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Page 1: Thinking about Reasoning about Knowledge

Thinking about Reasoning about Knowledge

HENRY E. KYBURG, JR.Departments of Philosophy and Computer Science, University of Rochester, Rochester, NY 14627,U.S.A. (email: [email protected])

Ronald Fagin, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi, Reasoningabout Knowledge, Cambridge, MA: MIT Press, 1995, xiii + 477 pp., $45.00 (cloth),ISBN 0–262–06162–7.

1. Introduction

This is a heavy book in a variety of senses. This is not to say that it is entirely lackingin the light touch, but that (apart from being physically weighty) it is devoted tothe detailed exploration of very complicated matters. Much that is in it is the resultof original research. It is also concerned with a wide range of issues. In the natureof things, this work can hardly be expected to be easily accessible to studentsor readers from a variety of domains of expertise. The authors hope, of course,that the material will be accessible to readers from several disciplines: computerscience, artificial intelligence, philosophy, and game theory; they even refer to the“non-mathematically-oriented” reader. While the book is liberally sprinkled withprose passages, I fear the non-mathematical reader who plugs through the bookwill be rare indeed. On the other hand, for the mathematically oriented reader inany of these disciplines, the book provides a rich and penetrating analysis of avariety of issues.

Much of the material has been taught in various courses at various institutionsby the authors. Nevertheless, it is hard to imagine very many instructors who couldget through more than a fraction of the topics, if they are to deal with them in aresponsible way. I fear the authors are being too optimistic if they are hoping forwide-scale adoptions.

The book is useful as a compendium of results in epistemic logic, includingresults about various complexity issues as well as results concerning the logicsthemselves. It is well organized, with good indices of both words and symbols. Butthe subject matter is in a state of flux and development, opinions are varied andchanging, and even the object of having such logics is unsettled. It could be arguedthat, except perhaps for the specialist, it is premature to devote so much effort to adetailed review.

So who is the book for? It is for serious readers in any of the disciplines men-tioned who want a background in the issues revolving around reasoning aboutknowledge. More than this: For readers in any of these disciplines who do possess

Minds and Machines 7: 103–112, 1997.c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

123271 MATHKAP

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that mysterious quality of “Mathematical maturity”, it is a very good introductionto a number of topics. One can go further: It is unique for the inclusion of a remark-able array of thoughtful, thought-provoking, substantive, text-supplementing, andinteresting exercises. Many of these exercises are essential to the text itself; forexample, proofs of theorems and the development of counterexamples are some-times referred to in the text, but actually appear as part of the well-guided exercisesequences. The volume is fine for self study by the mathematically mature profes-sional in any of the fields mentioned.

2. Content

So much for the audience, actual and intended. Let us now turn to the content. Aftera brief review in the present section of the topics covered, we will focus in the nexttwo sections more closely on a few of the prominent themes and assumptions. InSection 5, we will consider issues of style and scholarship, and conclude with abrief discussion of the difficulties that attend an enterprise such as this.

The basic machinery employed in the book is that of the possible-worlds seman-tics of modal logic. This is introduced, together with an example of a puzzle thatrequires reasoning about knowledge, in the first two chapters that introduce apossible-worlds model for knowledge. The puzzle is the muddy-children puzzle (itis also known by other names) in which a number of children are told that at leastone of them has mud on his forehead, and are asked whether they know if theyare among those with muddy foreheads. The first time the question is asked, anychild who looks around and sees only clean foreheads will know that his foreheadis dirty; therefore, if the question is asked a second time, the precocious childrenwill all know that at least two foreheads are dirty: Anyone who can see only onedirty forehead will know that his own forehead is dirty. And so on. This puzzlemakes frequent occurrences in the first seven chapters of the book. How can weformalize the process of reasoning that the ideally rational child goes through?

The third chapter, “Completeness and Complexity”, provides completeness andcomplexity results for the logic introduced in the second chapter, and extends thetheory (temporarily) to first-order logic in which the Barcan formula is assumedto hold. For the most part, propositional logic is taken to suffice for the topicsconsidered. The logic is a standard S5 modal logic, with operators Ki for 1 � i �n agents. (‘K’, of course, stands for Knowledge, not Knecessity.) We will return tothis matter in the next section.

Chapter 4, “Knowledge in Multi-Agent Systems”, introduces systems withmultiple agents and an environment. The idea of a run – a discrete (time is takento be discrete) path through the set of global states – turns out to be a powerfulinstrument of analysis.

Chapter 5, “Protocols and Programs”, connects protocols (that can be followedby an agent) and programs. Chapter 6, “Common Knowledge and Agreement”,pursues an issue brought up in the first chapters: that of “common knowledge”.

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Common knowledge turns out to be a key notion in many circumstances – forexample, in distributed systems of computers. It is an intrinsically infinitary notion:� is common knowledge provided that every agent knows�, and every agent knowsthat every other agent knows �, and every agent knows that. . . , and so on. It canbe achieved in a finitary way in special circumstances; finding approximations tocommon knowledge and exploring the impact of uncertainty in communicationare issues reserved for the last chapter of the book. Chapter 6 also concerns theproblems of coordinated attack, simultaneous Byzantine agreement, and, in general,the problem of attaining common knowledge when it can be attained or of showingthat it is unattainable when it is unattainable.1

Chapter 7, “Knowledge-Based Programming”, defends the idea that it is verynatural to adopt the metaphor of knowledge in dealing with distributed systems:For example, one can say that a sending agent should keep sending copies of amessage until it knows that the receiver knows what the message is. The ideahere is to take the language of knowledge as a high-level programming language;then, of course, we have to find a way of putting this program into some standardmachine-interpretable code.

Chapter 8, “Evolving Knowledge”, introduces time in a formal manner, addingtemporal modalities to the object language in which we are expressing knowledge:� is true until ; � is true from now on; � becomes true at some time; etc. Thisallows us to consider the evolution of knowledge (and its stability) over time.

Chapter 9, “Logical Omniscience”, is the chapter that will be most interesting tophilosophers. Here the authors attempt to alleviate what they see as a shortcomingof the standard S5 treatment of knowledge, namely its commitment to “logicalomniscience”. We will discuss their treatment of this issue in more detail in thenext two sections.

The tenth chapter, “Knowledge and Computation”, concerns the relation betweenknowledge and algorithms; it is one thing to “know” something implicitly, anotherto be able to make it explicit by computing it in a reasonable amount of time. Asalready mentioned, the notion of common knowledge – what everybody knowsthat everybody knows that everybody knows. . . – is the main topic of the eleventhchapter, “Common Knowledge Revisited”. A number of surprising and interestingconnections are made between common knowledge, simultaneity, and uncertainty.Under most of the circumstances considered earlier in the book, the achievement ofcommon knowledge is impossible. Yet, as the authors also argue, common knowl-edge is a prerequisite for practical agreement. A number of ways of mitigating thelimitations on achieving common knowledge are discussed.

3. “Modal Operators”

The semantics adopted for most of the book is a “possible world” semantics(Aumann structures – structures of events consisting of a set of events partitionedin a (possibly) different way for each agent – are also considered in Chapter 2,

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“A Model for Knowledge”). “Possible world” is, so far as I can tell, simply apicturesque way of saying “model”. The semantics used in the book is based on thestandard Kripke semantics for modal logic (Kripke, 1959). The authors construe“knows that” as a modal operator in a language consisting of primitive atomicpropositions, together with a set of n “modal operators”, K1, . . . , Kn, one for eachof n agents, and the two primitive connectives, : and ^.

I put the phrase “modal operators” in scare quotes, since modalities, such asnecessity and possibility, make no appearance in this text at all. This is just a matterof terminology, of course, but it has its costs. It suggests that the formal studyof modal logic began with Hintikka and von Wright, rather than with the Stoics.(In the notes to this chapter, the authors show that they do know the history ofmodal logic as well as the more recent logic of knowledge.) Lewis and Langford,the founders of modern modal logic, are credited with their axiom systems, butit is then unclear how axiom systems could be published in 1931 for ideas firstdeveloped formally in 1962.

The confusion of terminology is further compounded by the fact that the authorsuse the standard modal operators 2 and 3 as temporal modal operators: 2�meaning that � is true now and at all future times, and 3� meaning that � is trueat some time in the future.2

The models for the language are Kripke structures. These models have the formM = (S, �, K1, . . . , Kn ), where, S is a set of possible worlds, � is an interpretationthat assigns a truth value to each primitive proposition, and the Ki are binaryrelations on S – the accessibility relations of classical modal logic, one for eachagent. Throughout most of the book, the Ki are taken to be equivalence relations.

There are some interesting novelties in this approach. For example, we candefine “distributed knowledge”: A formula � is distributed knowledge in the groupG, if, roughly speaking, � can be derived when everyone pools their knowledge.Common knowledge requires that everyone in the group knows that everyone inthe group knows �, that everyone in the group knows that everyone in the groupknows that everyone in the group knows �, etc. This notion can be captured by afixed-point construction: In any model M, � is common knowledge in the group G,in symbols, ‘CG�’, if and only if everybody knows both � and CG�. It turns out,as we have already indicated, that common knowledge, construed in this way, is akey concept in a number of applications of epistemic logic to distributed systems.

It is assumed, in both Chapter 2 and Chapter 3, that knowledge implies truth:Ki�)�. While this is a conventional assumption about knowledge, and certainlyshould hold for the modality of necessity (in conventional terms, 2�) �), someof us (those who believe in induction) find it questionable. We shall return to thisissue shortly when we consider the problem of logical omniscience and its possiblecures.

Russell and Whitehead, in their monumental work on the foundations of mathe-matics, made one egregious blunder pertaining to English diction: They pronouncedthe symbol ‘�’ “implies”. An early attempt to make sense of this consisted in refer-

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ring to this “implication” as “material implication”. This has caused an inordinateamount of confusion and angry discussion in philosophy, much of it focussed onthe allegation that “a false proposition implies every proposition”. This went onfor perhaps fifty years. The discussion has now largely faded from view (thankgoodness), and most people are careful to refer to ‘�’ (or ‘)’ in this work) as “theconditional”. It is a sentential operator, often introduced, as it is in this book, as anabbreviation: �) is short for :(� ^ : ). We do not ordinarily read operatorsas verbs: We do not say “� conjunctions ”.

Although the authors of the present book are presumably well aware of all this,they reveal carelessness of both the reader’s sensibilities and perhaps understandingwhen they write that in standard modal logic, “logical and material implicationcoincide” (p. 310). They explain that � materially implies if the formula � ) is valid. This not only confuses history, but generates puzzles and mysterieswhere there should be none. The adjective “material” was introduced preciselyto distinguish between (logical) implication and the sentential operator that formsthe conditional. The authors appear to have adapted nearly the opposite sense,according to which ‘� ) ’ is an implication in the general case, but a materialimplication only if ‘� ) ’ is valid. So far as I know, the phrase ‘materialimplication’ has never been used in quite this sense before. On p. 311, when theauthors speak of “closure under material implication”, they use the term in thestandard way to refer simply to the conditional ‘�) ’.

The authors go on to consider nonstandard logics where “logical and materialimplication do not coincide” (p. 325). This sounds startling until you reflect that intheir usage “material implication” concerns the validity of a certain formula – onewhose main connective is a conditional – rather than any sort of relation. When theconditional operator is no longer taken to be truth functional, it is hardly surprisingthat it fails to “coincide” with logical implication, which is defined in terms oftruth: The implied proposition must be true whenever the implying proposition istrue.

Like C. I. Lewis (starting with Lewis, 1912) and many of his successors, theauthors appear to be concerned about the facts that p ) q is true whenever p isfalse, whatever q may be, and that (p ^ :p) ) q is a tautology (p. 325). One wayto alleviate such a concern is to introduce, as the authors do, a new, non-truth-functional, sentential connective characterized in terms of model satisfaction orprovability.

In view of the fact that these concerns can only arise if either one construes‘implies’ as a grammatical conjunction or one fails to distinguish between a sen-tence and the name of a sentence, it is hard to take them seriously. What one cantake seriously (and this is what the authors are clearly most concerned about) isthe possibility that non-truth-functional operators can be of importance in logic.This is an issue quite distinct from questions about the conditional. (Quine, 1940,provides an extremely clear review of this history up to 1940, at the end of section5. Quine, however, has little sympathy for non-truth-functional connectives.)3

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4. Logical Omniscience

The authors are seriously and properly motivated to investigate non-standard logicsby what they call the “problem of logical omniscience”. This is the theme of Chapter9, which we earlier mentioned as being of particular interest to philosophers andcognitive scientists as well as those working in artificial intelligence.

The problem of logical omniscience is really two problems. The approachesto reasoning about knowledge that are discussed in the first eight chapters of thebook all suppose that every agent knows all the logical consequences of what heknows, and “in particular, that they know all tautologies” (p. 309). This wouldrequire, among other things, unbounded resources, which real agents, whetherthey be human or not, don’t have. Chapter 10, in which the authors describe acomputational model of knowledge, deals with some approaches to this problem.

But there are other causes of lack of logical omniscience that are quite independent of computationalpower. For example, people may do faulty reasoning or refuse to acknowledge some of the logicalconsequences of what they know, even in cases where they do not lack the computational resourcesto compute those logical consequences. (p. 310.)

This is a somewhat vague characterization of the goals of Chapter 9, andin fact questions of “faulty reasoning” or the refusal to acknowledge “logicalconsequences”are not touched upon in the chapter. What is dealt with, among otherthings, are questions of closure. The gold standard is full logical omniscience:

� Given a class of structures, if the agent knows all of the formulas in the set , and logicallyimplies the formula �, then the agent also knows �. (p. 311.)

Three weaker conditions are mentioned:

� Knowledge of valid formulas: if � is valid, then agent i knows �.� Closure under logical implication: if agent i knows � and � logically implies , then agent i

knows .� Closure under logical equivalence: if agent i knows � and if � and are logically equivalent,

then agent i knows . (p. 311.)

Three forms of omniscience that are alleged not to follow from the gold standardare mentioned:

� Closure under material implication: if agent i knows � and if agent i knows �) , then agenti also knows .

� Closure under valid implication: if agent i knows � and if �) is valid, then agent i knows .

� Closure under conjunction: if agent i knows both � and , then agent i knows � ^ . (p. 311.)

The first two of these bullets refer to the ‘)’ connective, which may be given avariety of interpretations other than the standard truth-functional interpretation invarious systems; it is thus not surprising; that they do not follow from “full logicalomniscience”. The third form of omniscience is more surprising, in view of the

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fact that, under the standard interpretation of ‘^’, closure under conjunction doesfollow from full logical omniscience for standard structures. What changes here isnot the interpretation of ‘^’, but the structure itself.

This last constraint on knowledge is also particularly interesting since it seemsintuitively as if it ought to fail. It seems reasonable to say that an agent can haveperfectly adequate evidence for � and perfectly adequate evidence for withouthaving perfectly adequate evidence for their conjunction. This is certainly true forany “thresholding” approach to inference. (The relations among the constraintsembodied in the three bullets are explored to some extent in Kyburg, 1970.)

It is worth noting that the authors do not explore the possibilities of rejecting thethird constraint directly, though it fails to hold in some of the approaches to omni-science considered in Chapter 9. Of course, for the rejection of conjunctive closureto be useful and interesting, one might also want to reject the “knowledge axiom”:Ki�) �, an alternative that the authors also do not consider seriously, though intheir first two (very general) approaches to the problem of logical omniscience,they do allow the possibility of its being false.

The first approach to modelling knowledge without logical omniscience, thesyntactic approach, is pursued in x9.2.1. On this approach, we simply replace thetruth assignment � by an assignment � that assigns truth values to all formulas inall states – for example, one can assign “true” to both p and :p in a given state,since these are distinct formulas. (There is also the possibility, not dealt with bythe authors, of assigning more than the two standard truth values to formulas.)

On this approach, anything goes, in the sense that we can impose whateverproperties we want on knowledge: What an agent knows is just the set of proposi-tions to which truth is assigned in every state. By that very token, this is not terriblyinteresting.

The second approach is slightly more interesting: This is a semantic approachin terms of Montague–Scott structures, where such a structure is a set of states S,a truth assignment to each statement in each state, and a set of subsets of S, Ci (s),defined for each s in S and each individual i. Since a set of states corresponds to astatement, the set of sets of states Ci (s) corresponds to the set of statements that theagent knows in the state s. The authors offer a sound and complete axiomatizationfor validity in these structures: It involves a single class of axioms – all instancesof tautologies of propositional logic – and two rules of inference: modus ponens(note that the conditional ) here is standard) and closure under the biconditional, for knowledge: from �, to infer Ki�, Ki .

The subsequent sections of this chapter attempt to steer a middle course betweenthe logical omniscience required by the standard approach and the flexibility (andrelative emptiness) of the approaches just discussed. These approaches involveNonstandard Logic. We have already commented on relations between the condi-tional and implication. One classical approach has been to invent a new connectivewith a non-truth-functional definition that can be made to have “more intuitive”

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properties. This is sometimes done by rendering the treatment of negation nonclas-sical. This is the route followed by the authors.

The first version of this approach is to add to a standard modal structure anoperator � on states such that s�� = s, where s� is the state that gives the semanticsfor negation: :� holds at a state s just in case � does not hold in the adjunct states�. These structures yield a nonstandard behavior for the conditional ), but havethe drawback that no formulas are valid – i.e., true in all structures.

The second version of this approach is to introduce a new connective calledstrong implication such that � ,! holds at a state s in a model M, just in case(M, s) j= whenever (M, s) j= �. A sound and complete axiomatization of a systeminvolving ‘,!’ is given.

A third version involves choosing a subset W of S to function as “possible”states, and adopting a syntactic assignment of truth values with the stipulation thatthis assignment should behave in a standard way on the “possible” states W, andcan be defined arbitrarily on the rest of the states.

In general, this is an extremely interesting chapter, and will be of particularinterest to cognitive scientists. It is, unfortunately, one of the least documentedchapters. The authors refer frequently to “the” nonstandard logic in a number ofplaces, and even claim (p. 360) to have introduced the notion of strong implication,though the term strenge Implikation was introduced by Ackermann (1956), andvariants of the term have been used for many years. They do mention that theirsis only one of many “nonstandard” logics (p. 332), but they do not give a verycomplete picture of just how rich the field of nonstandard logic is. For exampleSusan Haack’s (1974) study of deviant logics contains references to a large numberof entailment logics, relevance logics, logics that introduce new truth values or thatleave truth value gaps, and so on. Not treated in Haack’s book are a large numberof relatively recent “paraconsistent” logics. (A new edition of this classic workwill include a treatment of paraconsistent logic.) The only one mentioned in thebook under review is that of Rescher and Brandom (1979), but others include agroup of Brazilian logicians led by the late Newton Da Costa (see Da Costa, 1974;a collection of relevant papers may be found in Arruda et al., 1977) and a group ofAustralian logicians, among whom may be numbered, in addition to the Routleys,Graham Priest (e.g., Priest, 1989).

I mention these facts, not because I think the authors should have includedreferences to all the important works on nonstandard logic (a heroic task!), butbecause I think that the authors should have conveyed some idea of the breadth ofthe nonstandard-logic literature. This is a difficult matter, of course: How do youconvey this without simply producing a dull catalog that is irrelevant to the themeof this particular book? And how can you be expected to know everything in anextremely broad area? The authors are computer scientists, not historians.

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5. Conclusion

This kind of book is a very difficult kind of book to write as well as to read. It isinevitable that some people will be disturbed by the failure of the authors to referto important and relevant material. Nevertheless, there is a brash quality in manyof the notes and remarks that sometimes makes it hard to be sympathetic. “The firstperson to talk in terms of ascribing mental qualities such as knowledge and beliefto machines was McCarthy” (p. 148). If you construe ‘machines’ narrowly enough,that may be true, though anyone who has listened to a garage mechanic talking toa balky motor might be inclined to disagree. There are a number of claims of theform “the first person to do X was [one of us]”. No doubt these claims are true – atleast I know no differently – but, carefully read, they are much narrower than theymay appear to be.

The feeling that the authors are not really taking anything that happened before1990 as more than an anticipation of what they themselves are doing is furthersuggested by the bibliography, in which, for example, Halpern is represented asfirst or second author 36 times. This is not unreasonable, since, as the authors pointout, the book is based on material that has been presented in numerous places overthe past decade. The notes at the end of each chapter are decidedly helpful and dogenerally provide a glimpse of the historical background of the topics covered inthat chapter. One does wish, though, that these notes either were more complete (butthat would require a multi-volume opus) or (far more feasible) that they soundedless authoritative. By this I do not mean to impugn their scholarship, which is prettyimpressive. What I do mean is that the tone – I don’t know how you put your fingeron a tone – suggests that they have covered (or attempted to cover) all the relevanthistory. Given the breadth of the areas they touch on, this is not a humanly feasibleambition. There are hundreds, if not thousands of articles and papers; dozens, ifnot hundreds of monographs and theses. To attempt to cover all the relevant historysmacks of hubris rather than scholarship, and is quite beside the point in a workthat is essentially original.4

Having delivered myself of that irritation, I would like to return to the contentof the book. There is a lot of detail in it – perhaps more than most people want –but it is a very useful thing to have. As a self-study book for the mathematicallymature reader, it is simply a wonderful book – made so by the well-crafted andnumerous exercises. It is hard to imagine that any reader who works through theexercises can fail to be rewarded with new understanding of a number of importantissues. All things considered, subject to some caveats concerning terminology andhistory, the volume is a remarkable and valuable one.5

Notes

1. Many of these problems are very closely tied to technical problems in distributed computing.The authors find a very fine line between generic agents and distributed computer systems. Thismay seem odd to some philosophers.

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2. The authors, in correspondence, have argued that there are many readers who “grew up” viewingmodality as applying more broadly than to necessity and possibility. Of course I don’t want toconfuse these younger readers, but I still think there is a point to defending classical grammaticaldistinctions, even in the face of common structure.

3. Here, again, here is disagreement; the authors claim that it is “quite standard” to treat “implies”as a sentence-forming operator, rather than as a relation. Perhaps I’m dated, but it seems to methat most logicians have abandoned that bizarre idea, though of course it surfaces in informaltalk among computer scientists and in some specialized technical work.

4. The authors, understandably, disagree. They see themselves as having tried, very hard, to coverall the important historical bases, and welcome any additions that are brought to their attention.That historical effort still seems to me to be beside the point in this work, but perhaps that is amatter of opinion.

5. I would like to express my gratitude for the patience with which the authors, and particularly JoeHalpern, have dealt with my intransigencies. We still do not entirely agree, but I have benefitedgreatly from extensive e-mail correspondence.

References

1. Ackermann, W. (1956), ‘Begrundung einer strengen Implikation’, Journal of Symbolic Logic 21,pp. 113–128.

2. Da Costa, Newton (1974), ‘On the Theory of Inconsistent Formal Systems’, Notre Dame Journalof Formal Logic 15, pp. 497–501.

3. Arruda, A. I., da Costa, N. C. A., and Chuaqui, R. (eds.) (1977), Non-Classical Logics, ModelTheory, and Computability: Proceedings of the 3rd Latin-American Symposium on MathematicalLogic (Campinas, Brazil; July 11–17, 1976), Amsterdam: North-Holland Publishing Co.

4. Haack, Susan (1974), Deviant Logic: Some Philosophical Issues, Cambridge, UK: CambridgeUniversity Press.

5. Kripke, Saul (1959), ‘A Completeness Theorem in Modal Logic’, Journal of Symbolic Logic 24,pp. 1–14.

6. Kyburg, Henry E., Jr. (1970), ‘Conjunctivitis’, in Marshall Swain (ed.), Induction, Acceptance,and Rational Belief, Dordrecht: D. Reidel, pp. 55–82.

7. Lewis, C. I. (1912), ‘Implication and the Algebra of Logic’, Mind 21, pp. 552–531.8. Priest, Graham (1989), ‘Reasoning about Truth’, Artificial Intelligence 39, pp. 231–244.9. Quine, W. V. (1940), Mathematical Logic, Cambridge, MA: Harvard University Press.

10. Rescher, Nicholas, and Brandom, Robert (1979), The Logic of Inconsistency: A Study in Non-Standard Possible-World Semantics and Ontology, Totowa, NJ: Rowman and Littlefield.

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