo not use the structural rules of dilution (thinning) and cut. Speci�cally,consider the sequent calculus whose basic sequents are of the form � : �,and whose rules are as follows. (�1;�2 means �1[�2; similarly, �; � means�[ f�g, and if something of this form occurs as a premise of a rule, it is tobe understood that � =2 �).

�; � : � � : �; �� : �;:� �;:� : �

�; � : � �; � : � �1 : �1; � �2 : �2; �

�; � ^ � : � �; � ^ � : � �1;�2 : �1;�2; � ^ �� : �; � � : �; � �1; � : �1 �2; � : �2

� : �; � _ � � : �; � _ � �1;�2; � _ � : �1;�2

29The restriction of Tennant's approach to the one-premise, one-conclusion, case obvi-ously gives Smiley's account. Smiley himself, handles the multiple-premise case, simplyby conjoining. As Tennant points out ([1984], p. 199), this generates a di�erent accountfrom his. It is not diÆcult to check that p _ q;:(p _ q) 6j= p ^ q for Smiley. (The con-joined antecedent is a contradiction; and any inference of which the conjoined form is asubstitution instance is not classically valid.) But it is valid for Tennant, since it is asubstitution instance of p _ q; r _ s;:(t _ q);:(r _ u) j= p ^ s.


Then we have � j= � i� the sequent � : � is provable. For the proof seeTennant [1984].30

Tennant's account of inference seems to capture very nicely what onemight call the `essential core' of classical inference. As an inference engineto be applied to inconsistent information/theories, it could not be appliedin the obvious way, however. This is because information is often heavilyredundant. For example, for Tennant's j=, we do not have p;:p _ q; q j= q.Yet given the information in the premises, it would certainly seem that weare entitled to infer q. Presumably, then, we would take � to follow from� i� for some �0 � �, �0 j= �.31 If we do this then more than transitivityfails; so does Adjunction. For :p; p _ q j= q and :q; p _ q j= p, hence bothp and q follow from f:p; p _ q;:qg = �. But for no subset of �, �0, do wehave �0 j= p ^ q. (� j= �, and if �0 is a proper subset of �, �0 6j=C p ^ q.)In this respect, Tennant's approach is similar to the next one that we willlook at.

4.2 Non-adjunction

All the other approaches that we will consider, except the last (algebraiclogics) accept validity as de�ned simply in terms of model-preservation.Thus, given some notion of interpretation, call it a model of a sentence if thesentence holds in the interpretation; an interpretation is a model of a set ofsentences if it is a model of every member of the set; and an inference is validi� every model of the premises is a model of the conclusion. In particular,then, if explosion is to be avoided, it must be possible to have models forcontradictions, which are not models of everything. Where the di�erencesin the following approaches lie is in what counts as an interpretation, andwhat counts as holding in it.

For the next approach, an interpretation, I , is a Kripke interpretationof some modal logic, say S5, employing the usual truth conditions. Eachworld in an interpretation may be thought of as the world according to someparty in a debate or discussion. This gives the approach its common name,discussive (or discursive) logic. I is a (discursive) model of sentence � i�� holds at some world in I , i.e., 3� holds in the model. Thus, � j= � i�� holds, discursively, in every discursive model of �, i.e., i� 3� j=S5 3�,where 3� is f3�;� 2 �g. This approach is that of Ja�skowski [1969].32 It isclear that discussive logic is paraconsistent, since we may have 3� and 3:�30The proof theory can be given a �ltered natural deduction form too. Essentially, clas-

sical deductions that have a certain \normal form" pass through the �lter. See Tennant[1980].31Though if we do this, symmetry suggests that we should take � to follow from � i�

for some �0 � � and �0 � �, �0 ` �0; in this case paraconsistency is lost since �;:� ` �.32Popper also seems to have had a similar idea in 1948. See his [1963], p. 321.

300 GRAHAM PRIEST

in an S5 interpretation, without having3�. For similar reasons, Adjunction(�; � j= � ^ �) fails. It should be noted, however, that � ^ :� j= �, so thelogic is not paraconsistent for conjoined contradictions.

A closely related approach can be found in Rescher and Brandom [1979].They de�ne validity as truth preservation in all worlds, but they augmentthe worlds of standard modal logic by inconsistent and complete worlds,constructed using operators _[ and _\. Speci�cally, worlds are constructedrecursively from standard worlds as follows. If W is a set of worlds, _[W is aworld such that � is true in _[W i� for some w in W , � is true in w; and _\Wis a world such that � is true in _\W i� for all w in W , � is true in w. As isintuitively clear, inconsistent worlds just provide another way of expressingwhat holds in a Ja�skowski interpretation. Incomplete worlds appear morenovel, but, in fact, add nothing. For if truth fails to be preserved in oneof these, it fails to be preserved in one of the ordinary worlds which gointo making it up. These ideas can be recast to show that the semantics ofRescher and Brandom, and of Ja�skowski are inter-translatable, and deliverthe same notion of validity.33

A notable feature of discussive logic is that � j= � i� for some � 2 �,� j=C �. (The proof from right to left is obvious. From left to right, supposethat for every � 2 �, � 6j=C �. Let w� be a classical world where � holds but� does not. If we take the interpretation whose worlds are fw�;� 2 �g thisis a counter-model for � j= �.) Thus, single-premise discussive inference isclassical, and there is no essentially multiple-premise inference. One wayto avoid the second of these features is to add an appropriate conditionalconnective. We will look at this later. Another way is to allow a certainamount of conjoining of premises. The question is how to do this in acontrolled way so that explosion does not arise.

One suggestion, made by Rescher and Manor, is, in e�ect, to allow con-joining up to maximal consistency.34 Given a set of premises, �, a max-imally consistent subset (mcs) is any consistent subset, �0, such that if

33Proof: Suppose that, discursively, � 6j= �. Then there is an interpretation such thatfor each � 2 �, there is some world, w�, such that � is true in w�, but � is not true inw�. Let w = _[fw�;� 2 �g, then w is a Rescher/Brandom counter-model. Conversely,suppose that � 6j= � for Rescher and Brandom. Then there is some world such that forevery � 2 �, � is true at w, but � is not. We show that there is a Ja�skowski counter-model. The result is proved by recusion on the construction of Rescher/Brandom worlds.If w is a standard world, the result is clear. So suppose that w = _\W , where the resultholds for all members of W . By de�nition, for every z 2W , and every � 2 �, � is true inz, but for some z, � is not true in z. Consider that z. This is a Rescher/Brandom counter-model to the inference. Hence, the result holds by recursion hypothesis. Alternatively,suppose that w = _[W , where the result holds for all members of W . By de�nition, forevery � 2 �, there is some w� 2W , such that � is true in w�, but � is not. By recursion,there must be a Ja�skowski countermodel for the inference �=�. � is true at some worldin this, but � is not. If we form the collection of worlds for all such �, this then gives usa Ja�skowski counter-model to the original inference.34Rescher and Manor [1970-1]. This takes o� from the earlier work of Rescher [1964].


� 2 ��0, �0 [f�g is inconsistent. We can now say that � follows from �i� for some mcs �0, �0 j=C �. In possible world terms, we can rephrase thisas follows. Let us say that an interpretation, I , respects � i� for every mcs�0, there is a world, w, in I such that �0 is true in w. Then it is not diÆcultto see that this policy is a variant of discussive logic: � j= � i� � holdsdiscussively in every interpretation that respects �. (If � follows classicallyfrom some �0, then it holds in every discussive interpretation that respects�. Conversely, suppose that it follows from no �0. Then for each �0 choosea world w�0 where �0 is true, but � is not. The interpretation containingall such w�0 is a countermodel.)

This policy is certainly stronger than simple discussive consequence. Forexample, it gives: p; q j= p ^ q. In fact, if � is (classically) consistentthen every classical consequence of � is a consequence. But it is still non-adjunctive: p;:p 6j= p ^ :p.35

A slightly di�erent way of proceeding is provided by Schotch andJennings.36 Given a �nite set, �, a partition is any family of disjoint sets,each of which is classically consistent, and whose union is �. The level of�, l(�), is the least n such that � can be partitioned into n sets (or, con-ventionally, 1 if there is no such n). � j= � i� l(�) =1 or, l(�) = n andfor any partition of � of size n, f�i; 1 � i � ng, there is an i such that�i j=C �. As with the previous approach, this de�nition can be convertedinto discussive terms, by taking our models to be those that respect thepremise set. But this time, an interpretation respects � i� for some parti-tion of the level of �, f�i; 1 � i � ng, and every i, there is a world in theinterpretation where �i is true.

Leaving aside the fact that Schotch and Jennings consider only �nitepremise sets, one di�erence between their approach and the previous oneconcerns the consequences of sets, �, with single inconsistent members.Such sets have no partitions, and so explode for Schotch and Jennings.They still have mcss though (e.g., �), and so do not explode for Rescherand Manor. If � has no single inconsistent member then Schotch and Jen-nings' consequence relation is included in that of Rescher and Manor. Forif f�i; i 2 ng is a partition of the premises, �, and for some i, �i j=C �,then �i can be extended to an mcs of �, and this classically entails �. Theconverse is not true, however. Let � = fp;:p; q; rg. This has two mcss,fp; q; rg and f:p; q; rg. Hence, for Rescher and Manor, q ^ r follows. But �has level 2, and one partition is ffp; qg; f:p; rgg. Neither of these classicallyentails q ^ r, so this does not follow for Schotch and Jennings (which seemswrong, intuitively).37

35Rescher and Manor also formulate a weaker policy of inference. � follows from � i�for all mcs �0, �0 j=C �. This logic is clearly adjunctive.36See their [1980], where they also discuss appropriate proof theories and modal

connections.37The same example shows that Schotch and Jennings' j=, unlike Rescher and Manor's,

302 GRAHAM PRIEST

Despite the di�erences, Schotch and Jennings' approach shares with thatof Rescher and Manor the following features: for consistent sets, conse-quence coincides with classical consequence; Adjunction fails. For Schotchand Jennings, like Ja�skowski, � ^ :� explodes. For Rescher and Manor, ithas no consequences (other than tautologies). The logics that we will lookat in subsequent sections are more discriminating concerning conjoined con-tradictions.

4.3 Interlude: Henkin Constructions

Before we move on to look at the other basic approaches to paraconsistentlogic, I want to isolate a construction that we will have many occasions touse. In a standard Henkin proof for the completeness of an explosive logic,we construct a maximally consistent set of sentences, and use this to de�nean evaluation. In the construction of the set, we keep something out of itby putting its negation in. As might be expected, these techniques do notwork in paraconsistent logic; but they can be generalised to do so. Whatplays the role of a maximally consistent set in a paraconsistent logic is aprime theory, where a set of sentences, �, is a theory i� it is closed underdeducibility; and it is prime i� � _ � 2 � ) (� 2 � or � 2 �). To keepsomething out in the construction of a prime theory, we have to exclude itexplicitly. I now show how.

Assume that the proof theory is to be given in natural deduction terms.For de�niteness I adopt the notational conventions of Prawitz [1965].38 Con-sider the following rules for conjunction and disjunction:

_I � �

� ^ �

^E � ^ ��

� ^ ��

_I �

� _ ��

� _ �

is non-monotonic, since we have q; r j= q ^ r.38In particular, something of the form:

�...�

in a rule indicates a subproof with � as one assumption|though there may be others|and conclusion �. If � is overlined, this means that the application of the rule dischargesit.


_E

� �...

...� _ �

Let ` be any proof theory that includes these rules. Write � ` � tomean that there are members of �, �1; :::; �n, such that � ` �1 _ ::: _ �n.Then if � 6` �, there are sets � � � and � � �, such that � 6` �, and� is a prime theory. To prove this, we enumerate the formulas of thelanguage: �0; �1; �2; :::, and de�ne a sequence of sets �n, �n (n 2 !) asfollows. �0 = �; �0 = �. If �n [ f�ng 6` �n, then �n+1 = �n [ f�ng and�n+1 = �n. Otherwise �n+1 = �n and �n+1 = �n [ f�ng. � =

Sn2! �n;

� =Sn2! �n.

It is not diÆcult to check by induction that for all n, �n 6` �n. (Supposethis holds for n; if �n [ f�ng 6` �n, the result for n + 1 is immediate. Sosuppose that �n [f�ng ` �n and �n+1 ` �n+1. Then �n ` f�ng[�n. Bya sequence of moves that amount to \cut", �n ` �n, contrary to inductionhypothesis.) By compactness, it follows that � 6` �.

It is also easy to check that � is a prime theory. Suppose that � ` �, but� 62 �. Then for some n, �n[f�g ` �n. Hence, � ` �. Next, suppose that� _ � 2 �, but � 62 � and � 62 �. Then for some m and n, �n [ f�g ` �n

and �m [ f�g ` �m. Hence � [ f� _ �g ` �, and so � ` �.

4.4 Non-truth-functionality

Let us now return to the other basic approaches to paraconsistent logics.On the �rst of these, explosion is invalidated by employing a non-truth-functional account of negation. Typically, this account of negation is im-posed on top of an orthodox account of positive logic. Thus, let an inter-pretation be a map, �, from the set of formulas to f1; 0g, satisfying just thefollowing conditions:

�(� ^ �) = 1 i� �(�) = 1 and �(�) = 1

�(� _ �) = 1 i� �(�) = 1 or �(�) = 1

In particular, the truth value of :� is independent of that of �. Validityis de�ned as truth preservation over all interpretations. It is obvious thatexplosion fails, since we may choose an evaluation that assigns both p and:p (and their conjunction) the value 1, whilst assigning q the value 0.

These semantics can be characterised very simply in natural deductionterms by just the rules _I , _E, ^I and ^E. Soundness is easy to check.For completeness, suppose that � 6` �. Then put � = f�g, and extend �

304 GRAHAM PRIEST

to a prime theory, �, with the same property, as in 4.3. De�ne a map, �,as follows:

�(�) = 1 if � 2 ��(�) = 0 if � =2 �

It is easy to check that � is an interpretation. Hence, we have the result.

This system contains no inferences that involve negation essentially. Forthis reason, : can hardly be thought of as a negation functor. Stronger para-consistent systems, where this is more plausibly the case, can be obtainedby adding conditions on the semantics. The following are some examples:39

(i) if �(�) = 0, �(:�) = 1

(ii) if �(::�) = 1, �(�) = 1

(iii) if �(�) = 1, �(::�) = 1

(iv) �(:(� ^ �)) = �(:� _ :�)(v) �(:(� _ �)) = �(:� ^ :�)

Sound and complete rule systems can be obtained by adding the correspond-ing rules, which are, respectively:

(i) � _ :�(ii)

::��

(iii)�::�

(iv):(� ^ �)

:� _ :�(v)

:(� _ �)

:� ^ :�

(Double underlining indicates a two-way rule of inference, and a zero premiserule, as in (i), can be thought of as an assumption that discharges itself.)The corresponding soundness and completeness proofs are simple extensionsof the basic arguments.

These additions give the ^;_;:-fragments of various systems in the lit-erature. (i) gives that of Batens' PI [1980]; (i) and (ii) that of da Costa'sC!;40 (i)-(v) that of Batens' PIS . In PIS every sentence is logically equiva-lent to one in Conjunctive Normal Form. This can be used to show thatPIS

39Some others can be found in Lopari�c and da Costa [1984], and Beziau [1990].40Semantics of the present kind for the da Costa systems were �rst proposed in da

Costa and Alves [1977].


is a maximal paraconsistent logic, in the sense that any logic that extendsit is not paraconsistent. (For details, see Batens [1980].)

Observe, for future reference, that if we add to PI or an extension thereofthe condition:

if �(�) = 1, �(:�) = 0

then all interpretations are classical, and so we have classical logic. As mayeasily be checked, adding this is sound and complete with respect to therule of inference:

� ^ :��

Another major da Costa system, C1, extends C! in accordance with thefollowing idea. It should be possible to express in the language the idea thata sentence, �, behaves consistently; and for consistent sentences classicallogic should apply. Let us write :(�^:�) as �0. Then it is natural enoughto suppose that �0 expresses the consistency of �. It does not, in any ofthe above systems, since we may have �^:�^�0 true in an interpretation.This is exactly what is ruled out by the condition:

(vi) �(�) = �(:�) then �(�0) = 041

(�(�) = �(:�) i� both are 1, by semantic condition (i). Note that (i) alsoguarantees the converse of (vi): �(�) 6= �(:�) then �(�0) = 1.)C1 is obtained by adding (vi) to C!, together with the following condition,

which requires consistency to be preserved under syntactic constructions:

(vii) if �(�0) = �(�0) = 1 then �((:�)0) = �((� ^ �)0) = �((� _ �)0) = 1

The deduction rules that correspond to (vi) and (vii) are, respectively:

(vi)� ^ :� ^ �0

�

(vii)�0

(:�)0�0 �0

(� ^ �)0�0 �0

(� _ �)0

Soundness and completeness of the extensions are easily checked.Now suppose that we have a piece of valid classical reasoning concerning

formulas composed of parameters p1; :::; pn. If we assume p01; :::; p0n then for

41Da Costa's actual condition is: if �(�0) = �(� � (� ^ :�)) = 1 then �(�) = 0. Thisis equivalent, given his account of the conditional.

306 GRAHAM PRIEST

every such formula, �, �0 follows by the appropriate applications of the rulesof (vii). Hence, whenever we have established � ^ :� we may apply rule(vi) to give �. But the addition of this inference is suÆcient to give classicallogic, as I have already observed. Hence any valid classical reasoning may berecaptured formally by adding the appropriate consistency assumptions.42

One �nal comment on treating negation non-truth-functionally. It is aconsequence of this that the substitutivity of provable equivalents breaksdown in general. For example, even though � is logically equivalent to� ^ � there is no guarantee that the negations of these formulas have thesame truth value in an interpretation.43

4.5 Many Values

The previous approach sticks with the traditional two truth values, and ob-tains a paraconsistent logic by making negation non-truth-functional. Thenext approach retains truth functionality, but drops the idea that thereare exactly two truth values. That is, such logics are many-valued.44 Amany-valued logic will be paraconsistent if it is possible for a formula andits negation both to take designated values (whilst not everything does). Anatural way of obtaining this is to have a designated value that is a �xedpoint for negation. The simplest such logic is a three valued one with values,t, b, and f , where t and b are designated, and the matrices are:

:t fb bf t

^ t b ft t b fb b b ff f f f

_ t b ft t t tb t b bf t b f

It will be noted that these are just the matrices of Lukasiewicz and Kleene's3-valued logics, where the middle value is normally thought of as undecid-able, or neither true nor false, and so not designated. It was the thoughtthat this value might be read as both true and false|a natural enoughthought, given dialetheism|and so be designated, that marks the startof many-valued paraconsistent logic. This was the approach proposed byAsenjo (see his [1966]), and others, e.g. Priest [1979], where the logic iscalled LP , a nomenclature that I will stick with here.

42It might be suggested that one ought not to take �0 as expressing consistency unlessit, itself, behaves consistently. This thought motivates the weaker da Costa systemC2, which is the same as C1, except that �0 is replaced everywhere by �0 ^ �00. Ofcourse, there is no reason to suppose that this expresses the consistency of � unless it,itself, behaves consistently. This thought motivates the da Costa system C3 where �0 isreplaced everywhere by �0 ^ �00 ^ �000. And so on for all the da Costa Systems Ci, for�nite non-zero i.43For a discussion of this in the context of da Costa's logics, see Urbas [1989].44For a general discussion of many-valued logics, see the articles on the topic in this

Handbook. See also, Rescher [1969].


LP may be generalised in various di�erent ways. One is as follows. If welet t = +1, b = 0 and f = �1 then the truth conditions of LP are:

�(:�) = ��(�)�(� ^ �) = minf�(a); �(�)g�(� _ �) = maxf�(a); �(�)g

The same conditions can be used for any set of integers, X , containing 0and closed under �. The designated values are the non-negative values. Letus call this a Sugihara generalisation, after the person who, in e�ect, �rstproposed a matrix of this kind, where X was the set of all integers.45

Any Sugihara generalisation, though semantically di�erent from LP , isessentially equivalent to it. Any LP countermodel is a Sugihara counter-model. But conversely, if we have a Sugihara countermodel, we can obtainan LP countermodel by mapping all positive values to +1, 0 to 0, and allnegative values to �1. A little thought is suÆcient to establish that themapping respects the matrices and preserves designated values, as required.

A di�erent way of generalising LP is as follows. If we let t = 1, b = 0:5and f = 0 then the truth conditions of LP are:

�(:�) = 1� �(�)�(� ^ �) = minf�(a); �(�)g�(� _ �) = maxf�(a); �(�)g

The same conditions can be used for any set of reals f0; 0:5; 1g � X � [0; 1],which is closed under subtraction (of a greater by a lesser). For suitablechoices of X , these are the matrices of the odd-numbered �nite Lukasiewiczmany-valued logics, and for X = [0; 1] they are the matrices of Lukasiewicz'continuum-valued logic. In Lukasiewicz' logics proper, the only designatedvalue is 1, which does not give a paraconsistent logic. But if one takesthe designated values to be fx; a < x � 1g (or fx; a � x � 1g) thenthe logic will be paraconsistent provided that 0 < a < 0:5 (or 0 < a �0:5). Let us call such logics Lukasiewicz generalisations. In a Lukasiewiczgeneralisation where the set of truth values is [0; 1], these may naturally bethought of as degrees of truth. Hence, such a logic is a natural candidatefor a paraconsistent fuzzy logic (logic of vagueness).46

It is not diÆcult to see that any Lukasiewicz generalisation is, in fact,equivalent to LP . As with the Sugihara generalisations, any LP coun-termodel is a Lukasiewicz countermodel; and conversely, any Lukasiewicz

45See Anderson and Belnap [1975], sect. 26.9.46A variation on this theme is given by Pe~na in a number of papers. (See, e.g., Pe~na

[1984].) Pe~na takes truth values to be an ordered set of more complex entities de�ned interms of the interval [0; 1].

308 GRAHAM PRIEST

countermodel can be collapsed into an LP countermodel by the mapping,f , de�ned thus:

f(x) = 1 if 1� a � x � 1= 0:5 if a < x < 1� a= 0 if 0 � x � a

or for the case where a is a designated value:

f(x) = 1 if 1� a < x � 1= 0:5 if a � x � 1� a= 0 if 0 � x < a

The generalisations of LP that we have considered in this section all,therefore, generate the same logic. What its proof theory is, we will see inthe next.

4.6 Relational Valuations

Standardly, semantic evaluations are thought of as functions from formulasto truth values, say, 0 and 1. Another way of invalidating explosion is totake them to be, not functions, but relations. A formula may then relateto both 0 and 1, another way of expressing the thought that a sentence isboth true and false. Assuming that negation behaves as usual, this meansthat both p and :p may relate to 1, whilst an arbitrary formula may not.A natural way of spelling out this idea is as follows.

If P is the set of propositional parameters, an evaluation, �, is a subsetof P � f0; 1g. The evaluation is extended to a relation for all formulas bythe familiar looking recursive clauses:

:��1 i� ��0

:��0 i� ��1

� ^ ��1 i� ��1 and ��1� ^ ��0 i� ��0 or ��0

� _ ��1 i� ��1 or ��1� _ ��0 i� ��0 and ��0

Let us say that a formula, �, is true in an interpretation, �, i� ��1, andfalse i� ��0; then validity may be de�ned as truth preservation in all inter-pretations. According to this account, classical logic is just the special casewhere multi-valued relations have been forgotten.


These semantics are the Dunn semantics for the logic of First DegreeEntailment, FDE.47 In natural deduction terms, FDE can be characterisedby the rules ^I , ^E, _I and _E, together with the rules:

:(� ^ �)

:� _ :�:� ^ :�:(� _ �)

�::�

Soundness is easily checked. For completeness, suppose that � 6` �. Extend� to a prime theory, �, with the same property, as in 4.3. Now de�ne aninterpretation, �, thus:

p�1 i� p 2 �p�0 i� :p 2 �

A straightforward (joint) induction shows that this characterisation extendsto all formulas. Completeness follows.

There are two natural restrictions that one may place upon Dunn evalu-ations:

#1 for every p, there is at most one x such that p�x#2 for every p there is at least one x such that p�x

Both conditions extend from propositional parameters to all formulas, by asimple induction. Thus, the �rst condition ensures that the relation is func-tional; the second that it is total. A relation that satis�es both conditionsis just a classical evaluation.

These extra conditions are sound and complete with respect to the extrarules:

� ^ :�� _ :�

respectively, as simple extensions of the completeness proofs demonstrate.We can express the relational semantics in functional terms by taking an

evaluation to be a function from formulas to subsets of f1; 0g, since thereis an obvious isomorphism between relations, �; and functions, �, given bythe condition:

��x i� x 2 �(�)

47Published in Dunn [1976], though he discovered them somewhat earlier than this. Inthe present context, it might be better to call the system `Zero Degree Entailment' sincethe language does not contain a conditional connective.

310 GRAHAM PRIEST

In this way, FDE can be seen as a many- (in fact, four-) valued logic.48

Restriction #2, which ensures that no formula takes the value �, gives athree-valued logic that is identical with LP . It is easy enough to check thatthe values f1g, f1; 0g, and f0g work the same way as t, b, and f , respectively.I will make this identi�cation in the rest of this essay. Restriction #1, whichensures that no formula takes the value f1; 0g, obviously gives an explosivelogic, which is, in fact, the strong Kleene three-valued logic. This is thereforea logic dual to LP .49

A feature of these semantics for LP and FDE is that they are monotonicin the following sense. Let �1 and �2 be functional evaluations. If for allpropositional parameters, p, �1(p) � �2(p) then for all �, �1(�) � �2(�).The proof of this is by a simple induction. One consequence of this for LPis worth remarking on. LP is clearly a sub-logic of classical logic, since ithas the classical matrices as sub-matrices. The consequence relation of LPis weaker than that of classical logic, since it is paraconsistent. But the setof logical truths of LP is identical with that of classical logic. For supposethat � is not valid in LP . Let � be an evaluation such that 1 =2 �(�).Let �0 be the interpretation that is the same as �, except that for everyparameter, p, if �(p) = f0; 1g, �0(p) = f0g. This is a classical evaluation;and by monotonicity, 1 =2 �0(�), as required.

Another feature of these semantics is the evaluation that assigns everypropositional parameter the value f1; 0g, vf1;0g; and, in the four-valuedcase, the evaluation that gives every parameter the value �, v�. A simpleinduction shows that these properties extend to all formulas. Thus, vf1;0gmakes all formulas true|and false|and v� makes every formula neither.In particular, then, FDE has no logical truths.50

4.7 Possible Worlds

Yet another, closely connected, way of invalidating explosion is to treatnegation as an intensional operator. This way was proposed by the Routleysin [1972]. A Routley interpretation is a structure, hW; �; �i, where W is a set(of worlds), � is a map from W to W , and � maps sets of pairs comprising aworld and propositional parameter to f1; 0g. (I will write �(w;�) as �w(�).)The truth conditions for conjunction and disjunction are the standard:

�w(� ^ �) = 1 i� �w(�) = 1 and �w(�) = 1

48In fact, the straight truth tables with values 1, 2, 3 and 4 were enunciated by Smiley.See Anderson and Belnap [1975], p. 161.49I will usually use the functional semantic representation for FDE and LP in the rest

of this essay. A word of warning, though: in the context of a dialetheic metatheory, thefunctional approach may have consequences that the relational approach, proper, doesnot have. See Priest and Smiley [1993], p. 49�.50Further interesting properties of LP and FDE are established in Pynko [1995a] and

[1995b].


�w(� _ �) = 1 i� �w(�) = 1 or �w(�) = 1

The truth conditions for negation are:

�w(:�) = 1 i� �w�(�) = 0

Note that if w� = w, these conditions just reduce to the classical ones. Anatural understanding of the � operator is a moot point.51 I will return tothe issue in a moment. Validity is de�ned in terms of truth preservation atall worlds of all interpretations.

In natural deduction terms, this system can be characterised by modify-ing that for FDE by dropping the rule for double negation, and replacingit with:

�...� :�:�

where, in the subproof, there are no undischarged assumptions other than�. Soundness is easily checked. For completeness, suppose that � 6` �.Extend � to a prime theory, �, with the same property, as in 4.3. Nowde�ne an interpretation, hW; �; �i, where W is the set of all prime theories,� is de�ned by the condition:

� 2 �� i� :� =2 �

and � is de�ned by:

��(p) = 1 i� p 2 � (#)

It is not diÆcult to check that if � is a prime theory, so is �� and hencethat � is well de�ned. First, suppose that � =2 �� and � =2 ��. Then :�and :� are in �. Since � is a theory, :� ^ :� 2 �, and so :(� _ �) 2 �.Hence, �_� =2 ��. Next, suppose that �� ` �, but � =2 ��. Then for some�1; :::; �n 2 ��, �1 ^ ::: ^ �n ` �. Hence, by contraposition and De Morgan:� ` :�1 _ ::: _ :�n. But :� 2 � ; hence :�1 _ ::: _ :�n 2 �. Since � isprime, for some 1 � i � n, :�i 2 �, i.e., �i =2 ��. Contradiction.

An easy recursion shows that (#) extends to all formulas. The resultfollows.51For some discussion and references, see the article on Relevance Logic and Entailment

in this Handbook.

312 GRAHAM PRIEST

The logic can be made stronger without (necessarily) ruining its paracon-sistency by adding further conditions on �. The most notable is: w�� = w.This is sound and complete with respect to the additional rule:

�::�

as a simple extension of the completeness argument demonstrates.These semantics are, in fact, very closely related to the those for FDE

of the previous section. Given an FDE interpretation, �, de�ne a Routleyevaluation on the worlds w and w�, as follows:

�w(p) = 1 i� 1 2 �(p)�w�(p) = 1 i� 0 =2 �(p)

A simple induction shows that these conditions follow for all formulas. Con-versely, we can turn the conditions into reverse. Given any Routley evalua-tion on a pair of worlds, w, w�, de�ne a Dunn evaluation by the conditions:

1 2 �(p) i� �w(p) = 10 2 �(p) i� �w�(p) 6= 1

Essentially the same induction shows that these conditions hold for all for-mulas. Hence, the two semantics are inter-translatable, and validate thesame proof theories.52 The translation also suggests a natural interpretationof the � operator. w� is that world characterised by the set of unfalsehoodsof w. (This is, of course, in general, distinct from the set of truths in afour-valued context.)

Under the above translation, the condition: 1 2 �(p) or 0 2 �(p), whichgives an LP interpretation, is equivalent to: �w(p) = 1 or �w�(p) 6= 1;imposing which condition on an intensional interpretation therefore givesan intensional semantics for LP .

4.8 Algebraic Semantics

Let us now turn to the �nal approach to paraconsistent logics that we willconsider, an algebraic one. In algebraic logic, an interpretation is a homo-morphism, �, from sentences into some algebraic structure, A = hA;^;_;:i;i.e., �(:�) = :�(�), �(� ^ �) = �(�) ^ �(�), etc. (I will use the same signsfor the connectives and the algebraic operations. Context, and the style ofvariable, will serve to disambiguate.) If the algebra is a lattice|as it usually

52Which shows that the contraposition rule is admissible in FDE, something that isnot at all obvious.


is, and will be in all the cases we consider|the consequence relation of thelogic is represented by the lattice order relation, de�ned in the usual way:a � b i� a ^ b = a. Thus, a logic will be paraconsistent if it is possible inthe algebra to have an a and b such that a ^ :a 6� b.

Several of the logics that we have looked at can be algebraicised. Con-sider, for example, FDE. If we take the four-valued semantics for this, wecan think of the values as a lattice whose Hasse diagram is as follows:

f1g% -

f1; 0g �- %

f0g(^ is lattice-meet; _ is lattice join; :f1; 0g = f1; 0g, and :� = �.) Thisgeneralises to a De Morgan algebra. A De Morgan algebra is a structure A =hA;^;_i, where hA;^;_;:i is a distributive lattice, and : is an involutionof period 2, i.e.:

::a = a

a � b) :b � :a

The structures take their name from the fact that in every such algebra:(a ^ b) = :a _ :b holds, as do the other De Morgan laws.

De�ne an inference �1; :::; �n=� to be algebraically valid i� for everyhomomorphism, �, into a De Morgan algebra, A, �(�1)^ :::^�(�n) � �(�).Then the algebraically valid inferences are exactly those of FDE. It iseasy to check that the rule system for FDE is sound with respect to thesesemantics. Completeness follows from completeness in the four-valued case.Alternatively, we can give a direct argument as follows.

Consider the relation � � �, de�ned by: � ` � and � ` �. One cancheck that this is an equivalence relation, and a congruence on the logicaloperators (i.e., if �1 � �1 and �2 � �2 then �1 ^ �2 � �1 ^ �2, etc.).53 IfF is the set of formulas, de�ne the quotient algebra, A = hF=�;^;_;:i,where, if [�] is the equivalence class of �, :[�] = [:�], [�] ^ [�] = [� ^ �],etc. One can check that A is a De Morgan lattice. Now, let � be thehomomorphism that maps every � to [�]. If �(�1) ^ ::: ^ �(�n) � �(�).Then [�1 ^ ::: ^ �n] � [�], i.e., [�1 ^ ::: ^ �n ^ �] = [�1 ^ ::: ^ �n]. Hence,�1 ^ :::^ �n ` �1 ^ :::^ �n ^ � and so �1 ^ :::^ �n ` �. Conversely, then, if�1 ^ ::: ^ �n 6` � then �(�1) ^ ::: ^ �(�n) 6� �(�), as required.

53The only tricky point concerns negation. For this, we need to appeal to the fact,which we have already noted, that if � ` � then :� ` :�. This can be establisheddirectly, by an induction on proofs.

314 GRAHAM PRIEST

It should be noted that not all the logics we have considered in previoussections algebraicise. In particular, the non-truth-functional logics resistthis treatment in general. This is for the same reason that the substitutivityof provable equivalents breaks down: the semantic value of :� is entirelyindependent of that of �. It cannot, therefore, correspond to any well-de�nedalgebraic operation.

The point can be made more precise in many cases. Suppose that A issome algebraic structure for a logic, and consider any interpretation, �, withvalues in the algebra, such that for some p, q and r, �(p) = �(q) 6= �(r).Then the condition �(�) = �(�) is a congruence relation on the set offormulas, and collapse by it gives a non-degenerate quotient algebra (i.e., analgebra that is neither a single-element algebra, nor the algebra of formulas).But many non-truth-functional logics can be shown to have no such thing.(See, e.g., Mortensen [1980].)

One �nal algebraic paraconsistent logic is worth noting. This is thatof Goodman [1981]. A Heyting algebra can be thought of as a distribu-tive lattice, with a bottom element, ?, and an operator, !, satisfying thecondition:

a ^ b � c i� a � b! c

(which makes ? ! ? the top element). We may de�ne :a as a! ?.Let T be a topological space. Then a standard example of a Heyting

algebra is the topological Heyting algebra hX;^;_;!;?i, where X is theset of open sets in T , ^ and _ are intersection and union, respectively, ? is�, and a ! b is (a _ b)o|overlining denotes complementation and o is theinterior operator of the topology. :a is clearly ao.

It is well known that for �nite sets of premises, Intuitionistic logic issound and complete with respect to the class of Heyting algebras, in fact,with respect to the topological Heyting algebras. That is, �1; :::�n j=I � i�for every homomorphism, �, into such an algebra, �(�1 ^ :::^�n) � �(�).54

The whole construction can be dualised in a natural way to give a para-consistent logic. A dual Heyting algebra is a distributive lattice, with a topelement, >, and an operator, , satisfying the condition:

a � b _ c i� a b � c

(which makes > > the bottom element). We may de�ne :a as > a. As may be checked, if T is a topological space, then the structurehX;^;_; ;>i is a dual Heyting algebra, where X is the set of closed setsof T , ^ and _ are intersection and union, respectively, > is the whole space,

54See, e.g., Dummett [1977], 5.3.


and a b is (a^ b)c|where c is the closure operator of the topology. :b isclearly b

c.

The logic generated by dual Heyting algebras is dual to Intuitionisticlogic. In particular, in Intuitionistic logic we have � ^ :� j= �, but not� j= � _ :�; and � j= ::�, but not ::� j= �. Thus in dual Intuitionistlogic, we have � j= � _ :� but not � ^ :� j= �; and ::� j= � but not� j= ::�. For a topological counter-model to the �rst, consider the real linewith its usual topology, and an interpretation, �, that maps p to [�1;+1],and q to �. Then �(p ^ :p) = f�1;+1g 6� � = �(q). (This illustrates howthe points in the set represented by p^:p may be thought of as the pointson the topological boundary between the set of points represented by p andthe set of points represented by :p.) For a counter-model to the second, let�(p) = f0g. Then �(::p) = � 6� �(p).

If j= � in dual Intuitionist logic, then j=C �, since the two-elementBoolean algebra is a dual Heyting algebra. Conversely, if � is any clas-sical tautology, its dual, �0, is a contradiction. Hence, j=C :�0. But thenby a result of Glivenko, j=I :�0, and so �0 j=I . Thus by duality, in dual Intu-itionist logic j= �. The logical truths of dual Intuitionist logic are thereforethe same as those of classical logic.

It is worth noting that just as Intuitionist logic can be given an inten-sional semantics, namely Kripke semantics, so can dual Intuitionist logic;we simply dualise the Kripke construction. For further details of all theabove, see Goodman [1981].55

5 CONDITIONAL CONNECTIVES

We have now looked at most of the basic techniques of paraconsistent logic,applied to languages containing only negation, conjunction and disjunction.56

I will call this language the basic language. Next, we will look at some im-portant extensions of these techniques (which do not ruin paraconsistency).In this part, we will start with the conditional, by which I mean some con-

55It is well known that in a certain well de�ned sense, Intuitionist logic can be seenas the \internal logic" of the category-theoretic structures called topoi. It is possible todualise the construction involved there to show that dual Intuitionist logic has an equallygood claim to that title. For details, see Mortensen [1995], who calls the ^, _, :-fragmentof a dual Heyting algebra a `paraconsistent algebra'.56There are others, such as the use of the techniques of combinatorial logic, but I will

not go into these here. For details, one can consult, e.g., Bunder [1984]. There ought to beyet more. The discussion of connexivism in 3.2 suggests that there ought to be a distinc-tive connexivist approach to paraconsistency. To date, this has not emerged. The mostarticulated modern connexivist logic is due to McCall (see sect. 29.8 of Anderson andBelnap [1975], which can also be consulted for references to other discussions). Althoughthis provides a connexivist treatment of the connective !, the logic of the basic languageis classical, and so explosive. Alternatively, one can formulate versions of relevant logicthat contain connexivist principles. See Routley [1978] and Mortensen [1984].

316 GRAHAM PRIEST

nective, ! (if necessary, added to the basic language), satisfying, at least,modus ponens: �; �! � j= �.

Although paraconsistency does not concern the conditional as such, manyof the paraconsistent logics that we have looked at have distinctive ap-proaches to the conditional. And this is no accident. If one identi�es �! �with the material conditional, � � �, de�ned in the usual way as :� _ �,then modus ponens reduces to the disjunctive syllogism. But in any logicwhere disjunction behaves normally and deducibility is transitive, the dis-junctive syllogism must fail, or explosion would arise, due to the \Lewisindependent argument". Speci�cally, in all the logics we have looked at ex-cept �lter logics and some of the non-adjunctive logics, the syllogism fails.In such logics, therefore, a distinct account of the conditional is required.For completeness' sake, we will start by considering the others.

5.1 ! as �

In �lter logics, we may simply identify ! with �. Things then proceed asbefore. A one-premise inference in this language, �=�, is prevalid i� it isclassically valid and F (�; �). It is valid i� it is a substitution instance of aprevalid inference.57

In the natural extension of Tennant's semantic approach, an inferencefrom � to � is prevalid i� � j=C � and for no proper subsets of � and�, �0 and �0, respectively, �0 j=C �0. The natural extension of the prooftheory is to add the conditional rules:

�; � : �;�

� : �! �;�

�1; � : �1 �;�2 : �2

�1;�2; �! � : �1;�2

Unfortunately, the equivalence between these two approaches now fails. For,semantically, p j= :p ! q (though the system is still paraconsistent); butwithout dilution there is no proof of the sequent p : :p! q. At this point,Tennant prefers to go with the proof theory rather than the semantics.He also prefers the intuitionist version, which allows at most one formulaon the right-hand side of a sequent. For further details, including naturaldeduction versions of the proof theory, see Tennant [1987], ch. 23.

In [1992] Tennant suggests modifying the rule for the introduction of !on the right.58 The � in the premise sequent is made optional, and thefollowing rule is added.

57One can modify this approach, invoking the �lter in the truth conditions of theconditional itself, to give logics of a more relevant variety. This is pursued in a number ofthe essays in Philosophical Studies 26 (1979), no. 2, a special issue on relatedness logics.58In fact, he gives the natural deduction rules. The sequent rules described are the

obvious equivalents.


�; � : �

� : �! �;�

The exact relationship between these rules and the above semantics is asyet unresolved.

In the non-adjunctive logics of Rescher and Manor, and Schotch andJennings: ! may again be identi�ed with �, producing no novelties. Themachinery of maximally consistent subsets and partitions carries straightover.

5.2 Discursive Implication

The situation is otherwise with discursive logic. Here a distinct approachis required, since, as we have already seen, the disjunctive syllogism failsdiscursively.

Given an S5 interpretation, Ja�skowski adds a conditional, ! (oftenwritten as �d, and called discursive implication), and de�nes � ! � as3� � �.59 It is easy to check that in discursive logic �; � ! � j= �,since 3�;3(3� � �) j=S5 3� (and so there are essentially multi-premiseinferences).

In fact, the logical truths of the pure ! fragment of discursive logic arethe same as those of the pure � fragment of classical logic. For let �� beany sentence containing only �s, and let �! be the corresponding sentencecontaining only !s. In an S5 interpretation with only one world, �� and3�! are equivalent. So if �� is not a classical logical truth, 3�! is nota discursive one. Conversely, suppose that �� is a classical logical truth.We need to show that 3�! is valid in every S5 model. As may easily bechecked, in S5, 3(3� � �) is logically equivalent to 3� � 3�. Hence,given 3�!, we may \drive the 3s inwards" to obtain a logically equivalentsentence where the modal operator applies only to propositional parameters.But this is a substitution instance of ��, and hence valid in S5. This resultdoes not carry over to the full language. For example, 6j= � ! (:� ! �),since, as may be checked, 6j=S5 3(3� � (3:� � �)).60

Full discursive logic can naturally be generalised in two obvious ways.The �rst is by using some modal logic other than S5. The second is bychanging the de�nition of what it is for a sentence, �, to hold discursivelyin an interpretation. We change this from3� holding to M� holding, whereM is some other modality (i.e., string of 3s and 2s). For references anddiscussion, see B laszczuk [1984] and Kotas and da Costa [1989].

59Given what amounts to Ja�skowski's identi�cation of truth with truth in some possibleworld, it might be more natural to de�ne �! � as 3�! 3�. This would have just thesame consequences.60The natural de�nition of the biconditional, � $ � , is (� ! �) ^ (� ! �). For

reasons not explained, Ja�skowski de�nes it as (� ! �) ^ (� ! 3�). This asymmetricand counter-intuitive de�nition would seem to have no signi�cant advantages.

318 GRAHAM PRIEST

5.3 Da Costa's C-systems

The natural way of extending the non-truth functional semantics of 4.4 toinclude a conditional connective, in keeping with the idea that such logics arejust the addition of a non-truth-functional negation to a standard positivelogic, is to give ! the classical truth conditions:

�(� ! �) = 1 i� �(�) = 0 or �(�) = 1

(Note that !, so de�ned, is distinct from �.) Adding this condition to thelogics of 4.4 (except, C! , which we will come to in a moment) gives the full(propositional) versions of the logics mentioned there; in particular it givesthe da Costa logic C1 (and the other Ci for �nite non-zero i). In each case,a natural deduction system can be obtained by adding the rules:

! E� �! �

�

(a)�

�! �

(b) � _ (�! �)

Soundness is proved as usual. The extension to the completeness proofamounts to checking that for a prime theory, �, � ! � 2 � i� � =2 � or� 2 �. From left to right, the result follows by (! E). From right to left:if � 2 � then the result follows from (a); if � =2 � then (�! �) 2 � by (b)and primeness.

If instead of (a) and (b), we add to any of these systems|except the oneswith a consistency operator; I will come to these in a second|the rule:

! I

�...�

�! �

we obtain, not classical positive logic, but intuitionist positive logic. (Theserules are well known to be complete with respect to this logic.) In particular,if we add ! I and ! E to the rule system for the basic language fragmentof C! we obtain da Costa's C!.

The intuitionist conditional is not, of course, truth functional, but avaluational semantics for C! can be obtained as follows. A semi-valuationis any function that satis�es the conditions for conjunction, disjunction andnegation, plus:


if �(�! �) = 1 then �(�) = 0 or �(�) = 1if �(�! �) = 0 then �(�) = 0

A valuation is any semi-valuation, �, satisfying the following condition. Let� be of the form �1 ! (�2 ! (�3:::! �n):::), where �n is not itself of theform � ! . Then if �(�) = 0 there is a semi-valuation, �0, such that forall 1 � i < n, �0(�i) = 1, and �0(�n) = 0. C! is sound and complete withrespect to this notion of valuation. For details, see Lopari�c [1986].61

Changing the deduction rules for ! to the intuitionist ones, makes nodi�erence for those logics that contain a consistency operator, and in par-ticular, the da Costa logics Ci for �nite i.62 The reason, in nuce, is thatthe consistency operator allows us to de�ne a negation with the propertiesof classical negation. As is well known, the addition of such a negation topositive intuitionist logic is not conservative, but produces classical logic.In more detail, the argument for C1 is as follows.63

De�ne :�� as :� ^ �o. Then it is easy to check that:

�(:��) = 1 i� �(�) = 0

In particular, then, :�� satis�es the rules for classical negation:

� _ :�� ^ :��

�

Given these, it is easy to show that � ! � a` :�� _ �. (Hint: from leftto right, assume � _ :�� and argue by cases. From right to left, assume �and :�� _ �, and argue to � by cases.) Hence, ! has the classical truthconditions.

5.4 Many-valued Conditionals

There are numerous ways to de�ne a many-valued conditional operator. Wewill just look at two of the more systematic.64

Given a Sugihara generalisation of LP , one can de�ne a conditional withthe following truth conditions:

61A Kripke-style semantics for C! can be found in Baaz [1986].62This was �rst observed, in e�ect, by da Costa and Guillaume [1965].63The argument for the other Cis is similar.64In the three-valued case, other de�nitions give the system of Asenjo and Tamburino

[1975], and the J systems of D'Ottaviano and da Costa [1970]. A natural many-valuedconditional, given the four-valued semantics of FDE, produces the system BN4 of Brady[1982].

320 GRAHAM PRIEST

�(�! �) = �(:� _ �) if �(�) � �(�)= �(:� ^ �) if �(�) > �(�)

This de�nition gives rise to \semi-relevant" logics, i.e., logics that avoid thestandard paradoxes of relevance, but are still not relevant.

In the case where the set of truth values is the set of all integers, thisgives the Anderson/Belnap logic RM . Proof-theoretically, RM is obtainedfrom the relevant logic R, which we will come to in the next section, byadding the \mingle" axiom:

` �! (�! �)

For details of proofs, see Anderson and Belnap [1975], sect. 29.3.In the 3-valued case, where the set of truth values is f�1; 0;+1g, the

conditions for ! give the matrix:

! +1 0 �1+1 +1 �1 �1

0 +1 0 �1�1 +1 +1 +1

and the stronger logic called RM3. This is sound and complete with respectto the axiomatic system obtained by augmenting the system R with theaxioms:

` (:� ^ �)! (�! �)` � _ (�! �)

For the proof, see Brady [1982].Turning to the second systematic approach, consider any Lukasiewicz

generalisation of LP . Lukasiewicz' truth conditions for his conditional, 7!,are as follows:

�(� 7! �) = 1 if �(�) � �(�)= 1� (�(�) � �(�)) if �(�) > �(�)

In the three-valued case, this gives the well known matrix:

7! 1 0.5 01 1 0.5 0

0.5 1 1 0.50 1 1 1


Now the most notable feature of the Lukasiewicz de�nition, given that 0:5 isdesignated, is that modus ponens fails. For example, consider a valuation, �,where �(p) = 0:5 and �(q) = 0. Then �(p 7! q) = 0:5. Hence p; p 7! q 6j= q.(Modus ponens is valid provided that the only designated value is 1, butthen the logic is not paraconsistent.)

Kotas and da Costa [1978] get around this problem by adding to thelanguage a new operator, �, with the truth conditions:

�(��) = 1 if �(�) is designated= 0 otherwise

and then de�ne a conditional, � ! �, as �� 7! �.65 They point out thesimilarity of this de�nition to Ja�skowski's de�nition of discursive implica-tion. (In fact, they use the symbol 3 instead of � because of this.)66

It is not diÆcult to check that modus ponens for ! holds. In fact, asKotas and da Costa point out, the ^, _, !-fragment of the logic is exactlypositive classical logic. The easiest way to see this is just to collapse thedesignated values to 1, and the others to 0, to obtain classical truth tables.

5.5 Relevant !s

Given a Routley interpretation (say one for FDE, though the other caseswill be similar), it is natural to treat ! intensionally. The simplest way isto give it the S5 truth conditions:

�w(�! �) = 1 i� for all w0 2 W (�w0(�) = 1) �w0(�) = 1)

Clearly, given an interpretation either � ! � is true at all worlds, or atnone. With the Routley � giving the semantics for negation, it follows thatthe same is true of negated conditionals. It also follows that �w(�! �) = 1i� �w�(� ! �) = 1 i� �w:(� ! �) 6= 1. Thus, the semantics validate therules:

LEM! (�! �) _ :(�! �)

EFQ!�! � :(�! �)

and so are unsuitable for serious paraconsistent purposes. Moreover, eventhough there may be worlds where �^:� is true, or where �_:� is false,

65In fact, their treatment is more general, since they consider the case in which theextension of � may be other than the set of designated values.66Pe~na [1984] de�nes an operator, F , on real numbers such that the value F� is 0 if

that of � is greater than 0, and 1 otherwise; and then de�nes a conditional operator,�C�, as F� _ �. The result is similar.

322 GRAHAM PRIEST

and so neither (�^:�)! � nor �! (�_:�) is valid, the system is not arelevant one since, e.g., j= p! (q ! q).

These facts may both be changed by modifying the semantics, by adding aclass of non-normal worlds. Thus, an interpretation is a structure hW;N; �; �i.The worlds in N are called normal; the worlds in W �N (NN) are callednon-normal. Truth conditions are the same as before, except that at non-normal worlds, the truth value of a conditional is arbitrary. Technically, �assigns to every pair of world and propositional parameter a truth value,as before, but for every w 2 NN and every conditional �! �, it now alsoassigns � ! � a value at w. This provides the value of � ! � at non-normal worlds (non-recursively). Validity is de�ned as truth preservationat all normal worlds of all interpretations.

If one thinks of the conditionals as entailments, then the non-normalworlds are those where the facts of logic may be di�erent. Thus, one maythink of non-normal worlds as logically impossible situations.67

The system described is called H in Routley and Lopari�c [1978].68 Itis sound and weakly complete (i.e., theorem-complete) with respect to thefollowing axiom system.

` �! �

` (� ^ �)! � ` (� ^ �)! �

` � ! (� _ �) ` � ! (� _ �)

` �$ ::�` (:� _ :�)$ :(� ^ �)

` (:� ^ :�)$ :(� _ �)

` (� ^ (� _ ))! ((� ^ �) _ (� ^ ))

If ` � and ` �! � then ` �If ` � and ` � then ` � ^ �If ` �! � and ` � ! then ` �!

If ` �! � then ` :� ! :�If ` �! � and ` �! then ` �! (� ^ )

If ` �! and ` � ! then ` (� _ �)!

Strong (i.e., deducibility-) completeness requires also the rules in disjunc-tive form.69 The disjunctive form of the �rst is: ` � _ and ` (�! �) _ then ` � _ . The others are similar.70

67For a further discussion of non-normality, see Priest [1992].68There are several other systems in the vicinity here. Some are obtained by varying

the conditions on �. Others, sometimes called the Arruda - da Costa P systems, areobtained by retaining the positive logic and adding a non-truth-functional negation. Fordetails, see Routely and Lopari�c [1978].69Which are known to be admissible anyway.70A sound and complete natural deduction system is an open question.


Soundness is proved as usual. The (strong) completeness proof is asfollows. We �rst show by induction on proofs that if � ` � then �_ ` �_ .It quickly follows that if � ` and � ` then � _ � ` . Now supposethat � 6` �. Extend � to a prime theory, �, with the same property, as in4.3. Call a set � a �-theory if it is prime, closed under adjunction, and� ! 2 � ) (� 2 � ) 2 �). Note that � is a �-theory. De�nethe interpretation hW;N; �; �i, where W is the set of �-theories; N = f�g,� 2 �� i� :� 62 � (which is well-de�ned). If � 2 NN then ��(� ! ) = 1i� � ! 2 �; and for all �:

��(p) = 1 i� p 2 �

Once it can be shown that this condition carries over to all formulas, theresult follows as usual. This is proved by induction. The only diÆcult caseconcerns ! when � = �. From right to left, the result follows from thede�nition of W . From left to right, the result follows from the followinglemma. If � ! 62 � then there is a �-theory, �, such that � 2 � and 62 �. To prove this, we proceed essentially as in 4.3, except that � ` �is rede�ned. Let �! be the set of conditionals in �; then � ` � is nowtaken to mean that there are �1; :::; �n 2 � and �1; :::; �m 2 � such that�! ` (�1 ^ :::^ �n)! (�1 _ :::_ �m). Now set � = f�g, and � = f�g, andproceed as in 4.3. The rest of the details are left as a (lengthy) exercise.71

If we add the Law of Excluded Middle to the axiom system:

` � _ :�

we obtain a logic that we will call HX . In virtue of the discussion in 4.7,one might suppose that this would be sound and complete if we add thecondition: for all w, and parameters, p, 1 = �w(p) or 0 = �w�(p). Thiscondition indeed makes � _ :� true in all worlds; but for just that reason,it also veri�es the irrelevant � ! (� _ :�). To obtain HX , we place thisconstraint on just normal worlds. The semantics are then just right, asmay be checked. For further details, see Routley and Lopari�c [1978]. Sincenormal worlds are now, in e�ect, LP interpretations, HX veri�es all thelogical truths of LP and so of classical logic.

A feature of this system is that substitutivity of equivalents breaks down.For example, as is easy to check, p$ q 6j= (r ! p)$ (r ! q). This can bechanged by taking the valuation function to work on propositions (i.e., set ofworlds), rather than formulas.72 The most signi�cant feature of semanticsof this kind is that there are no principles of inference that employ nested

71Details can be found in Priest and Sylvan [1992].72For details see Priest [1992].

324 GRAHAM PRIEST

conditionals in an essential way. This is due entirely to the anarchic natureof non-normal worlds. In e�ect, any breakdown of logic is countenanced.

One way of putting a little order into the anarchy without destroyingrelevance, proposed by Routley and Meyer,73 is by employing a ternary re-lation, R, to give the truth conditions of conditionals at non-normal worlds.An interpretation is now of the form hW;N;R; �; �i. All is as before, exceptthat � no longer gives the truth values of conditionals at non-normal worlds.Rather, for any w 2 NN , the truth conditions are:

�w(�! �) = 1 i� for all x; y 2W;Rwxy ) (�x(�) = 1) �y(�) = 1)

Note that this is just the standard condition for strict implication, exceptthat the worlds of the antecedent (x) and the consequent (y) have becomedistinguished. What, exactly, the ternary relation, R, means, is still amatter for philosophical deliberation. Validity is again de�ned as truthpreservation at all normal worlds.

These semantics give the basic system of aÆxing relevant logic, B. Anaxiom system therefor can be obtained by replacing the last two rules forH by the corresponding axioms:

` ((�! �) ^ (�! ))! (�! (� ^ ))` ((�! ) ^ (� ! ))! ((� _ �)! )

and adding a rule that ensures replacement of equivalents:

If ` �! � and ` ! Æ then ` (� ! )! (�! Æ)

The soundness and completeness proofs generalise those for H . Details canbe found in Priest and Sylvan [1992].

We may form the system BX proof theoretically by adding the Law ofExcluded Middle. Semantically, we proceed as with H , placing the appro-priate condition on normal worlds.

As with modal logics, stronger logics can be obtained by placing condi-tions on the accessibility relation, R. In this way, most of the logics in theAnderson/Belnap family can be generated. Details can be found in Restall[1993]. The strongest of these is the logic R, an axiom system for which isas follows:

` �! �` (�! �)! ((� ! )! (�! ))` �! ((�! �)! �)

73Initially, this was in Routley and Meyer [1973]. For further discussion of all thefollowing, see the article on Relevent Logic in this volume of the Handbook.


` (�! (�! �))! (�! �)` (� ^ �)! �; ` (� ^ �)! �` ((�! �) ^ (�! ))! (�! (� ^ ))` � ! (� _ �); ` �! (� _ �)` ((�! ) ^ (� ! ))! ((� _ �)! )` (� ^ (� _ ))! ((� ^ �) _ )` (�! :�)! (� ! :�)` ::�! �

with the rules of adjunction and modus ponens.The equivalence between the Dunn 4-valued semantics and the Routley

� operation that we noted in 4.7 suggests another way of obtaining anintensional conditional connective. In the simplest case, an interpretation isa structure hW; vi where W is a set of worlds and � is an evaluation of theparameters at worlds, but this time it is a Dunn 4-valued interpretation.The truth conditions for the basic language are as in 4.6, except that theyare relativised to worlds. Thus, using the functional notation:

1 2 �w(:�) i� 0 2 �w(�)0 2 �w(:�) i� 1 2 �w(�)

1 2 �w(� ^ �) i� 1 2 �w(�) and 1 2 �w(�)0 2 �w(� ^ �) i� 0 2 �w(�) or 0 2 �w(�)

1 2 �w(� _ �) i� 1 2 �w(�) or 1 2 �w(�)0 2 �w(� _ �) i� 0 2 �w(�) and 0 2 �w(�)

The natural truth and falsity conditions for ! are:

1 2 �w(�! �) i� for all w0 2W; (1 2 �w0(�)) 1 2 �w0(�))0 2 �w(�! �) i� for some w0 2W , 1 2 �w0(�) and 0 2 �w0(�)

These semantics do not validate the undesirable:

�! � :(�! �)

as their � counterparts do. But they are still not relevant. Relevant logicscan be obtained by adding a class of non-normal worlds. The semanticvalues of conditionals at these may either be arbitrary, as with H , or, aswith B, we may employ a ternary relation and give the conditions as follows:

326 GRAHAM PRIEST

1 2 �w(�! �) i� for all x; y 2W , Rwxy ) (1 2 �x(�))12 �y(�))0 2 �w(�! �) i� for some x; y 2W; Rwxy; 1 2 �x(�) and 0 2 �y(�)

As usual, extra conditions may be imposed on R. This constructionproduces a family of relevant logics distinct from the usual ones, and onethat has not been studied in great detail. One way in which it di�ersfrom the more usual ones is that contraposition of the conditional fails,though this can be recti�ed by modifying the truth conditions for ! byadding the clause: `and 0 2 �w0(�) ) 0 2 �w0(�)' (or in the case of non-normal worlds employing a ternary relation: `and 0 2 �x(�) ) 0 2 �y(�)').A more substantial di�erence concerns negated conditionals. Because ofthe falsity conditions of the conditional, all logics of this family validate� ^ :� j= :(� ! �). This is a natural enough principle, but absent frommany of the logics obtained using the Routley �.

The more usual relevant logics can be obtained with the 4-valued se-mantics, but only by using some ad hoc device or other, such as an extraaccessibility relation, or allowing only certain classes of worlds. For details,see Routley [1984] and Restall [1995].

5.6 ! as �

There is a very natural way of employing any algebra which has an orderingrelation to give a semantics for conditionals. One may think of the membersof the algebra as propositions, or as Fregean senses. The relation � on thealgebra can be thought of as an entailment relation, and it is then natural totake �! � to hold in some interpretation, �, i� �(�) � �(�). The problem,then, is to express the thought that � ! � holds in algebraic terms. Weobviously need an algebraic operator, !, corresponding to the connective;but how is one to express the idea that a! b holds when the algebra mayhave no maximal element?

A way to solve this problem for De Morgan algebras is to employ a desig-nated member of the lattice, e, and take the things that hold in the algebrato be those whose values are � e.74 While we are introducing new machin-ery, it is also useful algebraically to introduce another binary (groupoid) op-erator, Æ, often called`fusion', whose signi�cance we will come back to in a moment. We mayalso enrich the basic language to one containing a constant, e, and an op-erator, Æ, expressing the new algebraic features.

Thus, following Meyer and Routley [1972], let us call the structureA = hD; e;!; Æi a De Morgan groupoid i� D is a De Morgan algebra,hA;^;_;:i, and for any a; b; c 2 A:

74A di�erent way is to let T be a prime �lter on the lattice, thought of as the set of alltrue propositions. We can then require that a ! b 2 T i� a � b. For details, see Priest[1980].


e Æ a = aa Æ b � c i� a � b! cif a � b then a Æ c � b Æ c and c Æ a � c Æ ba Æ (b _ c) = (a Æ b) _ (a Æ c) and (b _ c) Æ a = (b Æ a) _ (c Æ a)

The �rst of these conditions ensures that e is a left identity on the groupoid.(Note that the groupoid may not be commutative.) And it, together withthe second, ensure that a � b i� e � a! b. The third and fourth ensure thatÆ respects the lattice operations in a certain sense. The sense is question inthat of a sort of conjunction, and this makes it possible to think of fusionas a kind of intensional conjunction.

An inference, �1; :::; �n=�; is algebraically valid i� for every homomor-phism, �, into a De Morgan groupoid, �(�1 ^ ::: ^ �n) � �(�), i.e., e ��((�1 ^ ::: ^ �n)! �)).75

These semantics are sound and complete with respect to the relevantlogic B of 5.5. Soundness is shown in the usual way, and completeness canbe proved, as in 4.8, by constructing the Lindenbaum algebra, and showingthat it is a De Morgan groupoid.

Stronger logics can be obtained, as usual, by adding further constraints.The condition: e � a_:a gives the law of excluded middle (and all classicaltautologies). Additional constraints on Æ give the stronger logics in the usualrelevant family, including R. Details of all the above can be found in Meyerand Routley [1972] (who also show how to translate between algebraic andworld semantics).76

Before leaving the topic of conditionals in algebraic paraconsistent logics,a �nal comment on dual intuitionist logic. Goodman [1981] proves that inthis logic there is no conditional operator (i.e., operator satisfying modusponens) that can be de�ned in terms of _;^ and :; and draws somewhatpessimistic conclusions from this concerning the usefulness of the logic. Suchpessimism is not warranted, however. Exactly the same is true in relevantlogic; this does not mean that a conditional operator cannot be added to thebasic language. And as Mortensen notes,77 given any algebraic structurewith top (>) and bottom (?) elements, the following conditions can alwaysbe used to de�ne a conditional operator:

�(�! �) = > if �(�) � �(�)= ? otherwise

75A di�erent notion of validity can be formulated using fusion thus: �(�1 Æ ::: Æ �n) ��(�), i.e., e � �((�1 ! (:::! (�n ! �):::).76See also Brink [1988]. A rather di�erent algebraic approach which produces a relevant

logic is given in Avro`n [1990]. This maintains an ordered structure, but dispenses withthe lattice. The result is a logic closely related to the intensional fragment of RM .77Mortensen [1995], p. 95.

328 GRAHAM PRIEST

Though this particular conditional is not suitable for robust paraconsistentpurposes since it satis�es: �! �;:(�! �) j= .

5.7 Decidability

Before we leave the topic of propositional logics, let me review, brie y, thequestion of decidability for the logics that we have looked at. Unsurprisingly,most (though not all) are decidable, as the following decision proceduresindicate. As will be clear, in many cases the procedures actually givencould be greatly optimised.

Any �lter logic is decidable if the �lter is. Given any inference, we cane�ectively �nd the set of all inferences of which it is a uniform substitutioninstance. Provided that the �lter is decidable, we can test each of these forprevalidity. If any of them is valid, the original inference is valid; otherwisenot.

Smiley's �lter is clearly decidable. So is Tennant's semantic �lter. Givenan inference with �nite sets of premises and conclusions, � and �, respec-tively, we can test the inference for classical validity. We may then testthe inferences for all subsets of � and �. (There is only a �nite numberof these.) If the original inference is valid, but its subinferences are not,it passes the test; otherwise not. Tennant's proof theory of 5.1 is also de-cidable. Anything provable has a Cut-free proof (since Cut is not a rule ofproof). Decidability then follows as it does in the case of classical logic.

Turning to non-adjunctive logics: Ja�skowski's discursive logic is decid-able; we may simply translate an inference into the corresponding one con-cerning S5, and use the S5 decision procedure for this. The same obviouslygoes for any generalisation, provided only that the underlying modal logicis decidable.

Rescher and Manor's logic is decidable in the obvious way. Given any�nite set of premises, we can compute all its subsets, the classical consis-tency of each of these, and hence determine which of the sets are maximallyconsistent. Once we have these, we can determine if any of them classicallyentails the conclusion. Similar comments apply to Schotch and Jennings'logic. Given any premise set, we can compute all its partitions, and so de-termine its level. For every partition of that size, we can test to see if oneof its members classically entails the conclusion.

Non-truth-functional logics are also decidable by a simple procedure.Given an inference, we consider the set of all subformulas of the sentencesinvolved (which is �nite). We then consider all mappings from these tof0; 1g, the set of which is also �nite. For each of these we go throughand test whether it satis�es the appropriate constraints in the obvious way.Throwing away all those that do not, we see whether the conclusion holds


in all that remain.78

All �nite many-valued logics are decidable by truth-tables. The in�nitevalued Lukasiewicz logics (and so their Kotas and da Costa augmentations)are not, in general, even axiomatisable, let alone decidable. (See Chang[1963].) This leaves RM . If there is a counter-model for an RM inference,there must be a number of maximum absolute value employed. Ignoring allthe numbers in the model whose absolute size is greater than this givesa �nite counter-model. Hence, RM has the �nite model property. Asis well known, any axiomatisable theory with this property is decidable.(Enumerate the theorems and the �nite models simultaneously. Eventuallywe must �nd either a proof of a countermodel.)

Dual intuitionist logic is decidable since intuitionist logic is. We justcompute the dual inference and test it with the intuitionist procedure.

This just leaves the logics of the relevant family. As we saw, the semanticsof these can take either a world form or an algebraic form. The question ofdecidability here is the hardest and most sensitive. The weaker logics in thefamily are decidable, and can be shown to be so by semantic methods (suchas �ltration arguments) and/or proof theoretic ones (such as Gentzenisationplus Cut elimination).79 The stronger ones, such as R, are not. Urquhart's[1984] proof of this fact contains one of the few applications of geometryto logic. A crucial principle in this context would seem to be contraction:(� ! (� ! �)) ! (� ! �) (or various equivalent forms, such as (� ^(�! �))! �). Speaking very generally, systems without this principle aredecidable; systems with it are not.

6 QUANTIFIERS

The novelty of paraconsistent logic lies, it is fair to say, almost entirely atthe propositional level. However, if a logic is to be applied in any seriousway, it must be quanti�cational. Most of the paraconsistent logics that wehave considered extend in straightforward ways to quanti�ed logics. In thissection I will indicate how. Let us suppose that the propositional language isnow augmented to a language, L, with predicates, constants, variables andthe quanti�ers 8 and 9 in the usual way. I will let the adicity of a predicatebe shown by the context. Propositional parameters can be identi�ed withpredicates of adicity 0. I will write �(x=t) to mean the result of substitutingthe term t for all free occurrences of x, any bound variables in � havingbeen relabelled, if necessary, to avoid clashes.

I will reserve the word `sentence' for formulas without free variables. Iwill always de�ne validity for inferences containing only sentences, thoughthe accounts could always be extended to ones employing all formulas, in

78For the method applied to the da Costa systems, see da Costa and Alves [1977].79See, respectively, Routley et al. [1982], sect. 5.9, and Brady [1991].

330 GRAHAM PRIEST

standard ways. Where quanti�ers have an objectual interpretation, andthe set of objects is D, I will assume|for the rest of this essay|that thelanguage has been augmented by a set of constants in such a way that eachmember of the domain has a name. In particular, I will always assume thatthe names are the members of D themselves, and that each object namesitself. This assumption is never essential, but it simpli�es the notation.

6.1 Filter and Non-adjunctive Logics

In �lter logics, we may simply take the �lter to be a relation on the extendedlanguage. Smiley's �lter works equally well, for example, when the notionof classical logical truth employed is that for �rst order, not propositional,logic. Similarly for Tennant's. In his case (without the conditional oper-ator), the semantics are sound and complete with respect to the sequentcalculus of 4.1 for the basic language, together with the usual rules for thequanti�ers:

� : �(x=c);�

� : 8x�;��; � : �

�;8x� : �

� : �;�

� : 9x�;��; �(x=c) : �

�; 9x� : �

where in the �rst and last of these, c does not occur in any formula in � or�. For proofs, see Tennant [1984]. (With the conditional operator added,the situation is di�erent, as we saw in 5.1.)

Non-adjunctive logic accommodates quanti�ers in an obvious way. Con-sider discursive logic. An inference in the quanti�ed language is discur-sively valid i� 3� j=CS5 3�, where CS5 is constant-domain quanti�ed S5.Clearly, any other quanti�ed modal logic could be used to generalise thisnotion.80

Rescher and Manor's approach and Schotch and Jennings' also gener-alise in the obvious way, the classical notion of propositional consequenceinvolved being replaced by the classical �rst-order notion. In the quanti�-cational case, the usefulness of these logics is moot, since the computationof classically maximally consistent sets of premises, or partitions, is highlynon-e�ective.

In all these logics, except Smiley's, the set of logical truths (in the ap-propriate vocabulary) coincides with that of classical quanti�er logic; hencethese logics are undecidable.81

80For details of quanti�ed modal logic, see the article on that topic in this Handbook.81I do not know whether Smiley's logic is decidable, though I assume that it is not.


6.2 Positive-plus Logics

Let us turn now to the logics that augment classical or intuitionist pos-itive logic with a non-truth-functional negation. Since the semantics ofthese are not truth functional, the most natural quanti�er semantics arenot objectual, but substitutional. Let me illustrate this with the simplestnon-truth-functional logic, with a classical conditional operator, but no se-mantic constraints on negation. Extensions of this to other cases are left asan exercise.

An interpretation is a pair hC; �i. C is a set of constants, and LC is thelanguage L augmented by the constants C. � is a map from the sentencesof LC to f1; 0g satisfying the same conditions as in the propositional case,together with:

�(8x�) = 1 i� for every constant of LC , c, �(�(x=c)) = 1�(9x�) = 1 i� for some constant of LC , c, �(�(x=c)) = 1

An inference is valid i� it is truth-preserving in all interpretations.The semantics are sound and complete with respect to the quanti�er

rules:

8I � _ �(x=c)

� _ 8x�provided that c does not occur in �, or in any undischarged assumption onwhich the premise depends.

8E 8x��(x=c)

9I �(x=c)

9x�

9E

�(x=c)...

9x� �

�

provided that c does not occur in � or in any undischarged assumption inthe subproof.

Soundness is proved by a standard recursive argument. For completeness,call a theory, �, saturated in a set of constants, C, i�:

9x� 2 � i� for some c 2 C, �(x=c) 2 �

332 GRAHAM PRIEST

8x� 2 � i� for every c 2 C, �(x=c) 2 �

It is easy to check that if � is a prime theory, saturated in C, then hC; �iis an interpretation, where � is de�ned by: �(�) = 1 i� � 2 �.

It remains to show that if � 6` � then � can be extended to a primetheory, �, saturated in some set of constants, C, with the same property;and the result follows as in the propositional case, using � to de�ne theinterpretation.

To show this, we augment the language with an in�nite set of new con-stants, C, and then extend the proof of 4.3 as follows. Enumerate theformulas of LC : �0, �1,... If 8x� or 9x� occurs in the enumeration, and theconstant c does not occur in any preceding formula, we will call �(x=c) awitness. Now, we run through the enumeration, as before, but this time, ifwe throw 9x� into the � side, we also throw in a witness; and if we throw8x� into the � side, we also throw in a witness. In proving that �n 6` �n,the only novelty is when a witness is present; and these can be ignored, by9E on the left, and 8I on the right. The rest of the proof is as in 4.3. Thesaturation of � in C follows from deductive closure and construction.

I observe that all the logics in this family contain positive classical quan-ti�er logic, and so are undecidable.

6.3 Many-valued Logics

Most of the many-valued logics with numerical values that we consideredin 4.5 and 5.4 had two particular properties. First, the truth value of aconjunction [disjunction] is the minimum [maximum] of the values of theconjuncts [disjuncts]. Second, the set of truth values is closed under greatestlower bounds (glbs) and least upper bounds (lubs), i.e., if Y � X thenglb(Y ) 2 X and lub(Y ) 2 X . Any such logic can be extended to a quanti�edlogic in a very natural way, merely by treating 8 and 9 as the \in�nitary"generalisations of conjunction and disjunction, respectively.

Speci�cally, a quanti�er interpretation adds to the propositional machin-ery, the pair hD; di where D is a non-empty domain of objects, d maps everyconstant into D, and if P is an n-place predicate, d maps P to a functionfrom n-tuples of the domain into the set of truth-values. Every sentence,�, can now be assigned a truth value, �(�), in the natural way. For atomicsentences, Pc1:::cn:

�(Pc1:::cn) = d(P ) hd(c1):::d(cn)i

The truth conditions for propositional connectives are as in the propositionallogic. The truth conditions for the quanti�ers are:


�(8x�) = glbf�(x=c); c 2 Dg�(9x�) = lubf�(x=c); c 2 Dg

Validity is de�ned in terms of preservation of designated values, as in thepropositional case.

I will make just a few comments about what happens when these de�ni-tions are applied to the many-valued logics we have looked at. The quanti-�ed �nite-valued logics of 4.5 all collapse into quanti�ed LP (which we willcome to in the next section), as extensions of the arguments given there,show. For a general theory of quanti�ed �nitely-many-valued logics, seeRosser and Turquette [1952]. Quanti�ed RM we will come to in a later sec-tion. In�nite-valued Lukasiewicz logics are proof-theoretically problematic.For a start, standard quanti�er rules may break down. In particular, 8x�may be undesignated, even though each substitution instance is designated.Thus, 8I may fail. (Similarly for existential quanti�cation.) Worse, as fortheir propositional counterparts, such logics are not even axiomatisable ingeneral.82

6.4 LP and FDE

The technique of extending a many-valued logic to a quanti�ed one canbe put in a slightly di�erent, and possibly more illuminating, way for thelogics with relational semantics, LP and FDE. An interpretation, I, is apair, hD; di, where D is the usual domain of quanti�cation, d is a functionthat maps every constant into the domain, and every n-place predicate intoa pair, hEP ; AP i, each member of which is a subset of the set of n-tuplesof D;Dn. EP is the extension of P ; AP is the anti-extension. For LPinterpretations, we require, in addition, that EP [ AP = Dn. Truth valuesare now assigned to sentences in accord with the following conditions. Foratomic sentences:

1 2 �(Pc1:::cn) i� hd(c1); :::; d(cn)i 2 EP

0 2 �(Pc1:::cn) i� hd(c1); :::; d(cn)i 2 AP

Truth/falsity conditions for connectives are as in the propositional case; andfor the quanti�ers:

1 2 �(8x�) i� for every c 2 D; 1 2 �(�(x=c))0 2 �(8x�) i� for some c 2 D, 0 2 �(�(x=c))

1 2 �(9x�) i� for some c 2 D; 1 2 �(�(x=c))

82See Chang [1963] for details.

334 GRAHAM PRIEST

0 2 �(9x�) i� for every c 2 D; 0 2 �(�(x=c))

An inference is valid i� it is truth-preserving in all interpretations. It shouldbe noted that if for every predicate, P , EP and AP are exclusive and ex-haustive, we have an interpretation of classical �rst order logic. All classicalinterpretations are therefore FDE (and LP ) interpretations.

These semantics are sound and complete if we add to the rules for LP orFDE, the rules 8I , 8E, 9I and 9E, plus:

8x:�:9x�

9x:�:8x�

Soundness is established by the usual argument. For completeness, supposethat � 6` �. Extend � to a set �, which is prime, deductively closed andsaturated in a set of new constants, such that � 6` �, as in 6.2. Then de�nean interpretation hD; di where D is the set of constants of the extended lan-guage, d maps any constant to itself, and for any predicate, P , its extensionand anti-extension are de�ned as follows:

hc1; :::; cni 2 EP i� Pc1:::cn 2 �hc1; :::; cni 2 AP i� :Pc1:::cn 2 �

We now establish that for all formulas, � :

1 2 �(�) i� � 2 �0 2 �(�) i� :� 2 �

The argument is a routine induction. Here are the cases for 8.

8x� 2 � , for all c; �(x=c) 2 � saturation, for all c; 1 2 �(�(x=c)) induction hypothesis, 1 2 �(8x�) truth conditions of 8

:8x� 2 � , 9x:� 2 � quanti�er rules, for some c; :�(x=c) 2 � saturation, for some c; 0 2 �(�(x=c)) induction hypothesis, 0 2 �(8x�) truth conditions of 8, 1 2 �(:8x�) truth conditions of :

The monotonicity property of the propositional logics LP and FDE car-ries over to the quanti�ed case. If I1 and I2 are any interpretations, withtruth value assignments �1 and �2, de�ne I1 � I2 to mean that I1 andI2 have the same domain, and for every predicate, P , the extension (anti-extension) of P in I1 is a subset of the extension (anti-extension) of P in


I2. A simple induction shows that if I1 � I2 then for all formulas, � (ina language with a name for every member of the domain), �1(�) � �2(�).As in 4.6, it follows that the set of logical truths of LP is exactly the sameas that of classical �rst order logic. And FDE has no logical truths (justconsider an interpretation that makes the extension and anti-extension ofevery predicate empty).

Since classical quanti�er logic is not decidable, neither is quanti�ed LP . IfP is any n-place predicate, let PLEM be the sentence: 8x1:::8xn(Px1:::xn _:Px1:::xn). If PLEM is true in an interpretation, then the extension andanti-extension of P , exhaust the n-tuples of the domain. If � is any formula,let �LEM be the conjunction of all formulas of the form PLEM , where Poccurs in �. It follows that �LEM j= � in FDE i� j= � in LP . Hence,quanti�ed FDE is undecidable too.

6.5 Relevant Logics

Turning to relevant logics, the issues are more complex. This is due to thefact that there are various approaches to these logics, the variety of thelogics themselves, and the intrinsic complexities of the stronger logics.

Let us start with the world semantics. As we saw in 5.5, a world semanticsfor a relevant logic with the Routely operator is a structure hW;N; �; � (; R)i,where W is a set of worlds, N is a subclass of normal worlds (the comple-ment being NN), � is the Routley operation (such that w = w��), and �assigns truth values to all propositional parameters at worlds. In the logicH , it also assigns values to conditionals at non-normal worlds. In strongerlogics, the ternary relation R is present, and is used to specify the values ofconditionals at non-normal worlds. When no constraints are placed on R,we have the logic B.

The simplest way of extending such semantics to those of a quanti�edlanguage is by removing � from the structure and adding a domain of quan-ti�cation, D, and a denotation function d. d speci�es a denotation for eachconstant (same at each world) and an extension for each n-place predicateat each world, dw(P ) � Dn. Truth conditions are given in the standardway. In particular, for the quanti�ers:

�(8x�) = 1 i� for every c 2 D; �(�(x=c)) = 1�(9x�) = 1 i� for some c 2 D, �(�(x=c)) = 1

An inference is valid i� it is truth-preserving in all normal worlds of allinterpretations.83

83More complex semantics can be employed in the usual variety of ways employed inmodal logic. (See the article on Quanti�ed Modal Logic in this Handbook.) In partic-ular, we might employ variable-domain semantics. This makes matters more complex.

336 GRAHAM PRIEST

Consider the following quanti�er axioms and rules (where ` is now takento indicate universal closure):

` 8x�! �(x=c)` �(x=c)! 9x�` � ^ 9x� ! 9x(� ^ �) x not free in �` 8x(� _ �)! (� _ 8x�) x not free in �

If ` 8x(�! �) then ` 9x�! � x not free in �If ` 8x(�! �) then ` �! 8x� x not free in �

It is easy to check that these axioms/rules are valid/truth-preserving forH . If they are added to the propositional axioms/rules for H , they are alsocomplete. For the proof, see Routley and Lopari�c [1980].84

If we strengthen the two rules to conditionals (so that the �rst of thesebecomes ` 8x(� ! �) ! (9x� ! �), etc.) and add them to the rulesfor B, they are also sound and complete. The same is true for a numberof the extensions of B, including BX . (For details, see Routley [1980a].)A notable exception to this fact is the system R. Though the system issound, it is, perhaps surprisingly, not complete.85 In fact, a proof-theoreticcharacterisation of constant domain quanti�ed R is still an open problem.The axioms and rules are complete for the stronger semi-relevant systemRM of 5.4.86

Since every relevant logic in the above family contains FDE, and this isundecidable, it follows that all the logics in this family are also undecidable.

6.6 Algebraic Logics

Given any algebraic logic, for which the appropriate algebraic structures arelattices, and in which conjunction and disjunction behave as lattice meetand join, there is, as with many-valued logics, a natural way to extend themachinery to quanti�ers. An algebra is complete i� it is closed under leastupper bounds (

W) and greatest lower bounds (

V), i.e., if the domain of the

algebra is A and B � A thenWB 2 A and

VB 2 A. If A is any algebraic

structure of the required kind, with domain A, then an interpretation isa triple hA; D; di, where D is the domain of quanti�cation, d maps everyconstant into D and every n-place predicate into a function from Dn into

(The philosophical gain, however, is dubious: world relativised quanti�ers can always bede�ned in constant-domain semantics, provided we have an Existence predicate.)84If one works with a free-variable notion of deducibility, as Routley and Lopari�c do,

one also has to add the rule of universal generalisation: if ` � then ` 8x�.85As Fine showed. Fine also produced a rather di�erent semantics with respect to

which it is complete. See Anderson et al. [1992], sects. 52 and 53.86See Anderson et al. [1992], sect. 49.2.


A. Algebraic values are then assigned to all formulas in the usual way. Inparticular, for quanti�ed sentences the conditions are:

�(8x�) =Vf�(�(x=c)); c 2 Dg

�(9x�) =Wf�(�(x=c)); c 2 Dg

I will comment on this construction for only two kinds of algebras. The�rst is when A is a De Morgan groupoid, or strengthening thereof. Inthis case, the above semantics clearly give quanti�ed relevant logics. Theirrelation to quanti�ed relevant logics based on the intensional semantics hasnot, as far as I am aware, been investigated.

The second is where A is a dual intuitionist algebra. In this case, thesemantics give a quanti�ed logic that is dual to quanti�ed intuitionist logic.For details, see Goodman [1981].87

6.7 A Brief Look Back

Now that we have surveyed a large number of paraconsistent logics up to aquanti�ed level|some very brie y|it would seem appropriate to look backfor a moment and put the systems into some sort of perspective.

The logics we have looked at fall roughly and inexactly into four cat-egories: non-transitive logics, non-adjunctive logics, non-truth-functionallogics and relevant logics. (The most interesting many-valued systems arezero degree relevant logic, FDE, or closely related to it, like LP , and somay be classed in this family.) The non-transitive logics seem to be goodfor extracting the essential juice out of classical inferences, but do not reallytake inconsistent semantic structures seriously. Non-adjunctive logics maybe just what one needs for certain applications (e.g., inferences in a database, where one would not necessarily want to infer �^:� from � and :�);they also take inconsistent structure seriously, though conjoined contradic-tions are handled indiscriminately, which makes them unsuitable for manyapplications. Non-truth-functional logics contain the whole of classical (orat least intuitionist) positive logic, and so are useful when strong canons ofpositive reasoning are required. However, this very strength is a weaknesswhen it comes to some important applications, as we shall see in connectionwith set theory. Undoubtedly the simplest and most robust paraconsistentlogic is the logic LP . When conditional operators are required, the relevantlogic BX is a good all-purpose paraconsistent logic. Its conditional operatoris satisfactory for many purposes, but may be considered relatively weak.It may be strengthened to give stronger relevant logics; but this, too, maycause a problem for some applications, as we shall see.

87There is also a topos-theoretic account of quanti�cation for dual intuitionistic logic.See Mortensen [1995], ch. 11.

338 GRAHAM PRIEST

7 OTHER EXTENSIONS OF THE BASIC APPARATUS

I now want to look at other extensions of the basic paraconsistent apparatus.One way or another, all the paraconsistent logics we have looked at canbe extended appropriately. However, it is tedious to run through everycase, especially when details are often obvious. Hence, I shall illustrate theextensions mainly with respect to just one logic. Since LP is simple andnatural, it recommends itself for this purpose. I will comment on otherlogics occasionally, when there is a point to doing so.

7.1 Identity and Function Symbols

LP|and all the other logics with objectual semantics that we have lookedat|can be extended to include function symbols and identity in the usualway. The denotation function, d, maps each n-place function symbol, f , toan n-place function on the domain. A denotation for every (closed) term,t, is then obtained by the usual recursive condition:

d(ft1:::tn) = d(f)(d(t1); :::; d(tn))

With functional terms present, the quanti�er rules of proof are extended toarbitrary (closed) terms in the usual way.

If we require the extension of the identity predicate to be fhx; xi ;x 2 Dgthen this is suÆcient to validate the usual laws of identity:

t = tt1 = t2 �(x=t1)

�(x=t2)

This does not require identity statements to be consistent. In LP the anti-extension of identity is any set whose union with the extension exhaustsD2, and so a pair can be in both the extension and the anti-extension ofthe identity predicate. In other logics, negated identities can be taken careof by whatever mechanism is used for negation. The completeness proof forquanti�ed LP can be extended to include function symbols and identity inthe usual Henkin fashion.

I note that description operators can be added in the obvious ways, withthe same panoply of options as in the classical case.88

7.2 Second-order Logic

Paraconsistent logics can also be extended to second order in the obviousways. Consider LP . We add (monadic) second order variables, X , Y ,::: to

88See the article of Free Logics in this Handbook.


the �rst-order language. Then, given an interpretation, hD; di, we extendthe language to one, LD, such that every member of the domain has a name,and for every pair E, A such that E [ A = D there is a predicate, P , withE and A as extension and anti-extension, respectively.89 The truth/falsityconditions for the second order universal quanti�er are then:

1 2 �(8X�) i� for every P in LD, 1 2 �(�(X=P ))

0 2 �(8X�) i� for some P in LD, 0 2 �(�(X=P ))

The truth/falsity conditions for the existential quanti�er are the dual ones.Appropriate monotonicity carries over to second order LP . Recall from

6.4 that if I1 and I2 are any interpretations, with truth value assignments�1 and �2, I1 � I2 means that I1 and I2 have the same domain, and forevery predicate, P , the extension (anti-extension) of P in I1 is a subset ofthe extension (anti-extension) of P in I2. The same sort of induction asin the �rst-order case shows that if I1 � I2 then for all formulas, �, inLD, �1(�) � �2(�). (The predicates added in forming LD have the sameextension/anti-extension in both interpretations; and thus atomic sentencescontaining them satisfy the condition.)

In the second order case, and unlike the �rst order case, the logical truthsof LP are distinct from their classical counterparts. For example, as is easyto check, in LP , j= 9X(Xa^:Xa) (just consider the predicate which has Das both extension and anti-extension).90 In fact, the logical truths of secondorder LP are inconsistent, since it is also a logical truth that 8X(Xa_:Xa),which is equivalent by quanti�er rules and De Morgan to :9X(Xa^:Xa).

7.3 Modal Operators

All the logics may have modal operators added to them in one way oranother. In the case of discursive logics, indeed, the semantics alreadyprovide for the possibility of alethic modal operators.

Adding modal operators to intensional logics where negation is handledby the Routley � operator is very natural, but su�ers problems similar tothose we witnessed at the start of 5.5 in connection with the conditional.Suppose we take an intensional interpretation and give the modal operatorsthe natural S5 conditions:

�w(2�) = 1 i� for every w0 2W; �w0(�) = 1

89This is the natural policy, since properties are characterised semantically by anextension/anti-extension pair. As in the classical case, there are other policies, e.g.,where only predicates corresponding to some restricted class of properties are added.90Second order FDE is constructed in the obvious way. The same sentence is a logical

truth of this, showing that, unlike the �rst order case, it has logical truths.

340 GRAHAM PRIEST

�w(3�) = 1 i� for some w0 2 W; �w0(�) = 1

(or evenN instead ofW ). Then the truth values of modalised statements arethe same at all worlds. Hence, �w(2�) = 1, �w�(2�) = 1, �w(:2�) =0. Hence 2�;:2� j= �, and so the logic is not suitable for serious paracon-sistent purposes. The problem does not arise if we attempt a modal logicweaker than S5, for then the truth conditions of modal operators are givenemploying a binary accessibility relation in the usual way, and the truthvalues of modal statements will vary across worlds. But, at least for somepurposes, an S5 modality is desirable.

These problems are avoided if we use the Dunn semantics for negation.The values of modalised formulas will still be the same at all worlds (in theS5 case), but we may now have both 2� and :2� true at a world. I willillustrate, again, with respect to LP . Let us start with the case where thebinary accessibility relation is arbitrary, the three-valued analogue of themodal system K.

An interpretation is now a structure hW;R; �i, where W is a set ofworlds; R is a binary relation on W ; and for each parameter, p, �w(p) 2ff1g; f1; 0g; f0gg. Truth/falsity conditions for the propositional connectivesare as in 5.5. The conditions for 2 are:

1 2 �w(2�) i� for every w0 such that wRw0, 1 2 �w0(�)0 2 �w(2�) i� for some w0 such that wRw0, 0 2 �w0(�)

and dually for 3.91

It is easy to check that at every world of an interpretation 2:� has thesame truth value as :3�, and dually. In fact, we can simply de�ne 3� as:2:�, and will do this in what follows.

To obtain a proof-theoretic characterisation for the logic, we add to therules for LP the following (chosen to make the completeness proof simple):

...� 2

2�

�...Æ 3�

3Æ

where there are no other undischarged assumptions in the sub-proofs.

2(� _ Æ)2� _3Æ

2 ^3�3( ^ �)

91If a conditional operator is required, we may add a class of non-normal worlds|andmaybe a ternary accessibility relation|and proceed as in 5.5.


2 1 ^ ::: ^ 2 n2( 1 ^ ::: ^ n)

3(Æ1 _ ::: _ Æn)

3Æ1 _ ::: _3ÆnSoundness is easily checked. For completeness, suppose that � 6` �. Extend� to a prime, deductively closed theory, �, with the same property, asin 4.3. De�ne an interpretation, hW;R; �i, where W is the set of primedeductively closed theories; �R� i� for all �:

2� 2 �) � 2 �

� 2 �) 3� 2 �

and � is de�ned by:

1 2 ��(p) i� p 2 �

0 2 ��(p) i� :p 2 �

All that remains is to show that these conditions extend to all formulas.Completeness then follows as usual. This is established by induction. Theonly diÆcult case is that for 2, which requires the following two-part lemma.

If 2� 62 � then there is a � 2W such that �R� and � 62 �. Proof: Let�2 = f ;2 2 �g and �3 = fÆ;3Æ 62 �g. Then �2 6` �;�3, by the �rstand second pair of rules, and a bit of �ddling with the third. Extend �2 toa prime, deductively closed set, �, with the same property, as in 4.3. Theresult follows.

If 3� 2 � then there is a � 2 W such that �R� and � 2 �. Proof:Let �2 and �3 be as before. Then �2; � 6` �3, by the �rst and secondpair of rules, and a bit of �ddling with the third. Extend �2; � to a prime,deductively closed set, �, with the same property, as in 4.3. The resultfollows.

We can now prove the induction step for 2:

2� 2 � , 8� s.t. �R�, � 2 � lemma in one directionde�nition of R in the other

, 8� s.t. �R�, 1 2 ��(�) induction hypothesis, 1 2 ��(2�)

:2� 2 � , 3:� 2 � de�nition of 3, 9�(�R� and :� 2 �) lemma in one direction

de�nition of R in the other, 9�(�R� and 0 2 ��(�)) induction hypothesis, 0 2 ��(2�)

342 GRAHAM PRIEST

Stronger modal logics can be obtained by placing conditions on R, andcorresponding conditions on the proof theory. But even if we make R uni-versal (so that for all x and y, xRy), and obtain the analogue of S5, westill do not get 2�;:2� j= �. To see this, merely consider the interpreta-tion with one world, w, which accesses itself; and where �w(p) = f1; 0g and�w(q) = f0g. It is easy to check that �w(2p) = �w(:2p) = f1; 0g. Hence,2p;:2p 6j= q.

The same treatment can be given to temporal operators. If we take these,as usual, to be F and G for the future, and P and H for the past, then(three-valued) tense logic gives F and G the same truth conditions as 3 and2, respectively; and P and H are the same, except that R is replaced by itsconverse, �R (where x �Ry i� yRx). Appropriate soundness and completenessproofs for the case where R is arbitrary are obtained by modifying thealethic modal argument,92 and stronger tense logics are obtained by addingconditions on R, in the usual way.93

Let me also mention conditional operators, >, of the Lewis/Stalnakervariety. These are modal (binary) operators, and can be given LP (or FDE)semantics in the same way that they are given a more usual semantics.For example, for the Stalnaker version, one extends interpretations with aselection function, f(w;�), thought of as selecting the nearest world to wwhere � is true. � > � is then true at w i� � is true at f(w;�). Detailsare left as a very non-trivial exercise.94

7.4 The Paraconsistent Importance of Modal Operators

Let me digress from the technical details to say a little about why modaloperators are important in the context of paraconsistency. The reason issimply that so many of the natural areas where one might want to apply aparaconsistent logic involve them.

Take alethic modalities �rst. Even though one might not think that thereare any true contradictions, one might still take them to be possible, in thesense of holding in some situations, such as �ctional or counterfactual ones.Thus, one might hold that for some p, 3(p ^ :p). This has a simple andobvious model in the above semantics. In this context, let me mention againthe importance for counterfactual conditionals of worlds where the impos-sible holds; \impossible worlds" are just what one needs to evaluate suchconditionals, according to the Lewis/Stalnaker semantics in whose directionI have just gestured.

Some have been tempted not just by the view that some contradictions

92See Priest [1982].93See the article on Tense Logic in this Handbook.94For a paraconsistent theory of conditionals of this kind, and of many other modal

operators, that employs the Routley � to handle negation, see Routley [1989].


are possible, but by the view that everything is possible.95 The valuation�f1;0g assigns every formula the value f1; 0g. (See 4.6). Hence, any inter-pretation that contains �f1;0g as one world will verify 3�, for all �, at anyworld that accesses it.96 If we interpret the modal operators 2 and 3 as thedeontic operators O (it is obligatory that) and P (it is permissible that), re-spectively, then the thesis that everything is possible becomes the nihilisticthesis that everything is permissible|what, according to Dostoevski, wouldbe the case if there is no God.

Less exotically, standard deontic logic su�ers badly from explosion.97

Since in classical logic �;:� j= � it follows that O�;O:� j= O�: if youhave inconsistent obligations then you are obliged to do everything. Thisis surely absurd. People incur inconsistent obligations; this may give riseto legal or moral dilemmas, but hardly to legal or moral anarchy.98 Andone does not have to believe in dialetheism to accept this. Unsurprisingly,deontic explosion fails, given the semantics of the previous section: justconsider the interpretation where there is a single world, w; R is universal;�w(p) = f1; 0g and �w(q) = f0g. It is not diÆcult to check that �w(Op) =�w(O:p) = f1; 0g, whilst �w(Oq) = f0g.

What is often taken to be the basic possible-worlds deontic logic (calledKD by Chellas [1980], p. 131) makes matters even worse, by requiring thatin an interpretation the accessibility relation be serial: for all x, there isa y such that xRy. This validates the inference O�=P�. It also validatesthe inference O:�=:O�. Hence we have, classically, O�;O:� j= O� ^:O� j= �; one who incurs inconsistent obligations renders the world trivial.Someone who believes that there are deontic dilemmas may just have tojettison the view that obligation entails permission, and so give up seriality.But on the above account one can retain seriality, and so both the aboveinferences; for O� ^ :O� 6j= �, as the countermodel of the last paragraphshows.99

Another standard way of interpreting the modal operator 2 is as anepistemic operator, K (it is known that), or a doxastic operator B (it isbelieved that). In these cases, classically, one would almost certainly want toput extra constraints on the accessibility relation, though what these shouldbe might be contentious: all can accept re exivity (xRx) for K (but notfor B) since this validates K� j= �. Whether one would want transitivity((xRy&yRz)) xRz) is much more dubious forB andK, since this gives the

95E.g., Mortensen [1989].96A similar, but slightly more complex, construction can be employed to the same

e�ect if the logic has a conditional operator.97See the article on Deontic Logic in this Handbook for details of Deontic Logic, in-

cluding the possible-worlds approach.98For further discussion, see Priest [1987], ch. 13.99We have just been dealing with some of the \paradoxes of deontic logic". There are

many of these. Arguably, all of them|or at least all the serious ones|are avoided byusing a paraconsistent logic with a relevant conditional. See Routley and Routley [1989].

344 GRAHAM PRIEST

highly suspect K� j= KK� and B� j= BB�. All this applies equally to thesemantics of the previous section. Moreover, the paraconsistent semanticssolve problems for doxastic logic of the same kind as for deontic logic. Itis clear that people sometimes have inconsistent beliefs (if not knowledge).Standard semantics give B�;B:� j= B�. Yet patently someone may haveinconsistent beliefs without believing everything.100

Observations such as this are particularly apt in the branch of AI knownas knowledge representation, where it is common to use epistemic operatorsto model the information available to a computer. (See, e.g., a number ofthe essays in Halpern [1986].) Such information may well be inconsistent.

Finally, to tense operators. Whilst one does not have to be a dialetheistto hold that inconsistencies may be believed, obligatory, or true in somecounterfactual situation, one does have to be, to believe that they were orwill be true. Such views have certainly been held, however. Following Zeno,the whole dialectical tradition holds that contradictions arise in a state ofchange. To see one of the more plausible examples of this, just consider astate described by p which changes instantaneously at time t0 to a statedescribed by :p. What is the state of a�airs at t0? One answer is that att0, p ^ :p is true. Indeed, the contradictory state is the state of change.101

This can be modeled by the paraconsistent interpretation hW;R; �i, whereW is the set of real numbers (thought of as times); R is the standard or-dering on the reals; and � is de�ned by the condition:

�t(p) = f1g if t < t0= f1; 0g if t = t0= f0g if t > t0

It is easy to check that this interpretation veri�es the inference: p^F:p=(p^:p) _ F (p ^ :p), which we might call `Zeno's Principle': change impliescontradiction.

7.5 Probability

Probability is not a modal notion. But it, too, has paraconsistent signi�-cance. One of the most natural ways of constructing a paraconsistent proba-bility theory is to extract one from a class of paraconsistent interpretations,in the manner of Carnap.102

100If you believe classical logic, then you might suppose that they are rationally com-mitted to everything, but that is quite di�erent. Even here, however, an explosive logicwould seem to go astray. Dialetheism aside, situations such as the paradox of the preface,as well as more mundane things, would seem to show that one can be rationally commit-ted to inconsistent propositions without being rationally committed to everything. SeePriest [1987], sect. 7.4.101See Priest [1982] and Priest [1987], ch. 11.102See Carnap [1950].


A probabilistic interpretation is a pair, hI; �i, where I is a class of in-terpretations for LP 103 and � a �nitely additive measure on I , that is, afunction from subsets of I to non-negative real numbers such that:

�(�) = 0�(X [ Y ) = �(X) + �(Y ) if X \ Y = �

If � is any sentence, let [�] = f� 2 I ; 1 2 �(�)g. For reasons that we willcome to, we also require that for all �, �([�]) 6= 0. There certainly are suchinterpretations and measures. For example, let I be any �nite class thatcontains the trivial interpretation, �f1;0g, where all sentences are true, andlet �(X) be the cardinality of X . Then this condition is satis�ed.

Given a probabilistic interpretation, we de�ne a probability function, p,by:

p(�) = �([�])=�(I)

It is easy to see that p satis�es all the standard conditions for a probabilityfunction, such as:

0 � p(�) � 1if � j= � then p(�) � p(�)if j= � then p(�) = 1p(� _ �) = p(�) + p(�)� p(� ^ �)

except, of course: p(:�) + p(�) = 1. Since we have p(� ^ :�) > 0, andp(� _ :�) = 1, it follows that p(�) + p(:�) > 1.

By the construction, we have, in fact, p(�) > 0 for all �. It might besuggested that a person whose personal probability function gives nothingthe value zero would have to be very stupid|or at least credulous. But sincep(�) may be as small as one wishes, this hardly seems to follow. Moreover,giving nothing a zero probability signals an open-minded and undogmaticpolicy of belief. Arguably, this is the most rational policy.

Given a probability function, conditional probability can be de�ned inthe usual way:

p(�=�) = p(� ^ �)=p(�)

A singular advantage of this paraconsistent probability theory over stan-dard accounts is that conditional probability is always de�ned, since thedenominator is always non-zero.103Again, many other paraconsistent logics could be used instead.

346 GRAHAM PRIEST

Perhaps the major application of probability theory is in framing anaccount of non-deductive inference. How, exactly, to do this is a moot ques-tion. But however one does it, a paraconsistent account of non-deductiveinference can be framed in the same way, employing paraconsistent prob-ability theory. For example, we may de�ne the degree of (non-deductive)validity of the inference �=� to be p(�=(�^ �)), where � is our backgroundevidence. As one would expect, deductively valid inferences come out ashaving maximal degree of inductive validity.

To compute the degree of validity of an inference, so de�ned, we wouldoften need to employ Bayes' Theorem. Let us look at the paraconsistenttwo-hypothesis version of this. Suppose that we have two hypotheses, h1and h2, that are exclusive and exhaustive, in the sense that j= h1 _ h2 andj= :(h1^h2), and that we wish to compute the probability of h1 on evidence,e, given the inverse probabilities of these hypotheses on the evidence (allrelative to some background evidence, �, which we will ignore).

Note �rst that p(h1=e) = p(h1^e)=p(e) = p(e=h1):p(h1)=p(e). It remainsto compute p(e). Since h1 _ h2 entails e _ h1 _ h2 we have :

1 = p(e _ h1 _ h2) = p(e) + p(h1 _ h2)� p(e ^ (h1 _ h2))= p(e) + 1� p(e ^ (h1 _ h2))

Hence:

p(e) = p(e ^ (h1 _ h2))= p((e ^ h1) _ (e ^ h2))= p(e ^ h1) + p(e ^ h2)� p(e ^ h1 ^ h2)= p(h1):p(e=h1) + p(h2):p(e=h2)� p(h1 ^ h2):p(e=(h1 ^ h2))

Thus:

p(h1=e) =p(e=h1):p(h1)

p(h1):p(e=h1) + p(h2):p(e=h2)� p(h1 ^ h2):p(e=(h1 ^ h2))This is the paraconsistent version of Bayes' Theorem. In the classical case,the last term of the denominator is zero, since j= :(h1 ^h2); but this is notso in the paraconsistent case. The theorem illustrates a general fact aboutparaconsistent probability theory: everything works as normal, except thatwe have to carry round some extra terms concerning the probabilities ofcertain contradictions which may be neglected in the classical case.

The extra complication may actually be a gain in some contexts. Letme mention one possible one; this concerns quantum mechanics. Quantummechanics is known to su�er from various phenomena often called `causalanomalies', a famous one of which is the two-slit experiment.104 In this, a104See, e.g., Haack [1974], ch. 8.


light is shone onto a screen through a mask with two slits. The intensityof light on any point on the screen is proportional to the probability thata photon hits it, �, given that it goes through one slit, �, or goes throughthe other, �. Let us write p(� _ �) as q. Then:

p(�=(� _ �)) = p(� ^ (� _ �))=p(� _ �)= p((� ^ �) _ (� ^ �))=q= p(� ^ �)=q + p(� ^ �)=q � p(� ^ � ^ �)=q

Classically, we know that :(� ^ �), and so the last term may be ignored.For similar reasons, q = p(� _ �) = p(�) + p(�), and by symmetry we canarrange for p(�) and p(�) to be equal. Hence:

p(�=(� _ �)) = p(� ^ �)=2p(�) + p(� ^ �)=2p(�)= 1

2 (p(�=�) + p(�=�))

Thus, the intensity of light on the screen should be the average of theintensities of light going each slit independently (which can be determinedby closing o� the other). Exactly this is what is not found.

Standard quantum logic105 avoids the result by rejecting the inference ofdistribution (i.e., the equivalence between � ^ (�_ �) and (� ^�)_ (� ^ �),and so faulting the second line of the above proof. A paraconsistent solutionis just to note that we cannot ignore the third term in the computation ofp(�=(� _ �)), even though we know that :(� ^ �). In qualitative terms,what this means is that the photon has a non-zero probability of doing theimpossible, and going through both slits simultaneously!

This application of paraconsistent probability theory to quantum me-chanics is highly speculative. Whether it could be employed to resolve theother causal anomalies of quantum theory, let alone to predict the observa-tions that are actually made, has not been investigated.106

7.6 The Classical Recapture

Most paraconsistent logicians have supposed that reasoning in accordancewith classical logic is sometimes legitimate. Most, for example, have takenit that classical logic is perfectly acceptable in consistent situations. Theyhave therefore proposed ways in which classical logic can be \recaptured"from a paraconsistent perspective.

The simplest such recapture occurs in non-adjunctive logics. As we notedin 4.2, single premise non-adjunctive reasoning is classical. Hence, classical

105See the article on this in the Handbook.106For more on the above issues, including the e�ects of paraconsistent probability

theory on con�rmation theory, see Priest [1987], sect. 7.6, and Priest et al. [1989], pp.376-9, 385-8.

348 GRAHAM PRIEST

reasoning can be regained simply by conjoining all premises. A di�erentstrategy is to employ a consistency operator, as is done in the da Costalogics Ci, for �nite non-zero i. As we saw in 5.3, this can be employed tode�ne a negation which behaves classically; hence classical reasoning cansimply be interpreted within the system. This approach has problems forsome applications, as we shall see when we come to look at set theory.

Yet another way to recapture classical reasoning, provided that a condi-tional operator is available, is to employ an absurdity constant, ?, satisfyingthe condition j= ? ! �, for all �. Such a constant makes perfectly goodsense paraconsistently. Algebraically, it corresponds to the minimal valueof an algebra (which can usually be added if it is not present already). Intruth-preservational terms, there are two ways of handling its semantics.One is to require that ? be untrue at every (world of every) evaluation. Itscharacteristic principle then holds vacuously. The other way (which may bepreferable if one objects to vacuous reasoning) is simply to assign ? (at aworld) the value of the (in�nitary) conjunction of all other formulas (at thatworld). A bit of juggling then usually veri�es the characteristic principle.(The de�nition itself guarantees it only when � does not contain ?.)

Now let C be the set of all formulas of the form (� ^ :�) ! ?. Thenan inference is classically valid i� it is enthymematically valid with C asthe set of suppressed premises, in most paraconsistent logics. For if everymember of C holds at (a world of) an interpretation, then the (world of the)interpretation is a classical one|or at least the trivial one|and hence if thepremises of a classically valid inference are true at it, so is the conclusion.Thus, we have an enthymematic recapture.

Let us write �� for �! ?. In classical (and intuitionist) logic, �� justis :�. It might therefore be thought that provided a logic possesses ?, wecould simply interpret classical logic in it by identifying :� with ��. Thisthought would be incorrect, though. In many paraconsistent logics, ��behaves quite di�erently from classical (and intuitionist) negation. Whatproperties it has depends, of course, on the properties of !. While it willalways be the case that �;�� j= �, it will certainly not be true in generalthat j= � _ ��, that �� j= �, or even that � j= �� . As an exampleof the last, consider an intensional interpretation for the logic H . (See 5.5.)Suppose that p is true at some normal world, w, but that at some non-normal world p ! ? is true (and ? is not). Then (p ! ?) ! ? fails atw.107

A �nal, and much less brute-force, way of recapturing classical logic startsfrom the idea that consistency is the norm. It is implicit in the paracon-sistent enterprise that inconsistency can be contained. Instead of spreadingeverywhere, inconsistencies can exist isolated, as do singularities in a �eld

107It might be thought that the existence of the explosive connective `�' would causeproblems for certain paraconsistent applications; notably, for example, for set theory.This is not the case, however, as we will see.


(of the kind found in physics, not agriculture). This metaphor suggests thateven if inconsistencies are present they will be relatively rare. If it is trueinconsistencies we are talking about, these will be even rarer|somethingthat the classical logician can readily agree with!108

This suggests that consistency should be a default assumption, in thesense of non-monotonic logic. Many non-monotonic logics can be formu-lated by de�ning validity over some class of models, minimal with respect toviolation of the default condition. In e�ect, we consider only those interpre-tations that are no more pro igate in the relevant way than the informationnecessitates. In the case where it is consistency that is the default condition,we may de�ne validity over models that are minimally inconsistent in somesense. I will illustrate, as usual, with respect to LP .109

Let I = hD; di be an LP interpretation. Let � 2 I! i� � is Pd1:::dn,where P is an n-place predicate and hd1; :::; dni 2 EP \AP in I. (Recall thatI am using members of the domain as names for themselves.) I! is a measureof the inconsistency of I. In particular, I is a classical interpretation i�I! = �. If I1 and I2 are LP interpretations, I will write I1 < I2, and saythat I1 is more consistent than I2, i� I1! � I2!. (The containment here isproper.) I is a minimally inconsistent (mi) model of � i� I is a model of� i� I is a model of � and if J < I;J is not a model of �. Finally, � isan mi consequence of � (� j=m �) i� every mi model of � is a model of �.

As is to be expected, j=m is non-monotonic. For if p and q are atomicsentences, it is easy to check that fp;:p_qg j=m q, but f:p; p;:p_qg 6j=m q.Moreover, since all classical models (if there are any) are mi models, andall mi models are models, it follows that � j= � ) � j=m � ) � j=C �.The implications are, in general, not reversible. For the �rst, note thatfp;:p _ qg 6j= q; for the second, note that fp;:pg 6j=m q. But if � isclassically consistent, its mi models are exactly its classical models, andhence we have � j=m �, � j=C �: classical recapture.

j=m has various other interesting properties. For example, it can beshown that if the LP consequences of some set is non-trivial, so are its miconsequences Reassurance. For details, see Priest [1991a].110

108Though this is not so obvious once one accepts dialetheism. For a defence of theview given dialetheism, see Priest [1987], sect. 8.4.109Though the �rst paraconsistent logician to employ this strategy was Batens [1989],

who employs a non-truth-functional logic. Batens also considers the dymanical aspectsof such default reasoning.110In that paper, in the de�nition of <, a clause stating that the domains of I1 andI2 are the same is added. With this clause, the result concerning classical recapture isfalse (and that paper is mistaken). For example, if � is 9xPx^9x:Px, then hD; di is anmi model, wher D = fag; EP = AP = fag, though this is not a classical model. (Thiswas �rst noted by Diderik Batens, in correspondence.) As < is de�ned here, f8x(Px ^:Px)g �m 9x8yx = y, which may be thought to be counter-intuitive. But if 8x(Px ^:Px) is all the information we have, and inconsistencies are to be minimised, perhaps it iscorrect to infer that there is just one thing. Note that f8x(Px^:Px);9xQx^9x:Qxg 6j=m

9x8yx = y. For hd; di is an mi model of the premises, where D = fa; bg; EP = AP =

350 GRAHAM PRIEST

8 SEMANTICS AND SET THEORY

The previous part gestured in the direction of various applications of para-consistent logic. I want, in the next two parts, to look at some other appli-cations in greater detail. These concern theories of particular mathematicalsigni�cance. In this part I will deal with semantics and set theory.

Semantic and set-theoretic notions appear to be governed by simple andapparently obvious principles. In semantics, these concern truth, T , satis-faction, S, and denotation, D, and are:

T -schema: T h�i $ �

S-schema: Sx h�i $ �(y=x)

D-schema: D htix$ x = t

where � is any sentence, � is any formula with one free variable, y, and tis any closed term. Angle brackets indicate a name-forming device. In settheory the principle is the schema of set existence:

Comprehension Schema: 9x8y(y 2 x$ �)

where � is any formula not containing x. What the connective $ is in theabove schemas, we will have to come back to.

Despite the fact that these schemas appear to be obvious, they all give riseto contradictions, as is well known: the paradoxes of self-reference, such as(respectively) the Liar Paradox, the Heterological Paradox, Berry's Paradoxand Russell's Paradox. The usual approaches to set theory and semanticsrestrict the principles in some way. Such approaches are all unsatisfactoryin one way or another, though I shall not discuss this here.111

A paraconsistent approach can simply leave the principles as they are,and allow the contradictions to arise. They need do no damage, becausethe logic is not explosive. Even so, not all paraconsistent logics are suit-able as the underlying logics of these theories. For a start, if the aboveschemas are formulated with the material � they give rise to a conjoinedcontradiction, so using a non-adjunctive logic (except Rescher and Manor's)explodes the theory.112 And in the da Costa systems, Ci, for �nite i, anoperator behaving like classical negation, :� can be de�ned (see 5.3). Theusual arguments establish contradictions of the form �^:��, and so again

fa; bg; EQ = fag; AQ = fbg. With the present de�nition, the proof of Reassurance forthe �rst-order case, appropriately modi�ed, still goes through.111See, for example, Priest [1987], chs. 1, 2.112Rescher and Brandom, [1980], p. 164, suggest splitting the biconditionals up into

two non-conjoined conditionals.


the theories explode. Fortunately, there are other paraconsistent logics thatwill do the job.113

8.1 Truth Theory in LP

Let us start with the semantic case. I will deal with truth; similar remarksand constructions hold for the other semantic notions, but I will leave read-ers to ponder these for themselves. The �rst question we need to address iswhat connective it is that occurs in the biconditional of the T -schema. The�rst possibility is that it is a material biconditional, �.114

Let us, then, suppose that we are dealing with the logic LP . We willneed some machinery to handle self reference; a straightforward option isto let this be arithmetic. Hence, we suppose the language, L, to be thatof �rst order arithmetic augmented by a one place predicate, T . To makethings easy, we will assume that L has a function symbol for each primitiverecursive function (and only those function symbols). Let T0 be the LPtheory in this language which comprises the truths of �rst order arithmeticplus the T -schema.

The assumption that T0 contains all of arithmetic is obviously a verystrong one, and means that the theory is not axiomatic. We could, instead,consider an axiomatic theory with some suitable fragment of arithmetic,but since a major part of our concern will be with what cannot be proved,it is useful to have the arithmetic part as strong as possible.

The �rst thing to note is that T0 is inconsistent. Given the resources ofarithmetic, for any formula,�, of one free variable, x, one can �nd, by theusual G�odel construction, a �xed-point formula, �, of the form �(x= h�i).115Now, let � be :Tx and let � be its �xed point. Then the T -schema givesus: T h�i � �, i.e., T h�i � :T h�i. Unpacking the de�nition of �, in termsof ^, _, and : and �ddling, gives exactly T h�i ^ :T h�i.116

Despite being inconsistent, T0 is non-trivial. An easy way to see thisis to observe, �rst of all, that if in any interpretation �(�) = f1; 0g then�(� � �) = f1; 0g. Hence, an LP model for T0 can be obtained by lettingthe denotations of the arithmetic language be that of the standard interpre-tation of arithmetic|so that, in particular, the domain is N , the naturalnumbers; recall that classical interpretations are just special cases of LP

113There are paraconsistent set theories based on da Costa's C systems. (See, e.g.,Arruda [1980], da Costa [1986].) In these theories, the schemas have to be constrained,as they are classically. This takes away much of the appeal of a paraconsistent approach.114It is natural to suppose that it ought to be a detachable conditional. Goodship [1996]

argues that it is only a material conditional. Whether or not this is the case, it is certainlyinteresting to explore the two possibilities.115See, e.g., Priest [1987], sect. 3.5.116It is worth noting that for the S-schema, the �xed point machinery is unnecessary

for the demonstration of inconsistency. For let � be :Sxx. Then an instance of theS-schema is: S h�i h�i � :S h�i h�i, and we can then proceed as before.

352 GRAHAM PRIEST

interpretations|and setting ET and AT , the extension and anti-extensionof T , both to N . Call this interpretation I0. In I0 every sentence of theform T h�i takes the value f1; 0g, and so by the observation concerning �,I0 is a model for the T -schema, and so of all of T0. The same interpreta-tion shows that if � is any arithmetic formula false in the standard model,T0 6j= �.T0 is a relatively weak theory. In particular, it does not legitimate the

two way rule of inference:

�

T h�i(just consider the south-north inference in I0, where � is an arithmeticsentence false in the standard model).117

Let the theory obtained by replacing the material T -schema of T0 withthis rule be called T1. T1 is inconsistent. For choose an � of the form:T h�i. The law of excluded middle gives T h�i _ :T h�i, i.e., T h�i _ �,which, applying the rule, gives T h�i and �, i.e., :T h�i.

We can construct a model for T1 as follows. If an interpretation assignsthe standard denotations to all arithmetical language let us call it arithmeti-cal. Any arithmetical interpretation is a model all of T1 except, perhaps,the T -schema. Let I1 and I2 be two arithmetical interpretations, with as-signment functions �1 and �2. De�ne �1 � �2 to mean that for all atomicsentences in the language; �:

�1(�) = t) �2(�) = t�1(�) = f ) �2(�) = f

If �1 � �2 then this condition extends to all formulas of L. For supposethat �1 � �2. If n is in the extension of T in I2 but not I1; then �2(Tn) = tor b, but �1(Tn) = f , violating the condition. Similarly for anti-extensions.Hence, I2 � I1. By monotonicity, for all �, �2(�) � �1(�). The conclusionfollows. For suppose that �2(�) 6= t. Then � is false (i.e., b or f) in I2;hence � is false in I1, i.e., �1(�) 6= t. The argument for f is similar.

This result is, in fact, just another version of monotonicity; I will call itthe Monotonicity Lemma.

Let I0 be any arithmetical interpretation, with evaluation function �0.We now de�ne a trans�nite sequence of arithmetical interpretations,

117Whether or not more follows with minimally inconsistent LP (see 7.6) is presentlyunknown. Another non-monotonic notion of inference also suggests itself here. Accordingto this, the things that follow are the things that hold in all minimally inconsistent modelswhere the arithmetic part is the standard model. Employing this would be appropriateif there were good reasons to believe that the only inconsistencies involve the truthpredicate.


hJi; i 2 Oni (On is the class of ordinals). I will make the constructionslightly more complex than necessary, for the bene�ts of the next section.It suÆces to de�ne the evaluation function �i of each interpretation. If i > 0and n is not the code of a sentence, then �i(Tn) = �0(Tn). We thereforeneed to consider only atomic formulas of the form T h�i. Let us say that� is eventually t by k i� 9i > 08j(i � j < k, �j(�) = t). Similarly for f .Then for k 6= 0:

�k(T h�i) = t if � is eventually t by k= f if � is eventually f by k= b otherwise

We can now establish that if 0 < i � k then �i � �k. The proof is bytrans�nite induction. Suppose that the result holds for all j < k. Weshow it for k. Since the truth values of atomic formulas other than onesof the form T h�i are constant, we need consider only these. So supposethat �i(T h�i) = t. Then � is eventually t by i. In particular, for some0 < j < i, �j(�) = t. By induction hypothesis, for all l such that j < l < k,�j � �l. Hence, by monotonicity �l(�) = t. Hence, � is eventually t by k,i.e., �k(T h�i) = t. The case for f is similar.

What this lemma shows is that once i > 0, and increases, sentences of theform T h�i can change their truth value at most once. If they ever attaina classical value, they keep it. Since there is only a countable number ofsentences of this form, there must be an ordinal, l, by which all the formulasthat change value have done so. Hence �l = �l+1. Call Jl, J�; and itscorresponding evaluation function ��. Then if ��(�) = t, ��(T h�i) = t.Similarly for f and b. Hence ��(�) = ��(T h�i), and so J� is a model of T1.For the same reason, J� also veri�es the two-way rule:

:�:T h�i

Yet the theory is not trivial: anything false in the standard model of arith-metic is untrue in J�, and so T1 6j= �.

It is not diÆcult to see that the construction used to de�ne J� is, in fact,just a dualised form of Kripke's �xed point construction for a logic withtruth value gaps using the strong Kleene three-valued logic.118 (Providedwe start with a suitable ground model, monotonicity is guaranteed from thebeginning, and so we can just set �k(T h�i) to t (or f) if � takes the value t(or f) at some i < k.) Hence, if any sentence is grounded in Kripke's sense,it takes a classical value in J�. In particular, if � is any false groundedsentence, T1 6j= �.

118See the article on Semantics and the Liar Paradox in this Handbook. One of the �rstpeople to realise that the construction could be dualised for this end was Dowden [1984].

354 GRAHAM PRIEST

8.2 Adding a Conditional

Although T1 validates the two-way inferential T -schema, it does not validatethe T -schema as formulated with a detachable conditional. This is for thesimple reason that LP does not contain such a conditional. A naturalthought is to augment the language with one to make this possible. Let theresulting language be L!. Not all conditionals are suitable here, however.This is due to Curry paradoxes. If the conditional satis�es the inferenceof contraction: � ! (� ! �) j= � ! �, then the theory collapses intotriviality. For consider the �xed-point formula, , of the form T h i ! ?(or if ? is not present, just an arbitrary �). The T -schema gives: T h i $(T h i ! ?). Contraction gives us: T h i ! ? and then a couple ofapplications of modus ponens give ?.119

This fact rules out the use of all the non-transitive logics we looked at(since they validate � $ (� ! �) j= �), all the da Costa logics and dis-cussive logic (using discussive implication for the T -schema), since thesevalidate contraction, and those relevance logics that validate contraction,such as R.120 A relevant logic without contraction can be used for thepurpose.

Let T2 be as for T0, except that the T -schema is formulated with !,and the underlying logic is BX (see 5.5, 6.5). T2 is inconsistent, since it isobviously stronger than T1. But it can be shown to be non-trivial. If we tryto generalise the proof for T1 in simple ways, attempts are stymied by thefailure of anything like monotonicity once ! is involved. However, there isa way of building on the proof.121 This requires us to move from objectualsemantics to simple evaluational semantics. For the purpose of this section(and this one only), an atomic formula will be any of the usual kind orany one of the form � ! �. Clearly, any sentence of the language can bebuilt up from atomic formulas using ^, _, :, 9 and 8. Call an evaluationof atomic formulas, �, arithmetical if it assigns to every identity its valuein the standard model of arithmetic. Given an arithmetical evaluation, itis extended to an evaluation of all sentences by LP truth conditions, usingsubstitutional quanti�cation.

A quick induction shows that any arithmetical evaluation assigns t to allthe arithmetic truths of the standard model (which do not contain ! orT ), and f to all the falsehoods. Moreover, for this notion of valuation, wedo have the Monotonicity Lemma. Finally, given any such evaluation, we

119An argument of this kind �rst appeared in Curry [1942]. Di�erent versions thatemploy close relatives of contraction, such as ` (� ^ (� ! �)) ! � (but not � ^ (� !�) ` �) can also be found in the literature. See, e.g., Meyer et al. [1979].120For good measure, it also rules out using Rescher and Manor's non-adjunctive ap-

proach. Using this, every consistent sentence would follow, since if � is consistent, so is�$ (�! �).121The following is taken from Priest [1991b], which simply modi�es Brady's proof for

set theory in [1989].


can construct a �xed point, ��, such that ��(�) = ��(T h�i), as in 8.1. Theconstruction is the same, except that in the de�nition of �k, we set Ts to tif s is a (closed) term that evaluates to the code of �, and � is eventuallyt by k. Similarly for f . (The values of atoms of the form � ! � do notchange in the process.)

An induction shows that if � � � then for all i, �i � �i. Supposethe result for all i < k. We show it for k. We need consider only thoseatomic formulas of the form Ts where s evaluates to the code of sentence �.�k(Ts) = t i� � is eventually t by k, for �. By induction hypothesis, thisimplies that � is eventually t by k for �. Hence, �k(Ts) = t, as required.The case for f is similar. From this result it obviously follows that if � � �then �� .

Let ) be the conditional connective of RM3 (see 5.4, identifying +1, 0,and �1 with t, b, and f , respectively). This also plays a role in the proof.Its relevant property is that if � � � then if � and � are formulas of L!and �(� ) �) = t, �(� ) �) = t. For if �(� ) �) = t then �(�) = f or�(�) = t. By monotonicity �(�) = f or �(�) = t. Hence, �(�) �) = t.

Let �0 be the arithmetical interpretation that assigns every sentence ofthe form Ts the value b. We now de�ne a trans�nite sequence of arithmeticvaluations, h�i; i 2 Oni, as follows. (I write (�j)

� as ��j .) For k 6= 0:

�k(�! �) = t if 8j < k, ��j (�) �) = t= f if 9 j < k, ��j (�) �) = f= b otherwise

And where � is of the form Ts, where s is any closed term which evaluatesto the code of a sentence:

�k(�) = t if 9i8j(i � j < k; ��j (�) = t)= f if 9i8j(i � j < k; ��j (�) = f)= b otherwise

We can now establish that if i � k then �i � �k. The proof is bytrans�nite induction. Suppose that the result holds for all j < k. We needto consider cases where a formula is of the form �! � or Ts, where s is aterm that evaluates to the code of a sentence. Take them in that order.

Suppose that �i(� ! �) = t. Then ��0 (� ) �) = t. By inductionhypothesis, for 0 < j < k, �0 � �j . Thus, ��0 � ��j . Hence, ��j (� ) �) = t,by the observation concerning ). Thus, �k(� ! �) = t, as required. Thecase for f is trivial.

For the other case, suppose that �i(�) = t. Then 9j < i, ��j (�) = t.By induction hypothesis, if j � l < k, �j � �l, and hence ��j � ��l . Bymonotonicity, ��l (�) = t. Thus, �k(�) = t. The case for f is similar.

356 GRAHAM PRIEST

What this lemma shows, as before, is that we must eventually reachan l such that �l = �l+1. Let this evaluation be ~�. Then ~� is a modelof all the extensional arithmetic apparatus. It also models the T -schema.For if i < l, ��i (�) = ��i (T h�i), and so ��i (� , T h�i) = t or b, and~�(� $ T h�i) = t or b. (For the same reason, ~� models the contraposedform: :�$ :T h�i. Since T h:�i $ :� is an instance of the T -schema, italso models T h:�i $ :T h�i.)

It remains to check that ~� models the axioms and respects the rules ofinference of BX . This requires no little checking. Most of it is routine. Here,for example, is one of the harder propositional axioms: ((� ! �) ^ (� ! )) ! (� ! ((� ^ )). Let the antecedent be ', and the consequent be .Then ~�(' ! ) = t or b i� for no i < l, ��i (' ) ) = f . Now, supposethat ��i (') ) = f . Then one of:

��i (') = t and (��i ( ) = b or ��i ( ) = f)

��i (') = b and ��i ( ) = f

In the �rst case, �i(� ! �) = �i(� ! ) = t. But then for all j < i,��j (� ) �) = ��j (� ) ) = t, in which case ��j (� ) (� ^ )) = t, and so�i(�! (� ^ )) = t, which is impossible. In the second case, �i(�! �) = tor b, and �i(� ! )) = t or b. But then for all j < i, ��j (� ) �) = t or b,and ��j (� ) ) = t or b, in which case ��j (� ) (� ^ )) = t or b, and so�i(�! ((� ^ )) = t or b, which is also impossible.

For further details, see Brady [1989].122 The construction shows that T1 isnon-trivial, since if � is any arithmetic sentence false in the standard model~�(�) = f . (Indeed, as with the previous construction, which is incorporatedin this, if � is any false grounded sentence, the same is true.)

8.3 Advantages of a Paraconsistent Approach

What we have seen is that it is possible to have a theory containing all themachinery of arithmetic, plus a truth predicate which satis�es the T -schemafor every sentence of the language|whether this is formulated as a materialbiconditional, a two-way rule of inference, or a detachable bi-conditional.It is inconsistent, but non-trivial; in fact, the inconsistencies do not spread

122Brady shows that the construction veri�es propositional logics that are a good dealstronger than BX. His treatment of identiy is di�erent, though. To verify the substi-tutivity rule of 7.1, it suÆces to show that if t1 = t2 holds in an interpretation then�(x=t1) and �(x=t2) have the same truth value. A quick induction shows that if this istrue for atomic � it is true for all �. Hence, we need consider only these. Next, show byinduction that if this holds for � it holds for all evaluations in the construction of ��, andso of �� itself. Finally, we show by induction that it holds for every �i in the hierarchy,and hence for ~�.


into the arithmetic machinery.123 Thus, it is possible to have a workable, ifinconsistent, theory which respects the central intuition about truth.

It is not my aim here to discuss the shortcomings of other standardapproaches to the theory of truth,124 but none can match this. All restrictthe T -schema in one way or another. The one that comes closest to havingthe full T -schema is Kripke's account of truth, which at least has it inthe form of a two way rule of inference. However, this account has thesingular misfortune of being self-referentially inconsistent. According to thisaccount, if � is the Liar sentence it is neither true nor false, and so not true,but the theory pronounces :T h�i itself neither true nor false. Accordingto T2, � is both true and false (i.e., has a true negation), and this is exactlywhat it proves: T h�i ^ :T h�i entails T h�i ^ T h:�i. It might also showthat � is not true (and so not both true and false). But paraconsistencyshows you exactly how to live with this kind of contradiction.

This is not unconnected with the matter of \strengthened" paradoxes. Ifsomeone holds the Liar sentence to be neither true nor false, one can invitethem to consider the sentence, �, `This sentence is not true' (as opposedto false). Whether � is true, false or neither, a contradiction arises. It issometimes suggested that a paraconsistent account of truth falls to the sameproblem, since � can have no consistent truth-value on this account either.It should be clear that this argument is just an ignoratio. A paraconsistentaccount does not require it to have a consistent truth-value. In fact, accord-ing to T2, T h:�i $ :T h�i; if this is right, there is no distinction betweenthe standard Liar and the \strengthened Liar" at all.125

Let me �nish with a word of caution. We can construct non-trivial the-ories which incorporate the S-schema of satisfaction and the D-schema ofdenotation, in exactly the same way as we did the T -schema. If, however,we try to add descriptions to a theory with self-reference and the D-schema,trouble does arise.

Suppose that we have a description operator, ", satisfying the Hilbertianprinciple: 9x� ` �(x="x�). If t is any closed term, t = t, and so bythe D-schema D hti t , and 9xD htix. Thus, by the description principle,D hti "xD htix, whence, by the D-schema again:

t = "xD hti x

123Nor does the T -schema have to be taken as axiomatic. One can give truth conditionsfor atomic sentences and then prove the T -schema in the usual Tarskian fashion. SeePriest [1987], ch. 7.124For this, see Priest [1987], ch. 2.125The advantages of a paraconsitent account of truth rub o� onto any account of modal

(deontic, doxastic, etc.) operators that treats them as predicates. For all such theories arejust sub-theories of the theory of truth. See Priest [1991b]. We will have an applicationof this concerning provability in 9.6.

358 GRAHAM PRIEST

Now in arithmetic, just as for any formula, �, with one free variable, x, wecan �nd a sentence, �, of the form �(x= h�i), so, for any term, t, with onefree variable, x, we can �nd a closed term; s, such that s is t(x= hsi). If fis any one place function symbol, apply this fact to the term f"yDxy, toobtain an s such that:

s = f"yD hsi y

Since s = "yD hsi y, it follows that s = fs: any function has a �xed point.This shows that the semantic machinery does have purely arithmetic con-sequences. In particular, for example, 9x x = x+ 1. Arithmetic statementslike this can be kept under control, as we will see later in the next part, butworse is to come.

Let f be the parity function, i.e.:

fx = 0 if x is odd= 1 if x is even

We have fs = 0 _ fs = 1. In the �rst case s = fs = 0, which is even, andso fs = 1. Thus, 0 = 1. Similarly in the second case. This is unacceptable,even for someone who supposes that there are some inconsistent numbers.

Where to point the �nger of suspicion is obvious enough. As we saw,the D-schema entails 9xD htix, for any closed term, t; and there is no rea-son why someone who subscribes to a paraconsistent account of semanticnotions must believe that every term has a denotation: in particular, inthe vernacular, `s' is `a number that is 1 if it is even and 0 if it is odd',which would certainly seem to have no denotation. This suggests that theD-schema should be subjected to the condition that 9xD hti x in some suit-able way. The behaviour of resulting theories is a particularly interestingunsolved problem.126

8.4 Set Theory in LP

Let us now turn to the second theory that we will look at, set theory. Thisis a theory of sets governed by the full Comprehension schema. This schemais structurally very similar to the T -schema, and many of the considerationsof previous subsections carry over to set theory in a straightforward manner.The major element of novelty concerns the other axiom, the Extensionalityaxiom.

Let us start with set theory in LP . The language here contains just thepredicates = and 2, and the axioms are:

126For a further discussion of all of these issues, see Priest [1997a].


9x8y(y 2 x � �)

8x(x 2 y � x 2 z) � y = z

where x does not occur free in �. Call this theory S0. S0 is inconsistent.For putting y =2 y for �, and instantiating the quanti�er we get: 8y(y 2 r �y =2 y), whence r 2 r � r =2 r. Cashing out � in terms of : and _ givesr 2 r ^ r =2 r.

In constructing models of S0, the following observation (due to Restall[1992]) is a useful one. First some de�nitions. Given two vectors of LPvalues, (gm;m 2 D), (hm;m 2 D), the �rst subsumes the second i� for allm 2 D; gm � hm. Now consider a matrix of such values (em;n;m;n 2 D).This is said to cover the vector (gm;m 2 D) i� for some n 2 D, the vector(em;n;m 2 D) subsumes it. A vector indexed by D is classical i� all itsmembers are t or f . (Recall that we are writing f1g, f1; 0g, f0g as t, b f ,respectively.)

Now the observation. Consider an LP interpretation, hD; di, and thematrix (em;n;m;n 2 D), where em;n = �(m 2 n). If this covers everyclassical vector indexed byD it veri�es the Comprehension principle. For let� be any formula not containing x, and consider the vector (�(�(y=m));m 2D). This certainly subsumes some classical vector; choose one such, and letthis be subsumed by (em;n;m 2 D). Now consider any formula of the formm 2 n � �(y=m). Where the two sides di�er in value, one of them has thevalue b. Hence, the value of the biconditional is either t or b. Thus the sameis true of 8y(y 2 n � �), and 9x8y(y 2 x � �).

Using this fact, it is easy to construct models for S0. Consider an LPinterpretation, hD; di, where D = fm;ng, and em;n is given by the followingmatrix:

2 m nm b tn b t

Each column is the membership vector of the appropriate member of D;and since that of m subsumes every classical vector indexed by D, thisveri�es the Comprehension axiom. In the Extensionality axiom, if y and zare the same, the axiom is obviously true. If they are distinct, one is n andthe other is m, and for each x, the value of x 2 n � x 2 m is b. Hence,8x(x 2 n � x 2 m) has the value b and Extensionality is veri�ed. In thismodel, m =2 n and n =2 n have the value f , as, therefore do 9y y =2 n and8x9y y =2 x. Hence, S0 is non-trivial.

A characterisation of what can be proved in S0 (and of what its minimallyinconsistent consequences are) is still an open question. There are, however,certainly theorems of Zermelo Fraenkel set theory, ZF , that are not provable

360 GRAHAM PRIEST

in S0. For example, in ZF there is provably no universal set: ZF ` 8x9y y =2x. But this is not a consequence of S0, as we have just seen.127

The simple model of S0 that we have just used to prove non-triviality isobviously pathological in some sense. An interesting question is what the\intended" interpretations of S0 are like. Whilst unable to give an answerto this, I note that for any classical model of ZF , M = hD; di, there is amodel of S0 which hasM as a substructure. Let a be some new object, letM+ = hD+; d+i, where D+ = D [ fag, d+ is the same as d, except thatfor every c 2 D+, the value of c 2 a is b; for every c 2 D, the value of a 2 cis f ; the value of a = a is t; and for every c 2 D, the value of a = c is f .M is clearly a substructure ofM+. The membership vector of a subsumesevery classical vector, and hence M is a model of Comprehension.

It remains to verify Extensionality: 8x(x 2 m � x 2 n) � m = n.If m and n are the same in M+, then the consequent is true, as is theconditional. So suppose that they are distinct. If they are both in D, then,by extensionality in M, there is some c 2 D such that c 2 m is t and c 2 nis f , or vice versa. Whichever of these is the case, c 2 m � c 2 n is f ,as is 8x(x 2 m � x 2 n). Hence the conditional is t. Finally, supposethat m 2 D and n is a (or vice versa, which is similar). Then if c 2 D+,every sentence of the form c 2 n is b. Hence, every sentence of the formc 2 m � c 2 n is b, as therefore is 8x(x 2 n � x 2 m). Hence, theconditional is true.

8.5 Brady's Non-triviality Proof

As a working set theory, S0 is rather weak. Since the Comprehension axiomis only a material one, we cannot infer that something is in a set from thefact that it satis�es its de�ning condition, and vice versa. This suggestsstrengthening the principle to a two-way rule of inference, as we did fortruth theory. This, in turn, requires the addition of set abstracts to thelanguage. So let us enrich the language with terms of the form fx;�g forany variable, x, and formula, �; and trade in the Comprehension principleof S0 for the two-way rule:

x 2 fy;�g�(y=x)

Call this theory S1. S1 is inconsistent. For let r be fx;x =2 xg. Then:

r 2 rr =2 r

The law of excluded middle then quickly gives us r 2 r ^ r =2 r.127For this, and some further observations in this direction, see Restall [1992].


The non-triviality of S1 is presently an open question. But even though itis probably non-trivial, as a working set theory, it is still rather weak. Thisis because we have no useful way of establishing that two sets are identical.Even if we can show that 8x(� � �), and so that 8x(x 2 fx;�g � x 2fx;�g), we cannot infer that fx;�g = fx;�g since Extensionality does notsupport a detachable inference.

We might hope to circumvent this problem by trading in the Extension-ality principle for the corresponding rule:

8x(� � �)

fx;�g = fx;�g

But if we do this, trouble arises.128 For let r be as before. Then sincer 2 r must take the value b in any interpretation, we have, for any �,8x(� � r 2 r), and so fx;�g = fx; r 2 rg. Thus, for any � and �,fx;�g = fx;�g; which is rather too much.

The problem arises because the Extensionality rule of inference allowsus to move from an equivalence that does not guarantee substitution (� ��; � � 6j=LP � � ) to one that does (identity). This suggests formu-lating Extensionality itself with a connective that legitimises substitution.So let us add a detachable connective to the language, !, and formulateExtensionality as:

8x(�$ �)

fx;�g = fx;�g

The trouble then disappears.

And now that we have a detachable conditional connective at our dis-posal, it is natural to formulate the Comprehension principle as a detachablebiconditional, as follows:

8y(y 2 fx;�g $ �(x=y))

We have to be careful about the conditional connective here. As with truth,any conditional connective that satis�es contraction would give rise to triv-iality. For let c be fx 2 x ! ?g. Then an instance of Comprehension isy 2 c $ (y 2 y ! ?). Instantiating with c, we get c 2 c $ (c 2 c ! ?),and we can then proceed, as with truth, to obtain ?. Even if the logic doesnot contain contraction, Curry-style paradoxes may still be forthcoming.For example, if we drop the contraction axiom from the relevant logic R

128There are other cases where the full Comprehension principle by itself is alright, butthrowing in extensionality causes problems; for example, set theory based on Lukasiewicz'continuum-valued logic. See White [1979].

362 GRAHAM PRIEST

and add the law of excluded middle, the Comprehension principle still givestriviality.129 Again however, a relevant logic without contraction will do thejob.

Consider the set theory with Extensionality and Comprehension formu-lated as just described, and based on the underlying logic BX (with freevariables, so that these may occur in the schematic letters of Extensional-ity and Comprehension). Call this S2: The �rst thing to note about S2 isthat identity can be de�ned in it, in Russellian fashion. Writing x = y for8z(x 2 z $ y 2 z), x = x follows. Substituting fw;�g for z, and usingthe Comprehension principle gives �(w=x) $ �(w=y). Hence, we need nolonger assume that = is part of the language.

Since the Comprehension principle of S2 gives the two-way deductionversion of S1, S2 is inconsistent. It is also demonstrably non-trivial, asshown by Brady [1989].130 To prove this, we repeat the proof for T2 of8.2 with three modi�cations. The �rst, a minor one, is that we add twopropositional constants t and f to the language; their truth values are alwayswhat the letters suggest. (This is necessary to kick-start the generation ofthe �xed point into motion. In the case of truth, this was done by thearithmetic sentences.) More substantially, in constructing �� we replace theclause for T by:

�k(s 2 fx;�g) = t if �(x=s) is eventually t by k= f if �(x=s) is eventually f by k= b otherwise

where s is any closed term, and � contains at most x free. The �nal modi�-cation is that in extending evaluations to all formulas, we use substitutionalquanti�cation with respect to the closed set abstracts.

Now; ~� veri�es all the theorems of S2, in the sense that if � is any closedsubstitution instance of a theorem, it receives the value t or b in ~�. Thisis shown by an induction on the length of proofs. That the logical axiomshave this property, and the logical rules of inference preserve this property,is shown as in 8.2. This leaves the set theoretic ones.

Given the construction of ~�, it is not diÆcult to see that it veri�es theComprehension principle. It is not at all obvious that Extensionality pre-serves veri�cation. What needs to be shown is that if 8x(�$ �) is veri�ed,so is anything of the form a 2 c$ b 2 c, where a is fx;�g and b is fx;�g.Let c be fy; g. Then, given Comprehension, what needs to be shown is that (y=a) $ (y=b) is veri�ed. If this can be shown for atomic , the resultwill follow by induction. Given the premise of the inference and Compre-hension, it is true if is of the form d 2 y. If it is of the form y 2 d, where

129See Slaney [1989]. Other classical principles are also known to give rise to trivialityin conjunction with the Comprehension schema. See Bunder [1986].130A modi�cation of the proof shows that the theory based on the logic B is, in fact,

consistent. See Brady [1983].


d is fz; Æg, we need to show that Æ(z=a)$ Æ(z=b) is veri�ed. We obviouslyhave a regress. In fact, the regression grounds out in a suitable way in theconstruction of ~�. For details, see Brady [1989].131

The non-triviality of S2 is established since there are many sentences thatare not veri�ed by ~�. It is easy to check, for example, that any sentenceof the form c 2 fx; fg takes the value f , as, therefore, does the formula8x9y y 2 x.

A notable feature of Brady's proof is the following. As formulated, theComprehension principle entails: 9y8x(x 2 y $ �), where y does notoccur in �. (The y in question is fx;�g and so cannot be a subformulaof �.) If we relax the restriction, we get an absolutely unrestricted versionof the principle. Brady's proof can be extended to verify this version too,by adding a �xed point operator to the language, and treating it suitably.Again, for details, see Brady [1989].

Finally, it is worth observing that the T -schema is interpretable in S2. If� is any closed formula, let us write h�i for fz;�g, where z is some �xedvariable. De�ne Tx to be a 2 x, where a is any �xed term. Then T h�i =a 2 fz;�g $ �. Moreover, the absolutely unrestricted Comprehensionprinciple gives us �xed points of the kind required for self-reference. Let �be any formula of one free variable, x. By the principle, there is a set, s, suchthat 8x(x 2 s$ �(x=fz; a 2 sg)). It follows that a 2 s$ �(x=fz; a 2 sg).Thus, if � is a 2 s, we have � $ �(x= h�i). S2 (with the absolutelyunrestricted Comprehension principle) therefore gives us a demonstrablynon-trivial joint theory of truth, sethood and self-reference.

8.6 Paraconsistent Set Theory

Despite the strong structural similarities between semantics and set theory,there is an important historical di�erence. Set theory is a well developedmathematical theory in a way that semantics is not. In the case of settheory, it is therefore natural to ask how a paraconsistent theory such as S2relates to this development.

To answer this question (at least to the extent that the answer is known),it will be useful to divide set theory into three parts. The �rst comprisesthat basic set-theory which all branches of mathematics use as a tool. Thesecond is trans�nite set theory, as it can be established in ZF . The third

131Brady's treatment of identity is slightly di�erent from the one given here. He de-�nes x = y as 8z(z 2 x $ z 2 y). Given Comprehension, this delivers the versionof Extensionality used here straight away. What is lost is the substitution principlex = y; �(w=x) ` �(w=y). Given the Comprehension principle, this can be reduced tox = y; x 2 z ` y 2 z (which follows from our de�nition of identity). Brady takes some-thing stronger than this as his substitutivity axiom: ` (x = y^z = z) ! (x 2 z $ y 2 z).Hence, his construction certainly veri�es the weaker principle. It is worth noting thatthe construction does not validate the simpler ` x = y ! (x 2 z $ y 2 z), which, in anycase, is known to be a Destroyer of Relevance. See Routley [1980b], sect. 7.

364 GRAHAM PRIEST

concerns results about sets, like Russell's set and the universal set, that donot exist in ZF . Let us take these matters in turn.

S2 is able to provide for virtually all of bread-and-butter set theory(Boolean operations on sets, power sets, products, functions, operationson functions, etc.), and so provide for the needs of working mathematics.132

For example, if we de�ne the Boolean operators, x \ y, x [ y and �x asfz; z 2 x^ z 2 yg, fz; z 2 x_ z 2 yg and fz; z =2 xg, respectively, and x � yas 8z(z 2 x ! z 2 y), then we can establish the usual facts concerningthese notions. Some care needs to be taken over de�ning a universal set, U ,and empty set, �, though. If we de�ne �, as fx;x 6= xg, we cannot showthat for all y, � � y, since the underlying logic is relevant and cannot provex 6= x! � for arbitrary �. (Dually for U .) If we de�ne � as fx;8z x 2 zg,this problem is solved, since 8z x 2 z ! x 2 y. (Dually for U .)

The reason for the quali�cation `virtually' in the �rst sentence of the lastparagraph, is as follows. The sets, as structured by union, intersection andcomplementation, are not a Boolean algebra, but a De Morgan algebra withmaximum and minimum elements. Though we can show that 8y y 62 x\ �x,we cannot show that x \ �x � �, since, relevantly, (� ^ :�) ! � fails.(Dually for U .) There are, in a sense, more than one universal and emptysets. Moreover, this is essential. If we had x \ �x � � then, taking fz;�gfor x, we get (� ^ :�) ! 8y z 2 y. Now take fz;�g for y, and we get(� ^ :�)! �; paraconsistency fails. In fact, Dunn [1988] shows that if theprinciples that there is a unique universal set, and a unique empty set, areadded to any set theory such as S2, full classical logic falls out.

Turning to the second area, the question of how much of the usual trans-�nite set theory can be established in S2 is one to which the answer iscurrently unknown. What can be said is that the standard proofs of a num-ber of results break down. This is particularly the case for results that areproved by reductio, such as Cantor's Theorem. Where � is an assumptionmade for the purpose of reductio, we may well be able to establish that(� ^ �) ! ( ^ : ), for some , where � is the conjunction of other factsappealed to in deducing the contradiction (such as instances of the Com-prehension principle). But contraposing and detaching will give us only:� _ :�, and we can get no further.133

Lastly, the third area: reasoning in S2, one can prove various results aboutsets that are impossible in ZF . For example, as usual, let fxg be fy; y = xg,fx; yg be fxg [ fyg and

Sx be fz; 9y 2 x; z 2 yg. r = fx;x =2 xg, and we

know that r 2 r and r =2 r. Then:134

(1) If x 2 r then fxg 2 r. For fxg 2 fxg or fxg =2 fxg. In the �rst case,

132Much of this is spelled out in Routley [1980b], sect. 8.133Interesting enough, however, it is possible to prove a version of the Axiom of Choice

using the completely unrestricted version of the Comprehension principle. See Routley[1980b], sect. 8.134The following is taken from Arruda and Batens [1982].


fxg = x, and so fxg 2 r. In the second case, fxg 2 r by de�nition.

(2) If x; y 2 r then fx; yg 2 r. For fx; yg 2 fx; yg or fx; yg =2 fx; yg. Inthe �rst case, fx; yg = x or fx; yg = y, and so fx; yg 2 r. In the secondcase, fx; yg 2 r by de�nition.

(3) ffx; rgg 2 r. For ffx; rgg 2 ffx; rgg or ffx; rgg =2 ffx; rgg. In the�rst case, ffx; rgg = fx; rg, hence, x = fx; rg = r. But then x; r 2 r sofx; rg 2 r, by (2), and, ffx; rgg 2 r, by (1). In the second case, ffx; rgg 2 rby de�nition.

(4) 8x x 2 Sr. For suppose that fx; rg 2 fx; rg. Then fx; rg = x orfx; rg = r. In the �rst case, fxg = ffx; rgg, so fxg 2 r, by (3). In thesecond, fx; rg 2 r. In either case x 2 S r. Suppose, on the other hand, thatfx; rg =2 fx; rg. Then fx; rg 2 r, by de�nition, and so x 2 S r.

ThatSr is universal, is hardly a profound result. But it at least illus-

trates the fact that there are possibilities which transcend ZF .

Let me end this section with a speculative comment on what all thisshows. The discussion of this section, and especially the part concerning thenon-Boolean properties of sets in S2, shows that it is impossible to recapturestandard set theory in its entirety in this theory. Sets are extensional entitiespar excellence; using an intensional connective in their identity conditions isbound to gum up the works. In fact, it seems to me that the most plausibleway of viewing S2 is as a theory of properties, where intensional identityconditions are entirely appropriate. But what you call these entities doesnot really matter here. The important fact is that they are not the sets ofstandard modern mathematical practice.

If we want a theory of such entities, the appropriate identity conditionsmust employ�, and this means that we are back with the proof-theoreticallyweak S0 (or S1). Since this does not contain ZF , how should someone whosubscribes to a paraconsistent theory of such sets view modern mathematicalpractice?

One answer is as follows. The standard model of ZF is the cumulativehierarchy. As we saw in 8.2, there are models of S0 which contain thishierarchy. We may thus take it that the intended interpretation of S0 is amodel of this kind (or if there are more than one, that they are all models ofthis kind). The cumulative hierarchy is therefore a (consistent) fragment ofthe set-theoretic universe, and modern set theory provides a description ofit. There is, however, more to the universe than this fragment. A classicallogician may well agree with that claim. For example, they may think thatthere are also non-well-founded sets. The paraconsistent logician agreeswith this: after all, r is not well-founded; but they will think that setsoutside the hierarchy may have even more remarkable properties: some ofthem are inconsistent.

366 GRAHAM PRIEST

9 ARITHMETIC AND ITS METATHEORY

In this part I want to look at the application of paraconsistent logic toanother important mathematical theory: arithmetic. The situation con-cerning arithmetic is rather di�erent from that concerning set theory andsemantics. There are no apparently obvious and intrinsically arithmeticalprinciples that give rise to contradiction, in the way that the Comprehen-sion principle and the T -schema do|or if there are, this fact has not yetbeen discovered. In the �rst instance, the paraconsistent interest in arith-metic arises because there is a class of inconsistent models of arithmetic.(It might be more accurate to say `models of inconsistent arithmetic'.) Itmay be supposed that these models are pathological in some sense.135 I willcome back to this matter later. But even if it is so, the models neverthelesshave an interesting and important mathematical structure, as do the clas-sical non-standard models of arithmetic|which are, in fact, just a specialcase, as we will see. And just as one does not have to be an intuitionistto �nd intuitionistic structures of intrinsic mathematical interest, so onedoes not have to be a dialetheist for the same to be true of inconsistentstructures. One thing this part illustrates, therefore, is the existence of anew branch of mathematics which concerns the investigation of just suchstructures.136

The existence of inconsistent models of arithmetic bears, as might beexpected, on the limitative theorems of Metamathematics. And whateverthe status of the inconsistent models themselves, many have held that thesetheorems have important philosophical implications. This part will alsolook at the connection between the inconsistent models and the limitativetheorems, and I will comment on the signi�cance of this for the philosophicalimplications of G�odel's incompleteness theorem.

9.1 The Collapsing Lemma

Let us start with a theorem about LP on which much of the followingdepends: the Collapsing Lemma.137

Let I = hD; di be any interpretation for LP . Let � be any equivalencerelation on D, that is also a congruence relation on the denotations of thefunction symbols in the language (i.e., if g is such a denotation, and di � eifor all 1 � i � n, then g(d1; :::; dn) � g(e1; :::; en)). If d 2 D let [d] be the

135Though this claim has certainly been queried. See Priest [1994].136On this, see further, Mortensen [1995]. Perhaps surprisingly, the �rst person to

investigate an inconsistent arithmetic was Nelson [1959], who gave a realisability-stylesemantics for the language of arithmetic, according to which the set of formulas realisedwas inconsistent (and closed under a logic somewhat weaker than intuitionist logic).137The theorem works equally well for FDE, but we will be concerned primarily with

models of theories that contain the law of excluded middle, and so where there are notruth-value gaps.


equivalence class of d under �. De�ne an interpretation, I� = hD�; d�i,to be called the collapsed interpretation, where D� = f[d]; d 2 Dg; if c is aconstant, d�(c) = [d(c)]; if f is an n-place function symbol:

d�(f)([d1]; :::; [dn]) = [d(f)(d1; :::; dn)]

(this is well de�ned, since � is a congruence relation); and if P is an n-placepredicate, its extension and anti-extension in I�, E�P and A�P , are de�nedby:

h[d1]; :::; [dn]i 2 E�P i� for all 1 � i � n, 9ei � di, he1; :::; eni 2 EP

h[d1]; :::; [dn]i 2 A�P i� for all 1 � i � n, 9ei � di, he1; :::; eni 2 AP

where EP and AP are the extension and anti-extension of P in I. It is easyto check that E�= is fh[d]; [d]i ; d 2 Dg, as required for an LP interpretation.

The collapsed interpretation, in e�ect, identi�es all members of an equiv-alence class to produce a composite individual that has the properties of allof its members. It may, of course, be inconsistent, even if its members arenot.

A swift induction con�rms that for any closed term, t, d�(t) = [d(t)].Hence:

1 2 �(Pt1:::tn) ) hd(t1); :::; d(tn)i 2 EP

) h[d(t1)]; :::; [d(tn)]i 2 E�P) hd�(t1); :::; d�(tn)i 2 E�P) 1 2 ��(Pt1:::tn)

Similarly for 0 and anti-extensions. Monotonicity then entails that for anyformula, �, �(�) � ��(�). This is the Collapsing Lemma.138

The Collapsing Lemma assures us that if an interpretation is a model ofsome set of sentences, then any interpretation obtained by collapsing it willalso be a model. This gives us an important way of constructing inconsistentmodels. In particular, if the language contains no function symbols, and I isa model of some set of sentences, then, by appropriate choice of equivalencerelation, we can collapse it down to a model of any smaller size. Thus wehave a very strong downward L�owenheim-Skolem Theorem: If a theory in alanguage without function symbols has a model, it has a model of all smallercardinalities.

I note that, since monotonicity holds for second order LP (section 7.2),the Collapsing Lemma extends to second order LP . Details are left as anexercise.

138The result is proved in Priest [1991a]. A similar result was proved by Dunn [1979].

368 GRAHAM PRIEST

9.2 Collapsed Models of Arithmetic

From now on, let L be the standard language of �rst-order arithmetic: oneconstant, 0, function symbols for successor, addition and multiplication, 0,+, and �, respectively, and one predicate symbol, =. If I is any interpreta-tion, let Th(I) (the theory of I) be the set of all sentences true in I. Let Nbe the standard model of arithmetic, and A = Th(N ). Let M = hM;di beany classical model of A|which is just special cases of an LP model. (Asis well known, there are many of these other than N .139) I will refer to thedenotations of 0, +, and � as the arithmetic operations ofM, and since noconfusion is likely, use the same signs for them.140

Let � be an equivalence relation on M , that is also a congruence relationwith respect to the interpretations of the function symbols. Then we mayconstruct the collapsed interpretation,M�. By the Collapsing Lemma,M�

is a model of A. Provided that � is not the trivial equivalence relation, thatrelates each thing only to itself, then M� will model inconsistencies. Forsuppose that �, relates the distinct members of M , n and m, then in M�,[n] = [m] and so h[n]; [m]i is in the extension of =. But since n 6= m inM; h[n]; [m]i is in the anti-extension too. Thus, 9x(x = x ^ x 6= x) holds inM�.

As an illustration of constructing an inconsistent model of A using theCollapsing Lemma, suppose that we partitionM into n+1 successive blocks,C0; :::; Cn+1, such that if x; z 2 Ci and x < y < z then y 2 Ci. And supposethat for 0 < i � n + 1, Ci is closed under the arithmetic operations ofM. (The existence of such a partition follows from a standard result in thestudy of classical models of arithmetic. See Kaye [1991], sect. 6.1.) Let1 � k 2 C0 [ C1 and de�ne x � y as:

(x; y 2 C0 and x = y) or

for some 0 < i � n+ 1, x; y 2 Ci and x = y mod k

where `x = y mod k' means that for some j 2M , x+ j � k = y, in M.

It is not diÆcult to check that � is an equivalence relation on M , and,moreover, that it is a congruence relation on the arithmetic operations ofM. Hence, we may use it to give a collapsed model. In this, C0 collapsesinto an initial tail of numbers, and each Ci (0 < i � n + 1) collapses intoa block of period k. For example, if M is the standard model, n = 1 andC0 = �, the collapsed model is a simple cycle of period k. The successorfunction in the model may be depicted as follows:

139See, e.g., Kaye [1991].140For a more detailed discussion of the material in this section, see Priest [1997a].


0 ! 1 ! ::: ! i" #

k � 1 ::: i+ 1

I will call such models cycle models. They were, in fact, the �rst inconsistentmodels to be discovered.141 If M is any model, n = 1; and k = 1, wehave a tail isomorphic to C0, and then a degenerate single-point cycle. Inparticular, if M is a non-standard model and C0 comprises the standardnumbers, we have the natural numbers with a \point at in�nity", :

0 ! 1 ! ::: -

9.3 Inconsistent Models of Arithmetic

Now that we have seen the existence of inconsistent models of arithmetic,let us look at their general structure.

Take any LP model of arithmetic,M = hM;di. I will call the denotationsof the numerals regular numbers. Let x � y be de�ned in the usual way,as 9z x + z = y. It is easy to check that � is transitive. For if i � j � kthen for some x, y, i + x = j and j + y = k. Hence (i + x) + y = k. But(i + x) + y = i + (x + y) (since it is a model of arithmetic). The resultfollows.

If i 2M , let N(i) (the nucleus of i) be fx 2M ; i � x � ig. In a classicalmodel, N(i) = fig, but this need not be the case in an inconsistent model.For example, in a cycle model the members of the cycle constitute a nucleus.If j 2 N(i) then N(i) = N(j). For if x 2 N(j) then i � j � x � j � i,so x 2 N(i), and similarly in the other direction. Thus, every member of anucleus de�nes the same nucleus.

Now, if N1 and N2 are nuclei, de�ne N1 � N2 to mean that for some (orall, it makes no di�erence) i 2 N1 and j 2 N2, i � j. It is not diÆcult tocheck that � is a partial ordering. Moreover, since for any i and j, i � jor j � i, it is a linear ordering. The least member of the ordering is N(0).If N(1) is distinct from this, it is the next (since for any x, x � 0 _ x � 1),and so on for all regular numbers.

Say that i 2M has period p 2M i� i+ p = i. In a classical model everynumber has period 0 and only 0. But again, this need not be the case inan inconsistent model, as the cycle models demonstrate. If i � j and i hasperiod p so does j. For j = i + x; so p + j = p + i + x = i + x = j. Inparticular, if p is a period of some member of a nucleus, it is a period of

141This was by Meyer [1978]. Things are spelled out in Meyer and Mortensen [1984].The idea of collapsing non-standard classical models is to be found in Mortensen [1987].Di�erent structures can be collapsed to provide inconsistent models of other kinds ofnumber, e.g., real numbers. See Mortensen [1995].

370 GRAHAM PRIEST

every member. We may thus say that p is a period of the nucleus itself. Italso follows that if N1 � N2 and p is a period of N1 it is a period of N2.

If a nucleus has a regular non-zero period, m, then it must have a mini-mum (in the usual sense) non-zero period, since the sequence 0; 1; 2; :::;m is�nite. If N1 � N2 and N1 has minimum regular non-zero period, p, then pis a period of N2. Moreover, the minimum non-zero period of N2, q, mustbe a divisor (in the usual sense) of p. For suppose that q < p, and that q isnot a divisor of p. For some 0 < k < q, p is some �nite multiple of q plus k.So if x 2 N2, x = x + q = x + p+ ::: + p+ k. Hence x = x+ k, i.e., k is aperiod of N2, which is impossible.

If a nucleus has period p � 1, I will call it proper. Every proper nucleusis closed under successors. For suppose that j 2 N with period p. Thenj � j0 � j + p = j. Hence, j0 2 N . In an inconsistent model, a numbermay have more than one predecessor, i.e., there may be more than one xsuch that x0 = j. (Although x0 = y0 � x = y holds in the model, we cannotnecessarily detach to obtain x = y.)142 But if j is in a proper nucleus, N ,it has a unique predecessor in N . For let the period of N be q0. Then(j + q)0 = j + q0 = j. Hence, j + q is a predecessor of j; and j � j + q0 = j.Hence, j + q 2 N . Next, suppose that x and y are in the nucleus, andthat x0 = y0 = j. We have that x � y _ y � x. Suppose, without loss ofgenerality, the �rst disjunct. Then for some z, x+ z = y; so j + z = j, andz is a period of the nucleus. But then x = x+ z = y. I will write the uniquepredecessor of j in the nucleus as 0j.

Now let N be any proper nucleus, and i 2 N . Consider the sequence:::;00 i;0 i; i;i0; i00:::. Call this the chromosome of i. Note that if i, j 2 N , the chro-mosomes of i and j are identical or disjoint. For if they have a commonmember, z, then all the �nite successors of z are identical, as are all its �nitepredecessors (in N). Thus they are identical. Now consider the chromosomeof i, and suppose that two members are identical. There must be memberswhere the successor distance between them is a minimum. Let these be jand j0:::0 where there are n primes. Then j = j + n, and n is a period ofthe nucleus|in fact, its minimum non-zero period|and the chromosomeof every member of the nucleus is a successor cycle of period n.

Hence, any proper nucleus is a collection of chromosomes, all of whichare either successor cycles of the same �nite period, or are sequences iso-morphic to the integers (positive and negative). Both sorts are possible inan inconsistent model. Just consider the collapse of a non-standard model,of the kind given in the last section, by an equivalence relation which leavesall the standard numbers alone and identi�es all the others modulo p. If pis standard, the non-standard numbers collapse into a successor cycle; if it

142In fact, it is not diÆcult to show that there is at most one number with multiplepredecessors; and this can have only two.


is non-standard, the nucleus generated is of the other kind.To summarise so far, the general structure of a model is a liner sequence

of nuclei. There are three segments (any of which may be empty). The�rst contains only improper nuclei. The second contains proper nuclei withlinear chromosomes. The �nal segment contains proper nuclei with cyclicalchromosomes of �nite period. A period of any nucleus is a period of anysubsequent nucleus; and in particular, if a nucleus in the third segment hasminimum non-zero period, p, the minimum non-zero period of any subse-quent nucleus is a divisor of p. Thus, we might depict the general structureof a model as follows (where m+ 1 is a multiple of n+ 1):

0; 1; :::

:::a! a0::::::b! b0:::

...

:::d0:::di e0:::ei" # " # :::dm:::d

0i em:::e

0i

:::

: : :f0:::fi g0:::gi" # " # :::fn:::f

0i gn:::g

0i

:::

Another obvious question is what possible orderings the proper nuclei canhave. For a start, they can have the order-type of any ordinal. To provethis, one establishes by trans�nite induction that for any ordinal, �, thereis a classical model of arithmetic in which the non-standard numbers canbe partitioned into a collection of disjoint blocks with order-type �, closedunder arithmetic operations. One then collapses this interpretation in sucha way that each block collapses into a nucleus.

The proper nuclei need not be discretely ordered. They can also havethe order-type of the rationals. To prove this, one considers a classical non-standard model of arithmetic, where the order-type of the non-standardnumbers is that of the rationals. It is possible to show that these canbe partitioned into a collection of disjoint blocks, closed under arithmeticoperations, which themselves have the order-type of the rationals. One canthen collapse this model in such a way that each of the blocks collapses intoa proper nucleus, giving the result. This proof can be extended to show thatany order-type that can be embedded in the rationals in a certain way, canalso be the order-type of the proper nuclei. This includes !� (the reverse of!) and !� + !, but not ! + !�. For details of all this, see Priest [1997b].

What other linear order-types proper nuclei may or may not have, is stillan open question.

9.4 Finite Models of Arithmetic

First-order arithmetic has many classical nonstandard models, but noneof these is �nite. One of the intriguing features of LP is that it permits

372 GRAHAM PRIEST

�nite models of arithmetic, e.g., the cycle models. For these, a completecharacterisation is known.

Placing the constraint of �nitude on the results of the previous section,we can infer as follows. The sequence of improper nuclei is either empty oris composed of the singletons of 0; 1; :::; n, for some �nite n. There must bea �nite collection of proper nuclei, N1 � ::: � Nm; each Ni must comprise a�nite collection of successor cycles of some minimum non-zero �nite period,pi. And if 1 � i � j � m, pj must be a divisor of pi.

143

Moreover, there are models of any structure of this form. To show this,we can generalise the construction of 9.2. Take any non-standard classicalmodel of arithmetic. This can be partitioned into the �nite collection ofblocks:

C0; C10 ; :::; C1k(1) ; :::Ci0 ; :::; Cik(i) ; :::; Cm0 ; :::; Cmk(m)

where C0 is either empty or is of the form f0; :::; ng, each subsequent blockis closed under arithmetic operations, and there are k(i) successor cycles inNi. We now de�ne a relation, x � y, as follows:

(x; y 2 C0 and x = y) orfor some 1 � i � m:

(for some 0 < j < k(i), x; y 2 Cij , and x = y mod pi) or( x; y 2 Ci0 [ Cik(i) and x = y mod pi)

One can check that � is an equivalence relation, and also that it is a con-gruence relation on the arithmetic operations. Hence we can construct thecollapsed model. � leaves all members of C0 alone. For every i it collapsesevery Cij into a successor cycle of period pi, and it identi�es the blocks Ci0and Cik(i) . Thus, the sequence Ci0 ; :::Cik(i)collapses into a nucleus of size

k(i). The collapsed model therefore has exactly the required structure.144

There are many interesting questions about inconsistent models, even the�nite ones, whose answer is not known. For example: how many modelsof each structure are there? (The behaviour of the successor function ina model does not determine the behavior of addition and multiplication,except in the tail.) Perhaps the most important question is as follows. Notall inconsistent model of arithmetic are collapses of classical models. LetM be any model of arithmetic; ifM0 is obtained from M by adding extrapairs to the anti-extension of =, call M0 an extension in M. If M0 is anextension ofM, monotonicity ensures that it is a model of arithmetic. Now,consider the extension of the standard model obtained by adding h0; 0i tothe anti-extension of =. This is not a collapsed model, since, if it were, 0would have to have been identi�ed with some x > 0. But then 1 would have

143It is also possible to show that each nucleus is closed under addition andmultiplication.144For further details, see Priest [1997a].


been identi�ed with x0 > 1. Hence, 00 6= 00 would also be true in the model,which it is not. Maybe, however, each inconsistent model is the extensionof a collapsed classical model. If this conjecture is correct, collapsed modelscan be investigated via an analysis of the classical models of arithmetic andtheir congruence relations.

9.5 The Limitative Theorems of Metamathematics

Let us now turn to the limitative theorems of Metamathematics in thecontext of LP . These are the theorems of L�owenheim-Skolem, Church,Tarski and G�odel. I will take them in that order.145 In what follows, letP be the set of theorems of classical Peano Arithmetic, and let Q be anynon-trivial theory that contains P .

According to the classical L�owenheim-Skolem, Q has models of everyin�nite cardinality but has no �nite models. Moving to LP changes thesituation somewhat. Q still has a model of every in�nite cardinality.146 Butit has models of �nite size too: any inconsistent model may be collapsed to a�nite model merely by identifying all numbers greater than some cut-o�.147

The situation with second order P is again di�erent in LP . Classically,this is known to be categorical, having the standard model as its only in-terpretation. But as I noted in 9.1, the Collapsing Lemma holds for secondorder LP . Hence, second order P is not categorical in LP . For example, ithas �nite models.

Turning to Church's theorem, this says, classically, that Q is undecidable.In LP , extensions of Q may be decidable. For example, letM be any �nitemodel of A (= Th(N )), and let Q be Th(M). Then Q is a theory thatcontains P . Yet Q is decidable, as is the theory of any �nite interpretation.In the language of M there is only a �nite number of atomic sentences;their truth values can be listed. The truth values of truth functions of thesecan be computed according to (LP ) truth tables, and the truth values ofquanti�ed sentences can be computed, since 9x� has the same truth value

145For a statement of these in the classical context, see Boolos and Je�rey [1974]. Thissection expands on the appendix of Priest [1994].146The standard classical proof of this adds a new set of constants, fci, i 2 Ig, to the

language, and all sentences of the form ci 6= cj , i 6= j, to Q. It then uses the compactnesstheorem. Things are more complex in LP , since the fact that ci 6= cj holds in aninterpretation does not mean that the denotations of these constants are distinct. Afterextending the language, we observe that ci = cj cannot be proved. We then construct aprime theory in the manner of 4.3, keeping things of this form out. This is then used tode�ne an appropriate interpretation.147Let us say that M is an exact model of a theory i� the truths of M are exactly the

members of the theory. Classically, for complete theories, there is no di�erence betweenmodelling and exact modelling. The situation for LP is more complex. It can be shownthat if Q has an in�nite exact model it has exact models of every greater cardinality.On the other hand, if Q has a �nite model, M, in which every number is denoted by anumeral, M can be shown to be the only exact model of Q (up to isomorphism).

374 GRAHAM PRIEST

as the disjunction of all formulas of the form �(x=a), where a is in thedomain of M; dually for 8.

Tarski's Theorem: this says that Q cannot contain its own truth predi-cate, in the sense that even if Q is a theory in an extended language, thereis no formula, �, of one free variable, x, such that �(x= h�i) � � 2 Q, forall closed formulas, �, of the language. This, too, fails for LP . Let M beany (classical) model of P , letM0 be any �nite collapse ofM, and let Q beTh(M0). By the Collapsing Lemma, Q contains P . Since Q is decidable,it is representable in (classical) P by a formula, �, of one free variable, x.That is, we have :

If � 2 Q then �(x= h�i) 2 PIf � 62 Q then :�(x= h�i) 2 P

By the Collapsing Lemma, `P ' may be replaced by `Q'. If � 2 Q, �(x= h�i) 2Q, and so �(x= h�i) � � 2 Q (since ; Æ j=LP � Æ); and if � 62 Q,:�(x= h�i) 2 Q, and so �(x= h�i) � � 2 Q (since :� 2 Q and : ;:Æ j=LP

� Æ).There is no guarantee that � and �(x= h�i) have the same truth value in

M0. In particular, then, � may not satisfy the T -schema in the form of atwo-way rule of inference. So it might be said that � is not really a truthpredicate. Whether or not this is so, we have already seen that there areQs where there is a predicate satisfying this condition (though this has tobe added to the language of arithmetic): the theory T1 of section 8.1.148

Finally, let us turn to G�odel's undecidability theorems. A statement of the�rst of these is that if Q is axiomatisable then there are sentences true in thestandard model that are not in Q. It is clear that this may fail in LP . LetM be any �nite model of arithmetic. Then if Q is Th(M), Q contains allof the sentences true in the standard model of arithmetic, but is decidable,as we have noted, and hence axiomatisable (by Craig's Theorem).

It is worth asking what happens to the \undecidable" G�odel sentencein such a theory. Let � be any formula that represents Q in Q. (Thereare such formulas, as we just saw.) Then a G�odel sentence is one, �, ofthe form :�(x= h�i). If � 2 Q then :�(x= h�i) 2 Q, but �(x= h�i) 2 Qby representability. If � 62 Q then :�(x= h�i) 2 Q by representability,i.e., � 2 Q, so �(x= h�i) 2 Q by representability. In either case, then,

148The construction of 8.1 can be applied to any model of arithmetic|not just thestandard model|as the ground model. However, if we apply it to a �nite model care needsto be exercised. The construction will not work as given, since di�erent formulas maybe coded by the same number in the model, which renders the de�nition of the sequenceof interpretations illicit. We can switch to evaluational semantics, as in 8.2, though theconstruction then no longer validates the substitutivity of identicals. Alternatively, wecan refrain from using numbers as names, but just augment the language with names forall sentences.


� ^ :� 2 Q.G�odel's second undecidability theorem says that the statement that canon-

ically asserts the consistency of Q is not in Q; this statement is usually takento be :�(x= h�0i), where �0 is 0 = 00, and � is the canonical proof predicateof Q. This also fails in LP .149 Let Q be as in the previous two paragraphs.Then Q is not consistent. However, it is still the case that �0 62 Q (pro-vided that the collapse is not the trivial one). Consider the relationship:n is (the code of) a proof of formula (with code) m in Q. Since this isrecursive, it is represented in A by a formula Prov(x; y). If � is provablein Q then for some n, Prov(n; h�i) 2 A (where n is the numeral for n);thus, 9xProv(x; h�i) 2 A and so Q. If � is not provable in Q then for all n,:Prov(n; h�i) 2 A; thus, 8x:Prov(x; h�i) 2 A (since A is !-complete) and:9xProv(x; h�i) 2 A and so Q. Thus, 9xProv(x; y) represents Q in Q. Inparticular, since �0 62 Q, :9xProv(x; h�0i) 2 Q, as required.

9.6 The Philosophical Signi�cance of G�odel's Theorem

People have tried to make all sorts of philosophical capital out of the nega-tive results provided by the limitative theorems of classical Metamathemat-ics. As we have seen, all of these, save the L�owenheim-Skolem Theorem,fail for arithmetic based on a paraconsistent logic. Setting this theoremaside, then, nothing can be inferred from these negative results unless onehas reason to rule out paraconsistent theories. At the very least, this addsa whole new dimension to the debates in question.

This is not the place to discuss all the philosophical issues that arise inthis context, but let me say a little more about one of the theorems by way ofillustration. Doubtless, the incompleteness result that has provoked mostphilosophical rumination is G�odel's �rst incompleteness theorem: usuallyin a form such as: for any axiomatic theory of arithmetic (with suÆcientstrength, etc.), which we can recognise to be sound, there will be an arith-metic truth|viz., its G�odel sentence|not provable in it, but which we canestablish as true.150 This is just false, paraconsistently. If the theory isinconsistent, the G�odel sentence may well be provable in the theory, as wehave seen.

An obvious thought at this point is that if we can recognise the theoryto be sound then it can hardly be inconsistent. But unless one closes thequestion prematurely, by a refusal to consider the paraconsistent possibility,this is by no means obvious. What is obvious to anyone familiar with thesubject, is that at the heart of G�odel's theorem, is a paradox. The paradoxconcerns the sentence, , `This sentence is not provable', where `provable'

149It is worth noting that there are consistent arithmetics based on some relevant logics,notably R, for which the statement of consistency is in the theory. See Meyer [1978].150For example, the theorem is stated in this form in Dummett [1963]; it also drives

Lucas' notorious [1961], though it is less clearly stated there.

376 GRAHAM PRIEST

is not to be understood to mean being the theorem of some axiom systemor other, but as meaning `demonstrated to be true'. If is provable, thenit is true and so not provable. Thus we have proved . It is therefore true,and so unprovable. Contradiction. The argument can be formalised withone predicate, B, satisfying the conditions:

` B h�i ! �

If ` � then ` B h�i

for all closed �|including sentences containing B. For if is of the form:B h i, then, by the �rst, ` B h i ! :B h i, and so ` :B h i, i.e., ` .Hence, ` B h i, by the second.

And we do recognise these principles to be sound. Whatever is provableis true, by de�nition; and demonstrating � shows that � is provable, and socounts as a demonstration of this fact.151

B is a predicate of numbers, but we do not have to assume that B isde�nable in terms of 0, + and � using truth functions and quanti�ers. Theargument could be formalised in a language with B as primitive. As we sawin the previous part in connection with truth, it is quite possible to havean inconsistent theory with a predicate of this kind, where the sentencesde�nable in terms of 0, + and � using truth functions and quanti�ers behavequite consistently.

Of course, if B is so de�nable, which it will be if the set of things wecan prove is axiomatic, then the set of things that hold in this language isinconsistent. And there are reasons for supposing that this is indeed thecase.152 Even this does not necessarily mean that the familiar natural num-bers behave strangely, however. As the model with the \point at in�nity" of9.2 showed, it is quite possible for inconsistent models to have the ordinarynatural numbers as a substructure.153 There are just more possibilities inHeaven and Earth than are dreamt of in a consistent philosophy.

10 PHILOSOPHICAL REMARKS

In previous parts I have touched occasionally on the philosophical aspectsof paraconsistency. In this section I want to take up a few of the philosoph-ical implications of paraconsistency at slightly greater length. Its major

151The paradox is structurally the same as a paradox often called the `Knower paradox'.In this, B is interpreted as `It is known that'. For references and discussion of this paradoxand others of its kind, see Priest [1991b].152See Priest [1987], ch. 3. This chapter discusses the connection between G�odel's

theorem, the paradoxes of self-reference and dialetheism at greater length.153Though whether the theory of that particular model is axiomatisable is currently

unknown.


implication is very simple. As I noted in 3.1, the absolute unacceptabilityof inconsistency has been deeply entrenched in Western philosophy. It isan assumption that has hardly been questioned since Aristotle. Whilst thelaw of non-contradiction is a traditional statement of this fact, it is ECQwhich expresses the real horror contradictionis: contradictions explode intotriviality. Paraconsistency challenges exactly this, and so questions anyphilosophical claim based on this supposed unacceptability. This does notmean that consistency cannot play a regulative function: it may still bean expected norm, departure from which requires a justi�cation; but it canno longer provide a constraint of absolute nature. Given the centrality ofconsistency to Western thought, the philosophical rami�cations of para-consistency are bound to be profound, and this is hardly the place to takethem all|or even some|up at great length. What I will do here is considervarious objections to employing a paraconsistent logic, and explore a littlesome of the philosophical issues that arise in this context. In the processwe will need to consider not only the purposes of logic, but also the naturesof negation, denial, rational belief and belief revision.154

10.1 Instrumentalism and Information

Why, then, might one object to paraconsistent logic? Logic has many uses,and any objection to the use of a paraconsistent logic must depend on whatit is supposedly being used for. One thing one may want a logic for is todraw out consequences of some information in a purely instrumental way.In such circumstances one may use any logic one likes provided that it givesappropriate results. And if the information is inconsistent, an explosivelogic is hardly likely to do this.

Referring back to the list of motivations for the use of a paraconsistentlogic in 2.2, drawing inferences from a scienti�c theory would fall into thiscategory if one is a scienti�c instrumentalist. Drawing inferences from theinformation in a computer data base could also fall into this category. If thelogic gives the right results|or at least, does not give the wrong results|useit.

The only objection that there is likely to be to the use of a paraconsistentlogic in this context is that it is too weak to be of any serious use. One mightnote, for example, that most paraconsistent logics invalidate the disjunctivesyllogism, a special case of resolution, on the basis of which many theorem-provers work.155 This objection carries little weight, however. Theorem-

154Other philosophical aspects of paraconsistency are discussed in numerous places, e.g.,da Costa [1982], Priest [1987], Priest et al. [1989], ch. 18.155It is worth noting, however, that some theorem-provers that use resolution are not

complete with respect to classical semantics. For example, to determine whether � fol-lows from the information in a data base, some theorem-provers employ a heuristic thatrequires them resolve :� with something on the data base, and so on recursively. Em-

378 GRAHAM PRIEST

provers can certainly be based on other mechanisms.156 Moreover, theinferential moves of the standard programming language PROLOG can allbe interpreted validly in many paraconsistent logics (when `:-' is interpretedas `a').

One will often require a logic for something other than merely instru-mental use. This does not mean that one is necessarily interested in truth-preservation, however. One might, for example, require a logic whose validinferences preserve, not truth, but information. The computer case couldalso be an example of this. Other natural examples of this in the list of 2.2are the �ctional and counterfactual situations. By de�nition, truth is notat issue here.157

Information-preservation implies truth preservation, presumably, but theconverse is not at all obvious, and not even terribly plausible. The informa-tion that the next ight to Sydney leaves at 3.45 and does not leave at 3.45would hardly seem to contain the information that there is life on Mars. Aparaconsistent logic is therefore a plausible one in this context.

What information, and so information-preservation, are, is an issue thatis currently much discussed. One popular approach is based on the situationsemantics of Barwise and Perry [1983].158 This takes a unit of information(an infon) to be something of the form hR; a1; :::; an; si, where R is an n-place relation, the ai's are objects, and s is a sign-bit (0 or 1). A situation isa set of infons. The situations in question do not have to be veridical in anysense. In particular, they may be both inconsistent and incomplete. In fact,it is easy to see that a situation, so characterised, is just a relational FDEevaluation. This approach to information therefore naturally incorporatesa paraconsistent logic, which may be thought of as a logic of informationpreservation.159

10.2 Negation

Another major use of logic (perhaps the one that many think of �rst) isin contexts where we want inference to be truth-preserving; for example,

ploying this procedure when the data base is fp;:pg and the query is q will result in anegative answer. Such inference engines are therefore paraconsistent, though they do notanswer to any principled semantics that I am aware of.156For details of some automated paraconsistent logics, see, e.g., Blair and Subrahma-

nian [1988], Thistlewaite et al. [1988].157One might also take the other example on that list, constitutions and other legal

documents, to be an example of this. Such documents certainly contain information.And one might doubt that this information is the sort of thing that is true or false: itcan, after all, be brought into e�ect by �at|and may be inconsistent. However, if it isthat sort of thing, legal reasoning concerning it would seem to require truth-preservation.158See, e.g., Devlin [1991].159It is worth noting that North American relevant logicians have very often|if not

usually|thought of the FDE valuations information-theoretically, as told true and toldfalse. See, e.g., Anderson et al. [1992], sect. 81.


where we are investigating the veridicality of some theory or other. Andhere, it is very natural to object to the use of a paraconsistent logic. Sincetruth is never inconsistent a paraconsistent logic is not appropriate.

A paraconsistent logician who thinks that truth is consistent may agreewith this, in a sense. We have already seen in 7.6 how a paraconsistent logic,applied to a consistent situation, may give classical reasoning. However, adialetheist will object; not to the need for truth preservation, but to theclaim that truth is consistent: some contradictions are true: dialetheias.

This is likely to provoke the �ercest objections. Let me start by divid-ing these into two kinds: local and global. Global objections attack thepossibility of dialetheias on completely general grounds. Local objections,by contrast, attack the claim that some particular claims are dialetheic ongrounds speci�c to the situation concerned.

Let us take the global objections �rst. Why might one think that di-aletheias can be ruled out quite generally, independently of the consid-erations of any particular case? A �rst argument is to the e�ect that acontradiction cannot be true, since contradictions entail everything, andnot everything is true. It is clear that in the context where the use of aparaconsistent logic is being defended, this simply begs the question.

Of more substance is the following objection. The truth of contradic-tions is ruled out by the (classical) account of negation, which is manifestlycorrect. The amount of substance is only slightly greater here, though: theclaim that the classical account of negation is manifestly correct is just plainfalse.

An account of negation is a theory concerning the behaviour of somethingor other. It is sometimes suggested that it is an account of how the particle`not', and similar particles in other languages, behaves. This is somewhatnaive. Inserting a `not' does not necessarily negate a sentence. (The nega-tion of Àll logicians do believe the classical account of negation' is not Àlllogicians do not believe the classical account of negation'.) And `not' mayfunction in ways that have nothing to do with negation at all. Consider,e.g.: Ì'm not a Pom; I'm English', where it is connotations of what is saidthat are being rejected, not the literal truth.

It seems to me that the most satisfactory understanding of an accountof negation is to regard it as a theory of the relationship between state-ments that are contradictories. Note that this by no means rules out aparaconsistent account of negation.160 Even supposing that we characterisecontradictories as pairs of formulas such that at least one must be trueand at most one can be true|with which an intuitionist would certainlydisagree|it is quite possible to have both 2(�_:�) and :3(�^:�) validin a paraconsistent logic, as we saw in 7.3.

Anyway, whatever we take a theory of negation to be a theory of, it is but

160As Slater [1995] claims.

380 GRAHAM PRIEST

a theory. And di�erent theories are possible. As we have already observed,Aristotle gave an account of this relationship that was quite di�erent fromthe classical account as it developed after Boole and Frege. And modernintuitionists, too, give a quite di�erent account. Which account is correctis to be determined by the usual criteria for the rational acceptability oftheories. (I will say a little more about this later.) The matter is not at allobvious.

Quine is well known for his objection to non-classical logic in general,and paraconsistent logic in particular, on the ground that changing thelogic (from the classical one) is `changing the subject', i.e., succeeds only ingiving an account of something else ([1970], p. 81). This just confuses logic,qua theory, with logic, qua object of theory. Changing one's theory of logicno more changes what it is one is theorising about|in this case, relation-ships grounding valid reasoning|than changing one's theoretical geometrychanges the geometry of the cosmos. Nor does it help to suppose that logic,unlike geometry, is analytic (i.e., true solely in virtue of meanings). Whetheror not, e.g., `There will be a sea battle tomorrow or there will not' is ana-lytic in this sense, is no more obvious than is the geometry of the cosmos.And changing from one theory, according to which it is analytic, to another,according to which it is not, does not change the facts of meaning.161

How plausible a paraconsistent account of negation is depends, of course,on which paraconsistent account of negation is given. As we saw in part 4,there are many. One of the simplest and most natural is provided by therelational semantics of 4.5. This is just the classical account, except thatclassical logic makes the extra assumption that all statements have exactlyone truth value. And logicians as far back as Aristotle have questioned thatassumption.162

10.3 Denial

Another global objection to dialetheism goes by way of a supposed connec-tion between negation and denial. It is important to be clear about thedistinction between these two things for a start. Negation is a syntacticand/or semantic feature of language. Denial is a feature of language use:it is one particular kind of force that an utterance can have, one kind ofillocutionary act, as Austin put it. Speci�cally, it is to be contrasted withassertion.163 Typically, to assert something is to express one's belief in, oracceptance of, it (or some Gricean sophistication thereof). Typically, todeny something is to express ones rejection of it, that is, one's refusal to ac-

161The analogy between logic and geomety is discussed further in Priest [1997a].162The topics of this section and the next are discussed at greater length in Priest [1999].163Traditional logic usually drew the distinction, not in terms of saying, but in terms of

judging. It can be found in these terms, for example, in the Port-Royal Logic of Arnauldand Nicole.


cept it (or some Gricean sophistication thereof). Clearly, if one is uncertainabout a claim, one may wish neither to assert nor to deny it.

Although assertion and denial are clearly di�erent kinds of speech act,Frege argued, and many now suppose, that denial may be reduced to theassertion of negation.164 If this is correct, then dialetheism faces an obviousproblem. Even if some contradictions were true, no one could ever endorsea contradiction, since they could not express an acceptance of one of thecontradictories without expressing a rejection of the other.165

Frege's reduction has no appeal if we take the negation of a statementsimply to be its contradictory. In asserting `Some men are not mortal', Iam not denying Àll men are mortal'. I might not even realise that theseare contradictories, and neither might anyone else. And if this does notseem plausible in this simple case, just make the example more complex,and recall that there is no decision procedure for contradictories.

The reduction takes on more plausibility if we identify the negation ofa sentence as that sentence pre�xed by Ìt is not the case that'. But evenin this case, the claim that to assert a negation is invariably to deny thesentence negated appears to be false. Dialetheists who asserts both, e.g.,`The Liar sentence is true' and Ìt is not the case that the Liar sentence istrue', are not expressing the rejection of the former with the latter: theyare simply expressing their acceptance of a certain negated sentence.166 Itmay well be retorted that this reply just begs the question, since what is atissue is whether a dialetheist can do just this. This may be so; but it maynow be fairly pointed out that the original objection just begs the questionagainst the dialetheist too.

In any case, there would appear to be plenty of other examples whereto assert a negation is not to deny. For example, we may be brought tosee that our views are inconsistent by being questioned in Socratic fashionand thus made to assert an explicit contradiction. When this happens, weare not expressing the rejection of any view. What the questioning exposesis exactly our acceptance of contradictory views. We may, in the light ofthe questioning, come to reject one of the contradictories, and so revise ourviews, but that is another matter.167

To assert a negated sentence is not, then, ipso facto to deny the sentencenegated. Some, having taken this point to heart, object on the other side ofthe street: dialetheists have no way of expressing some of their views, specif-ically their rejection of certain claims: we need take nothing a dialetheist

164See Frege [1919].165For an objection along these lines, see Smiley in Priest and Smiley [1993].166And even those who take negation to express denial must hold that there is more to

the meaning of negation than this. It cannot, for example, perform that function whenit occurs attached to part of a sentence.167Some non-dialetheists have even argued that it may not even be rational to revise

our views in some contexts. See, e.g., Prior on the paradox of the preface [1971], pp. 84f.

382 GRAHAM PRIEST

says as a denial.168

This objection is equally awed. For a start, even if to assert a negatedsentence is to deny it, it is certainly not the only way to deny it. One cando so by a certain shake of the head, or by the use of some other bodylanguage. A dialetheist may deny in this way. Moreover, just because theassertion of a negated sentence by a dialetheist (or even a non-dialetheist,as we have seen) may not be a denial, it does not follow that it is not. Indenial, a person aims to communicate to a listener a certain mental state,that of rejection; and asserting a negated sentence with the right intonation,and in the right context, may well do exactly that|even if the person is adialetheist.169

10.4 The Rational Acceptability of Contradictions

This does not exhaust the possible global objections to dialetheism,170 butlet us move on to the local ones. These do not object to the possibil-ity of dialetheism in general, but to particular (supposed) cases of it. Wenoted in 2.2 that a number of these have been proposed, which include legaldialetheias, descriptions of states of change, borderline cases of vague pred-ications and the paradoxes of self-reference. Though the detailed reasonsfor endorsing dialetheism in each case are di�erent, their general form is thesame: a dialetheic account of the phenomenon in question provides the mostsatisfactory way of handling the problems it poses. A local objection maytherefore be provided by producing a consistent account of the phenomenon,and arguing this is rationally preferable. The precise issues involved herewill, again, depend on the topic in question; but let us examine one issuein more detail. This will allow the illustration of a number of more generalpoints.

The case we will look at is that of the semantic paradoxes. The back-ground to this needs no long explanation, since a logician or philosopher whodoes not know it may fairly be asked where they have been this century.Certain arguments such as the Liar paradox, and many others discoveredin the middle ages and around the turn of this century, appear to be soundarguments to the e�ect that certain contradictions employing self-referenceand semantic notions are true. A dialetheic account simply endorses thesemantic principles in question, and thus the contradictions to which thesegive rise. A consistent account must �nd some way of rejecting the reason-ing, notably by giving a di�erent account of how the semantic apparatus

168Objections along these lines can be found in Batens [1990], and Parsons [1990]. Areply can be found in Priest [1995].169That the same sentence may have di�erent forces in di�erent contexts is hardly a

novel observation. For example, an utterance of `Is the door closed', can be a question,a request or a command, depending on context, intonation, power-relationships, etc.170Some others, together with appropriate discussion, can be found in Sainsbury [1995],

ch. 6.


functions. This account must both do justice to the data, and avoid thecontradictions.

Many such accounts have, of course, been o�ered. But they are all wellknown to su�er from various serious problems. For example, they mayprovide no independent justi�cation for the restrictions on the semanticprinciples involved, and so fail to explain why we should be so drawn to thegeneral and contradiction-producing principles. They are often manifestlycontrived and/or y in the face of other well established views. Perhapsmost seriously, none of them seems to avoid the paradoxes: all seem to besubject to extended paradoxes of one variety or another.171 If the globalobjections to dialetheism have no force, then, the dialetheic position hereseems manifestly superior.172

It might be said that the inconsistency of the theory is at least a primafacie black mark against it. This may indeed be so; but even if one of theconsistent theories could �nd plausible replies to it problems, as long as thetheory is complex and �ghting a rearguard action, the dialetheic accountmay still have a simplicity, boldness and mathematical elegance that makesit preferable.

As orthodox philosophy of science realised a long time ago, there are manycriteria which are good-making for theories: simplicity, adequacy to thedata, preservation of established problem-solutions, etc.; and many whichare bad-making: being contrived, handling the data in an ad hoc way, and,let us grant, being inconsistent, amongst others.173 These criteria are usu-ally orthogonal, and may even pull in opposite directions. But when appliedto rival theories, the combined e�ect may well be to render an inconsistenttheory rationally preferable to its consistent rival.

General conclusion: a theory in some area is to be rationally preferred toits rivals if it best satis�es the standard criteria of theory choice, familiarfrom the philosophy of science. An inconsistent theory may be the onlyviable theory; and even if it is not, it may still, on the whole, be rationallypreferable.174

171All this is documented in Priest [1987], ch. 2.172One strategy that may be employed at this point is to argue that a dialetheic theory

is trivial, and hence that any other theory, even one with problems, is better. As wehave seen, dialetheic truth-theory is non-trivial, but one might nonetheless hope to provethat it is trivial when conjoined with other unobjectionable apparatus. Such argumentshave been put forward by Denyer [1989], Smiley, in Priest and Smiley [1993], and Everett[1995] and elsewhere. Replies can be found in, respectively, Priest [1989b], Priest andSmiley [1993], and Priest [1996]. Since my aim here is to illustrate general features of thesituation, I will not discuss these arguments.173Though one might well challenge the last of these as a universal rule. There might be

nothing wrong with some contradictions at all. See Priest [1987], sect. 13.6, and Sylvan[1992], sect. 2.174For a longer discussion of the relationship between paraconsistency and rationality,

see Priest [1987], ch. 7.

384 GRAHAM PRIEST

10.5 Boolean Negation

Another sort of local objection to some dialetheic theories is based on theclaim that, whatever one says about negation, there is certainly an operatorthat behaves in the way that Boolean negation does|call it what you like.Some paraconsistent logicians may even agree with this. (As we saw in 5.3,such an operator is de�nable in some of the da Costa systems.) And ifthe point is correct, it suÆces to dispose of any dialetheic account of thesemantic paradoxes which endorses the T -schema; similarly, any accountof set theory that endorses the Comprehension schema. For as I observedin the introduction to part 8, these schemas will then generate Booleancontradictions, and so entail triviality.

Someone who endorses such an account of semantics or set theory musttherefore object to the claim that there is an operator that behaves as doesBoolean negation. Why, after all, should we suppose this?175 It mightbe suggested that we can simply de�ne an operator, �, satisfying the prooftheoretic principles of Boolean negation, and in particular: �;�� ` �. Sucha suggestion would fail: the reason is simply that there is no guaranteethat a connective, so characterised, has any determinate sense. The pointwas made by Prior [1960], who illustrated it with the operator \tonk", �,supposedly characterised by the rules � ` ��, �� ` �. Such an operatorinduces triviality and can make no sense. Similarly, a paraconsistent logicianwho endorses the T -schema may fairly point out that the supposition thatthere is an operator satisfying the proof-theoretic conditions of Booleannegation induces triviality, and so makes no sense.176

The claim is theory laden, in the sense that it presupposes that the T -schema is correct. (The addition of such an operator need not producetriviality if only more limited machinery is present.) But any claim aboutwhat makes sense is bound to be theory-laden in a similar way. Prior'sargument, for example, presupposes the transitivity of deducibility, whichmay be questioned, as we have seen. The thought that Boolean negation ismeaningless may initially be somewhat shocking. But the point has beenargued by intuitionist logicians for many years. And though the groundsare quite di�erent,177 the paraconsistent logician sides with the intuitionistagainst the classical logician on this occasion.

Can we not, though, characterise Boolean negation semantically, and soshow that it is a meaningful connective? The answer is, again, no; notwithout begging the question. How one attempts to characterise Booleannegation semantically will depend, of course, on one's preferred sort of se-

175The following material is covered in more detail in Priest [1990].176There may, of course, be operators that behave like Boolean negation in a limited

domain. That is another matter.177The intuitionist reason is that meaningful logical operators cannot generate state-

ments with recognition-transcendent truth conditions, which Boolean negation does. See,e.g., Dummett [1975].


mantics. Let me illustrate the matter with the Dunn semantics. Similarconsiderations apply to others. With these semantics, the natural attemptto characterise Boolean negation is:

��1 i� it is not the case that ��1��0 i� ��1

And such a characterisation makes perfectly good semantic sense. However,it does not entail that � satis�es the Boolean proof-theoretic principles.Why should one suppose, crucially, that it validates �;�� j= �? From thecharacterisation, it certainly follows that for all �, it is not the case that��1 and ��1; but to infer from this that for all �, if ��1 and ��1 then��1 (which states that the inference is valid), just employs the principle ofinference that a conditional is true if the negation of its antecedent is. Andno sensible paraconsistent conditional validates this.

In other words, to insist that �, so characterised, is explosive, just begsthe question against the paraconsistentist. And if it is claimed that thenegation in the statement of the truth conditions is itself Boolean, and sothe inference is acceptable, this again begs the question: whether there is aconnective satisfying the Boolean proof-theoretic conditions is exactly whatis at issue.178

10.6 Logic as an Organon of Criticism

We have now noted three reasons why one might employ a logic: as apurely instrumental means of generating consequences, as an organon ofinformation preservation, and as an organon of truth preservation. Thisdoes not exhaust the uses for which one might employ a logic. Anothervery traditional one is as an organon of criticism, to force others to revisetheir views. One may object to the use of a paraconsistent logic in thiscontext as follows. If one subscribes to a paraconsistent logic, then there isnothing to stop a person from accepting any inconsistency to which theirviews lead. Hence, paraconsistency renders logic useless in this context.179

The move from the premise that contradictions do not entail everythingto the claim that there is nothing to stop a person subscribing to a contra-diction is a blatant non-sequitur. The threat of triviality may be a reason

178I have sometimes heard it said that the logic of a metatheory must be classical. Thisis just false, as the existence of intuitionist metatheories serves to remind. For certainpurposes a dialetheist may, in any case, use a classical metatheory. If, for example, weare trying to show a certain theory to be non-trivial, it suÆces to show all the theoremshave some property which not all sentences have. This might well be shown using ZF .As we saw in 8.6, ZF makes perfectly good dialetheic sense.179An objection of this kind is to be found in Popper [1963], pp. 316-7. The following

is discussed at greater length in Priest [1987], ch. 7.

386 GRAHAM PRIEST

for revision; it is not the only reason. This is quite obvious in the case ofnon-dialetheic paraconsistency. If a contradiction is entailed by one's views,then even though they do not explode into triviality, they are still not true.One will still, therefore, wish to revise. One may not, as in the classical case,have to revise immediately. It may not be at all clear how to revise; andin the meantime, an inconsistent but non-trivial belief set is better than nobelief set at all. But the pressure will still be there to revise in due course.

The situation may be thought to change if one brings dialetheism intothe picture. For the contradiction may then be true, and the pressure torevise is removed. Again, however, the conclusion is too swift. It is certainlytrue that showing that a person's views are inconsistent may not necessarilyforce a dialetheist to revise, but other things may well do so. For example, ifa person is committed to something of the form �! ?, and their views areshown to entail �, there will be pressure to revise, for exactly the classicalreason.180

Even if a dialetheist's views do not collapse into triviality, the inferenceto the claim that there is no pressure to revise is still too fast. The fact thatthere is no logical objection to holding on to a contradiction does not showthere are no other kinds of objection. There is a lot more to rationality thanconsistency. Even those who hold consistency to be a constraint on ratio-nality hold that there are many other such constraints. In fact, consistencyis a rather weak constraint. That the earth is at, that Elvis is alive andliving in Melbourne, or, indeed, that one is Kermit the Frog, are all viewsthat can be held consistently if one is prepared to make the right kinds ofmove elsewhere; but these views are manifestly irrational. For a start, thereis no evidence for them; moreover, to make things work elsewhere one hasto make all kinds of ad hoc adjustments to other well-supported views. Andwhatever constraints there are on rational belief|other than consistency|these work just as much on a dialetheist, and may provide pressure to revise.Not, perhaps, pressure of the stand 'em up - knock 'em down kind. But suchwould appear to be illusory in any case. As the history of ideas has shown,rational debates may be a long and drawn out business. There is no magicstrategy that will always win the debate|other than employing (or at leastshowing) the instruments of torture.181

180Provided that one is not a person who believes that everything is true, then asserting�!? is a way of denying �. A dialetheist might do this for a whole class of sentences,and so rule out contradictions occurring in certain areas, wholesale.181Avicenna, apparently, realised this. According to Scotus, he wrote that those who

deny the law of non-contradiction `should be ogged or burned until they admit that itis not the same thing to be burned and not burned, or whipped and not whipped'. (TheOxford Commentary on the Four Books of the Sentences, Bk. I, Dist. 39. Thanks toVann McGee for the reference.)


11 CONCLUSION

Let me conclude this essay by trying to put a little perspective into the de-velopment of paraconsistent logic. Paraconsistent and explosive accounts ofvalidity are both to be found in the history of logic. The revolution in logicthat started around the turn of the century, and which was constituted bythe development and application of novel and powerful mathematical tech-niques, entrenched explosion on the philosophical scene. The application ofthe same techniques to give paraconsistent logics had to wait until after thesecond world war.

The period from then until about the late 1970s saw the development ofmany paraconsistent logics, their proof theories and semantics, and an ini-tial exploration of their possible applications. Though there are still manyopen problems in these areas, as I have indicated in this essay, the sub-ject was well enough developed by that time to permit the beginning of asecond phase: the investigation of inconsistent mathematical theories andstructures in their own rights. Whereas the �rst period was dominated bya negative metaphor of paraconsistency as damage control, the second hasbeen dominated by a more positive attitude: let us investigate inconsis-tent mathematical structures, both for their intrinsic interest and to seewhat problems|philosophical, mathematical, or even empirical|they canbe used to solve.182

Where this stage will lead is as yet anyone's guess. But let me speculate.Traditional wisdom has it that there have been three foundational crisesin the history of mathematics. The �rst arose around the Fourth CenturyBC, with the discovery of irrational numbers, such as

p2. It resulted in

the overthrow of the Pythagorean doctrine that mathematical truths areexhausted by the domain of the whole numbers (and the rational numbers,which are reducible to these); and eventually, in the development of anappropriate mathematics. The second started in the Seventeenth Centurywith the discovery of the in�nitesimal calculus. The appropriate mathemat-ics came a little faster this time; and the result was the overthrow of theAristotelian doctrine that truth is exhausted by the domain of the �nite (orat least the potential in�nite, which is a species of the �nite). The thirdcrisis started around the turn of this century, with the discovery of appar-ently inconsistent entities (such as the Russell set and the Liar sentence)in the foundations of logic and set theory|or at least, with the realisationthat such entities could not be regarded as mere curiosities. This provideda major|perhaps the major|impetus for the development of paraconsis-tent logic and mathematics (as far as it has got). And the philosophicalresult may be the overthrow of another Aristotelian doctrine: that truth is

182It must be said that both stages have been pursued in the face of an attitudesometimes bordering on hostility from certain sections of the establishment logico-philosophical, though things are slowly changing.

388 GRAHAM PRIEST

exhausted by the domain of the consistent.183

University of Queensland, Australia.

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183I am very grateful to those who read the �rst draft of this essay, and gave mevaluable comments and suggestions: Diderik Batens, Ross Brady, Ot�avio Bueno, Newtonda Costa, Chris Mortensen, Nicholas Rescher, Richard Sylvan, Koji Tanaka, Neil Tennantand Matthew Wilson.


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INDEX

F (�), 270G(�), 270HRM(i1; : : : ; in), 261PRM(i1; : : : ; in)-�-terms, 265R!, 229T!, 229[Y=x]X , 241��, 268

!, modality in linear logic, 118�-normal form, 242�-redex, 242�-reduction, 241, 242��-normal form, 242��-reduction, 242Æ, combination of information, 63Æ, fusion, 12�-normal form, 243�-redex, 242�-reduction, 2428-elimination, 148-introduction, 14 , see disjunctive syllogism�I-calculus, 19�-depth, 270�-terms, 240! as �, 326�, necessity, 13^, relational converse, 115dapn(�), 270dcp(�), 270dn(�), 270do(�), 270 Lukasiewicz generalisations, 307 Lukasiewicz, J., 295, 306

abstraction, 240accessibility relation, 138{141, 146,

190, 194, 200

accessible, 139, 146Ackermann, W., 16adjunction, 10algebraic, 147, 190algebraic BZL-realization, 199algebraic logics, 336algebraic realization, 137, 141, 143,

145, 147, 148, 174, 191,194

algebraic realization for �rst-orderOL, 172

algebraic semantics, 137, 147, 148,152, 153, 176, 189, 195,198, 312

algebraically adequate, 145algebraically complete realization,

162Amemiya, I., 164Amemiya{Halperin Theorem, 165Amemiya{Halperin's Theorem, 166Anderson, A. R., 1{5, 7, 9, 11, 14{

17, 86, 88Anderson, C. A., 295angle bisecting condition, 171application, 231arithmetic, 366

Peano, 42relevant, 41{44

Arruda, A., 295Asenjo, F. G., 295assertion, 8atom, 167atomic, 170atomic types, 234atomicity, 170attribute, 172Austin, J. L., 4

396 INDEX

Avron, A., 18axiom, 86axiomatizaable logic, 158

B, the logic, 76{78, 82B, 155B-realizaation, 155B-realization, 156, 157B+, the logic, 12, 75, 77Barwise, J., 114basic logic, 193, 213, 214, 216, 222basic orthologic, 219, 222Batens, D., 289, 349Battilotti, G., 213, 214, 224, 226Bell, J. L., 164, 165Belnap extension lemma, 40Belnap, N. D., 1{5, 7, 9, 11, 14{

17, 86, 88, 100Beltrametti, E., 135, 179Bennett, M, K., 184Birkho�, G., 129{132, 134Birkho�, S., 179Bo-realization, 157Bohr, N., 134Boolean algebra, 130, 148, 153,

187Boolean negation, 384Boolean-valued models, 177, 178Born probabilty, 180Born rule, 180bound variables, 241BQ, the logic, 82bracket abstraction, 233Brady, R. T., 100branching counter machine, 94Brouwer{Zadeh lattice, 183Brouwer{Zadeh poset, 183Brouwer{Zaden logics, 193Bugajski, S., 179Buridan, J., 294Busch, P., 179BZ3-lattice, 202BZ-frame, 195BZ-lattice, 193, 195

BZ-poset, 183

calculus for orthologic, 220canonical model, 160, 162, 163,

170, 200, 208canonical realization, 163Cassinelli, G., 135Cattaneo, G., 179, 183, 196Cauchy sequences, 131Chang, C. C., 185, 210Chovanec, F., 184Church, A., 7, 8, 16, 19classical relevance logics, 7Clinton, Bill, 118closed subspace, 131{133, 135{137,

146, 176, 179, 203closed term, 241coincidence lemma, 174collinearity, 102combinators, 229combinatory completeness, 233combinatory logic, 232compatibility, 148, 160compatible, 148, 167, 170complete, 154, 170complete Boolean algebra, 177complete BZ-lattice, 198complete extension, 153complete involutive bounded lat-

tice, 182complete ortholattice, 165, 166complete orthomodular lattice, 163,

177completeness, 175completeness theorem, 160, 161,

175, 191, 196, 200, 208completion by cuts, 166conditional, 2, 354

counterfactual, 6conditional connectives, 315con�guration, 158con�nement, 14confusion

about `Æ', 112

INDEX 397

about the logic B, 76between kinds of consequence,

74use{mention, 3

connexive logic, 32consecution calculus, 86consequence, 141, 172, 194, 195,

210consequence in a realization, 138,

141, 174conservative extension, 34consistency, 159consquence in a given realization,

205constant domain, 173context, 213, 221continuous lattice, 60contraction, 86, 87, 100, 218, 219contradiction, 7converse

of a relation, 115Correspondence Thesis, 4Costa, N. C. A. da, 289counterfactual conditional, 149covering property, 170covers, 170cube, 217cube of logics, 213, 218Curry's Lemma, 88Curry{Howard isomorphism, 237cut rule, 47, 87cut-elimination, 87, 213, 221, 222cut-elimination theorem, 213cut-formula, 221Cutland, N., 213, 223

Da Costa's C-system, 318da Costa's C1, 305da Costa's C! , 304da Costa, N. C. A., 178Dalla Chaiara, M. L., 204Dalla Chiara, M. L., 155, 178, 184Davies, E. B., 179de Morgan monoid, 60

word problem, 101decidability, 100, 157decidability (of paraconsistent logic),

328decidable, 196deduction

`OÆcial', 17relevant, 15

deduction theorem, 8, 14, 15, 153,161

modal, 17modal relevant, 18relevant, 16

deductive closure, 159de�nition of the canonical model,

175demodaliser, 9denial, 380derivability, 159, 207derivable, 159derivation, 159, 207description operator, 204dialetheism, 291Dickson's theorem, 92di�erence poset, 184discursive implication, 317Dishkant, H., 138disjunctive syllogism ( )

admissibility in R, 36disjunctive syllogism ( ), 1, 7, 31,

33, 47admissibility RQ, 39admissibility in R!, 41admissibility in R]], 43admissibility in R, 34, 35, 39,

45and Boolean negation, 82in the metalanguage, 36inadmissibility in R], 43the Lewis `Proof', 32

Disjunctive syllogism (DS), 294display logic, 100distribution

398 INDEX

of conjunction over disjunc-tion, 10, 29, 37, 66, 93,95, 97, 118

distributive laws, 137distributivity, 133, 153domain, 172, 173domain of individuals, 173domains of certainty, 196double negation, 12Dummett, M., 213, 223Dunn, J. M., 46, 48, 196Dvure�censkij, A., 184Dwyer, R., 10

E, the logic, 1, 2, 5, 7, 9, 11{13, 16, 19{21, 23, 25{27,30, 31, 34{36, 39, 47{49,75, 77, 80, 84{86, 92, 93,100, 102, 108, 109, 113,116, 117

E, the logicundecidability, 102

E!, the logic, 9{11, 17{19Efde, the logic, 27, 54e�ect, 190, 203e�ect algebra, 184, 189, 205e�ect algebras, 204e�ects, 180, 181, 192, 209Einstein, A., 134elementary (�rst-order) property,

162elementary class, 163elementary substructure, 164elimination of the cuts, 220elimination theorem, 87entailment, 2, 3, 151

non-transitive, 32entailment in the algebraic seman-

tics, 151entailment-connective, 151epistemic logics, 155EQ, the logic, 13, 47, 48, 82event-state systems, 135events, 130

ex contradictione quodlibet (ECQ),288

excluded middle principle, 130existential quanti�er, 14expansion, 13experiemental proposition, 130experimental proposition, 129, 131,

132, 136, 179exponentials, 224

f , the false constant, 12Faggian, C., 213, 226FDE, 333�ltration, 297Finch,P. D., 149Fine, K., 66, 163�nite model property, 157, 196Finkelstein, A., 135�rst-degree entailment, 60formula

signed, 100formulas-as-types isomorphism, 237Foulis, D. J., 135, 136, 179, 184free variable, 241full cut, 223functional character, 229fusion, 12fuzzy, 180fuzzy accessibilty relation, 194fuzzy complement, 183, 193, 197fuzzy intuitionistic logics, 193fuzzy negation, 182, 196, 200fuzzy-accessible, 194fuzzy-intuitionistic semantics, 196fuzzy-like negation, 193fuzzy-orthogonal set, 194

G�odel's Theorem, 375generalisation, 14generalized complement, 205Gentzen system, 86, 88, 89, 92Gentzen, G., 158, 220Gibbins, P., 213, 223Girard's linear logic, 193

INDEX 399

Girard's linear negation, 213Girard's negation, 221Girard, J.-Y., 118, 213, 219Giuntini, R., 183{185, 189, 194,

202, 204Goldblatt, R. H., 155, 158, 162,

163, 165grammar, 3Greechie diagram, 167Greechie, R. J., 135, 166, 167, 182Grice, H. P., 6Gudder, S. P., 135, 182, 189guinea pigs, 5

H, the logic, 7, 12, 77H+, the logic, 12H!, the logic, 10, 15, 17, 22Haack, S., 6Halperin, I., 164Hardegree, G. H., 149Hegel, G., 292Henkin constructions, 302Heraklitus, 292Herbrand{Tarski, 153, 161heredity, 64heriditary right maximal terms, 261Heyting algebra, 314Hilbert lattice, 171Hilbert quantum logic, 166Hilbert space, 131, 135{137, 145,

150, 165, 170, 180, 190,203, 209

Hilbert space lattice, 135Hilbert systems, 7Hilbert{Bernays, 158Hilbert-space realizations, 190Hilbertian space, 171

identity and function symbols (para-consistent logic), 338

identity axiom, 86implication, 2implication-connective, 146{148, 150import-export condition, 148

incompatible quantities, 133, 136indistinguishability relation, 178individual, 173individual concept, 173, 176inductive extension, 174inference ticket, 9in�mum, 136, 137, 172, 182in�nitary conjunction, 197in�nitary disjunction, 198information, 63inhabitant, 234, 243inner product, 131, 132, 146, 164interpretation, 172, 177intuitionistic accessibility relation,

194intuitionistic complement, 183, 197intuitionistic logic, 12, 139, 155,

213intuitionistic logic (H), 7intuitionistic orthogonal set, 195intuitionistically-accessible, 194involutive bounded lattice, 189, 191involutive bounded poset, 183, 188involutive bounded poset (lattice),

181irreducibility, 170irreducible, 170

Jauch, J., 135Ja�skowski, S., 295

K, the logic, 45{47Kopka, F., 184Kalmbach, G., 147, 166KB, 191Keating, P., 115Keller, H. A., 171Klaua, D., 196Kleene condition, 181Kleene, S. C., 306KR, the logic, 102{104, 107, 108KR, the logic

undecidability, 102Kraus, S., 179

400 INDEX

Kripke, S., 86Kripke-style semantics, 138Kripkean models, 155Kripkean realization, 139{143, 145,

149, 174, 194Kripkean realization for (�rst-order)

OL, 173Kripkean realizations, 155Kripkean semantics, 138, 139, 143,

147, 148, 150, 152, 156,160, 190, 191, 194{197

Lahti, P., 179lambda calculus, 229lambda reductions, 247lambda terms, 240lattice, 182

de Morgan, 59LC, the logic, 79Leibniz' principle of indiscernibles,

179Leibniz-substitutivity principle, 178lemma of the canonical model, 161,

176, 201Lemmon, E. J., 22length of a lattice, 170Lewis, C. I., 293Lewis, D. K., 6Lindenbaum, 153Lindenbaum property, 154line, 102linear combinations, 131linear logic, 46, 118, 222, 224linear orthologic, 213lnf, see long normal formlogic

deviant, 6the One True, 15

logic R, 324logical consequence, 138, 141, 143{

145, 152, 156, 172, 174,194, 195, 205, 210

logical theorem, 159

logical truth, 138, 141, 143, 156,172, 174, 194, 195, 205,210

long, 270long normal form, 246LP, 333LR, the logic, 93{95, 99LR+, the logic, 93LRW, the logic, 99, 118Ludwig, G., 179 Lukasiewicz axiom, 187 Lukasiewicz quantum logic, 209 Lukasiewicz' in�nite many-valued

logic, 193 Lukasiewicz' in�nite-many-valued

logic, 185

Mackey, G., 134MacNeille completion, 163, 166,

182, 183, 190Mangani, P., 186many-valued conditionals, 319many-valued logics, 332Martin, C., 294Martin, Errol, 10, 48maximal Boolean subalgebras, 167maximal �lter, 153McLaren, 166metavaluation, 37metrically complete, 131metrically complkete, 165Meyer, R. K., 10, 46, 48, 86, 101Minari, P., 144mingle, 13minimal quantum logic, 136minimally inconsistent, 349minimally inconsistnet, 349Mittelstaedt, 179Mittelstaedt, P., 135, 149mix rule, 87mixtures, 135modal deduction theorem, 17modal interpretation, 155, 157modal operators, 339

INDEX 401

modal relevant deduction theorem,18

modal translation, 156, 191modality, 2model, 138, 141, 152modular lattice, 105modus ponens, 20, 32modus ponens, 15{17, 19, 31, 37{

39for the material conditional,

47Moh Shaw-Kwei, 8, 16monotonicity lemma, 352Morash, R.P., 171Mortensen, C., 288multisets, 214, 215MV-algebra, 185{188, 210, 211

natural deduction, 21, 158necessity, 2, 11, 13necessity operator, 198negation, 7, 12, 378

Boolean, 12, 81, 82, 102minimal, 12

negative domain, 197, 203Nicholas of Cusa, 292Nishimura, H., 213Nistic�o, G., 183, 196non-adjunction, 299non-adjunctive logics, 330normal form, 239normal worlds, 322NR, the natural deduction sys-

tem, 21null space, 133

observable, 146`OÆcial deduction', 17Once (j1; : : : ; jk), 255Once+(j1; : : : ; jk), 256Once�(j1; : : : ; jk), 255One True Logic, 15operational, 214operational rule, 86

Organon, 293Orlov, 295Orlov, I., 7orothvalued universe, 177ortho-pair realization, 197, 203ortho-pair semantics, 199ortho-valued (set-theoretical)universe

V , 177ortho-valued models, 177orthoalgebra, 184, 185, 204, 205orthoarguesian law, 167orthocomplement, 132, 137, 139,

213orthocomplemented orthomodular

lattice, 133orthoframe, 139, 140, 150, 155, 156,

165, 166, 194, 197, 198,201

orthogonal complement, 132ortholattice, 137, 142, 147, 153,

162, 164, 166, 182, 191,197

orthologic, 136, 189, 205, 213, 219,221, 223, 224

orthomodular, 143, 144, 148, 165,174

orthomodular canonical model, 162orthomodular lattice, 142, 143, 147,

148, 153, 154, 162, 167,176, 178

orthomodular poset, 181, 204, 205orthomodular property, 191orthomodular quantum logic, 136orthomodular realization, 157, 161orthomodularity, 162orthopair semantics, 199orthopairproposition, 197, 198, 200,

201, 203orthopairpropositional conjunction,

197orthopairpropositional disjunction,

197othoposet, 181Oxford English Dictionary, 5

402 INDEX

P�W, the logic, 10PA, Peano arithmetic, 42PA+, positive Peano arithmetic,

42paraconsistent modal operators, 342paraconsistent quantum logic, 189,

219, 220paraconsistent set theory, 363permutation, 7, 86

total, 102Perry, J., 114phase-space, 129{131Philo-law, 147physical event, 179physical property, 179physical qualities, 130Piron, C., 135Piron{McLaren's coordinationiza-

tion theorem, 170Piron{McLaren's Theorem, 171point, 102polarities, 59polynomial conditionals, 147, 150positive conditionals, 146positive domain, 197, 203positive laws, 149positive logic, 139, 146, 153positive paradox, 7, 10, 22positive-plus logics, 331possibility operator, 198possible worlds, 138Powers, L., 10Pratt, V., 222Prawitz, D., 66pre-Hilbert space, 164predicate-concept, 173pre�xing, 7premisses, 214Priest, G., 349principal formula, 215, 221principal type scheme (PTS), 235principle quasi-ideals, 141prinicpal ideal, 182Prior Analytics, 293

probabiity measure, 135probability, 344projection, 133, 150, 179, 180, 193projective space, 102projetion, 183proof, 15proof reductions, 247proper �lter, 153proposition, 138, 139, 141, 144,

146, 149, 151, 164, 165,179, 195

propositional quanti�cation, 11pseudo canonical realization, 163pseudocomplemented lattice, 147,

148Pulmannov�a, S., 184pure state, 130, 132, 133, 145, 146,

150, 190pure states, 129, 131, 146, 203

Q-combinators, 237Q-combinatory logic, 238Q-de�nable, 252Q-logic, 238Q-terms, 237Q+, the logic, 102Q-translation algorithm, 249QMV-algebra, 185, 187, 188, 209,

210quanti�cation

propositional, 11quanti�er, 14quanti�ers (in paraconsistent logic),

329quantum events, 132quantum logical approach, 135, 171quantum logical implication, 158quantum MV-algebra, 185quantum proposition, 157quantum set theory, 178quantum-logical natural numbers,

177quantum-sets, 178quasets, 178

INDEX 403

quasi-consequence, 152quasi-ideal, 145, 147, 165quasi-linear, 188quasi-linear QMV-algebra, 189, 209quasi-model, 152, 154quasisets, 178Quesada, M., 288

R], the arithmetic, 41{44R]], the arithmetic, 41, 43, 44R, the logic, 2, 5, 7, 9{14, 17{21,

23, 25{27, 30, 31, 34{39,41, 42, 49, 54, 55, 57,58, 60{63, 65{82, 84{89,92, 93, 95{102, 104, 108,109, 111{114, 116{118

R, the logicundecidability, 102

R+, the logic, 10{12R�, the logic, 13R!, the logic

decidability, 86R!, the logic, 7{10, 16{22, 64,

65, 86, 87R:!, the logic, 45

R!;^, the logiccomplexity, 93

Rfde, the logic, 27{29, 54, 59R!;^;_, the logic, 10R!;^, the logic, 93, 94R-theory, 36RA, the logic, 116, 117Ramsey-test, 290Randall, C., 135, 136, 179rational acceptability of contradic-

tions, 382realizability, 152realizable, 154realization, 209realization for, 205realization for a strict-implication

language, 151reassurance, 349reductio, 12

reduction, 231reference, 173, 175re ection, 215regular, 182, 191regular involutive bounded poset,

183regular paraconsistent quantum logic,

191, 195regular symmetric frame, 194regularity, 181regularity property, 190relation

binary, 80ternary, 68, 78, 80, 102

relation algebra, 115relational valuations, 308relevance, 2, 6relevant arithmetic, 41{44relevant arrows, 321relevant deduction, 15relevant implication, 2relevant logics, 193, 335relevant predication, 118representation theorem, 135, 165residuation, 12resource consciousness, 18Restall, G., 100restricted-assertion, 8, 9rigid, 173rigid individual concepts, 175RM, the logic, 7, 13, 18, 78{80,

101RM3, the logic, 95RMO!, the logic, 18Routley interpretation, 310Routley, R., 63, 292Routley, V., 292RQ, the logic, 13, 14, 39{42, 70,

82, 83rule, 158, 195, 199, 205, 214, 215

S5, 196, 202S4, 155S!, the logic, 10

404 INDEX

S4, the logic, 9, 13, 17, 21, 27, 30,77, 79

S5, the logic, 48, 78, 79Sambin, G., 213, 214Sasaki projection, 167Sasaki-hook, 149satis�cation, 173Sch�utte, K., 46, 47Schr�odinger, E., 178Scott, D., 60Scotus, 294second-order (paraconsistent) logic,

338secondary formula, 215self-distribution, 8

permuted, 9self-implication, 7semantics, 48

algebraic, 49operational, 63Routley{Meyer, 6, 57, 118semi-lattice, 64, 66set-theoretical, 49, 55, 63

semantics and set theory, 350semi-transparent e�ect, 180, 190separable, 131separable Hilbert space, 165separation theorem, 86sequent, 46

contraction of, 87right handed, 46

sequent calculus, 158, 213, 214sequents, 214set theory in LP , 358set-theory, 177set-up, 57, 68sharp, 179, 180, 190sharp property, 193, 203SheÆeld Shield, The, 114�-complete orthomodular lattice,

135, 170signed formula, 100siilarity logics, 155situation theorey, 114

Slomson, A. B., 164, 165Socrates, 118Sol�er, M. P., 171soundness, 175, 196, 208soundness theorem, 160, 161, 199,

200Stalnaker, , 149Stalnaker, R., 6Stalnaker-function, 149standard orthomodular Kripkean

realization, 162standard quantum logic, 136standard realization, 162statistical operator, 180, 181statistical operators, 135Stone theorem, 153strict implication, 150, 151strict-implication operation, 150strong Brouwer{Zadeh logic, 193strong partial quantum logic, 204,

205structural, 214structural rule, 86, 216, 217, 223subspaces, 154, 166substructural logics, 1, 223, 229substructure, 164suÆxing, 8Sugihara countermodel, 307Sugihara generalisation, 307Sugihara Matrix, 101summetry, 215superposition principle, 131Suppes, P., 135supremum, 133, 136, 137, 172, 182,

207Sylvan, R., 293symmetric frame, 190, 191symmetry, 221syntactical compactness, 159syntactical compatibility, 159syntactically compatible, 175

T , the true constant, 12

INDEX 405

T, the logic, 11, 19, 77, 78, 92,108, 109

t, the true constant, 11t, the logic, 77T�W, the logic, 10T+, the logic, 11T!, the logic, 9{11Takeuti, G., 177Tamura, S., 213Tennant, N., 32ternary relation, see relation, ternarythe classical recapture, 347the limitative theorems of meta-

mathematics, 373theorems, 37theorey of descriptions, 176theories of quasisets, 178theory, 35thinning, 86Thomason, R., 6title, 1told values, 60Toraldo di Francia, G., 178total space, 133TQ, the logic, 82transfer rule, 219translations, 229true, 177truncated sum, 186truth, 138, 141, 143, 156, 172, 174,

194, 195, 205TW, the logic, 77, 78, 106{108TW+, the logic, 12, 77TW!, the logic, 10TWQ, the logic, 82type, 229type assignment, 234, 243

uncertainty principles, 133undecidability, 102unitary vector, 131, 145universal quanti�er, 14unsharp, 190unsharp approach, 180

unsharp orthoalgebra, 184unsharp partial quantum logic, 204unsharp physical properties, 183unsharp property, 193Urbas, I., 289urelements, 178Urquhart, A., 63, 93, 102

vacuous quanti�cation, 14validity, 64valuation, 173valuation-function, 137, 172Varadarajan, V. S., 135, 164, 170veri�ability, 152, 153veri�able, 154veri�cation, 173visibility, 213, 215, 221von Neumann's collapse of the wave

function, 150von Neumann, J., 129{132, 134,

179

W, the logic, 10, 99wave functions, 131Way Down Lemma, 35Way Up Lemma, 35weak Brouwer{Zadeh logic, 193weak consequence, 152weak equality, 232weak implication calculus, 7weak import-export, 149weak Lindenbaum theorem, 160,

161, 174weak non monotonic behaviour, 150weak partial quantum logic, 204,

205, 207weakening, 86, 218, 219weaker sets of combinators, 229weakly linear, 188Wittgenstein, L., 287, 292Wolf, R. G., 1world valuation, 173

X, the logic, 14, 15, 77

406 INDEX

Zermelo{Fraenkel, 178zero degree entailment, 309

handbook of philosophical logic vol 6

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