models for a paraconsistent set theory · keywords: naive set theory; topological models;...

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Journal of Applied Logic 3 (2005) 15–41

www.elsevier.com/locate/ja

Models for a paraconsistent set theory

Thierry Libert

Département de Mathématique, Université Libre de Bruxelles, CP211,Boulevard du Triomphe, 1050 Brussels, Belgium

Abstract

In this paper the existence of natural models for a paraconsistent version of naive set thdiscussed. These stand apart from the previous attempts due to the presence of some non-mingredients in the comprehension scheme they fulfill. Particularly, it is proved here that allowiequality relation in formulae defining sets, within an extensional universe, compels the use omonotonic operators. By reviewing the preceding attempts, we show how our models can natuobtained as fixed points of some functor acting on a suitable category (stressing the use of fixearguments in obtaining such alternative semantics). 2004 Elsevier B.V. All rights reserved.

MSC:03E70; 03B53

Keywords:Naive set theory; Topological models; Paraconsistent logic

1. Introduction

It is known from Russell’s paradox that the first-order axiomatization of the naivtheory of Cantor and Frege is inconsistent in classical logic. More precisely, some pesets ‘{x|ϕ}’ provided by specificϕ-instances of thecomprehension scheme, i.e. ‘∀x(x ∈{x|ϕ} ↔ ϕ)’, lead to triviality if the underlying logic is classical. The most popularthose is the so-called Russell set,R := ‘{x|x /∈ x}’, for by the law of excluded middle onimmediately gets ‘R ∈ R ∧ R /∈ R’, an apparentcontradiction. Naive set theory is thus on

E-mail address:[email protected](T. Libert).

1570-8683/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jal.2004.07.010

http://www.elsevier.com/locate/jal

mailto:[email protected]

16 T. Libert / Journal of Applied Logic 3 (2005) 15–41

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it mustare not

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of the simplest examples of an intuitively correct theory leading so readily and despeto such a contradiction. Of course not so desperately as it might seem to be.

Although it fortunately appeared that these contradictory sets are by no measential for the foundations of mathematics, certain logicians have expressed athat such inconsistent objects be handled and studied within suitable theories, npara(in)consistentones. After all,non-well-founded sets(also calledhypersets), whichare no more indispensable for the foundations of mathematics, have subsequentlyinteresting applications in modeling circular phenomena, notably in computer scienc1

There are many examples in mathematics where the introduction ofimaginary/idealobjects, though giving some advantage to deal with them, has also forced us to gsome basic properties or principles.2 Obviously, the price to be paid here concernslogic in which the theory is embedded, and its possible debilitating effects on clareasoning and mathematical practice.

For a paraconsistent logician, the motivating consideration is that “minor inconscies” should not be allowed to lead, as in classical logic, to irrelevant conclusions.regard to set theory, it is hardly arguable that ‘R ∈ R ∧ R /∈ R’ could be considered asminor inconsistency on the sole (yet actual) fact that a significant amount of mathemcan be developed so unwisely within the naive theory of sets. Certainly it would beeloquent to exhibitnon-trivial models of paraconsistent universes of sets which, at lcontain thecumulative hierarchywithin their classical part.

However expressive such “non-classical” responses to the paradoxes might be,be admitted that in general the techniques used to produce alternative semanticsdevoid of mathematical interest, especially those involving afixed-point argument. In orderto illustrate this, we are going to present our models as fixed points of some functorF(·)’acting on a suitable category.

It has already been shown that models for some non-classical set theoriesobtained in that way from solutions to reflexive domain equations ‘X � F(X)’. Partic-ularly, models for non-well-founded sets were so constructed by computer scientiststechniques of domain theory (see[1,26]). In turn, the most interesting solutions to tconsistency problem forpositive comprehensionprinciples, to which in fact the models we propose are intimately related, were characterized by such reflexive equatwell. These structures, subsequently calledhyperuniverses, are topological models ofset theory (which strongly contradicts the axiom of foundation) that is based on a gcomprehension scheme restricted to thosepositiveformulae in which the classical negatiodoes not occur “explicitly”.3 Accordingly, the universal classV := ‘{x|x = x}’ should be

1 For a comprehensive introduction to this topic and its applications, we recommend Barwise and Mos[8]. The interested reader should also consult Peter Aczel’s book[3].

2 Adding an ‘imaginary’ numberi such thati2 = −1 to the reals forces one to sacrifice theorderedfieldstructure, just as adding non-well-founded sets to ZFC interferes with the formulation of inductive definetc.

3 For precise references on “positive comprehension”, the reader should consult[17] or [23], where a briefhistorical account of the subject is given.

T. Libert / Journal of Applied Logic 3 (2005) 15–41 17

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a set, and this is a singular departure from the limitation of size doctrine of ZF and rset theories.4

In positive set theory, beside the universal classV , the complement of Russell’s clasnamelyR := ‘{x|x ∈ x}’, is also a set. The next step is to deal with the Russell set itso we make the move in this paper by showing how aweak negationcan be incorporateinto positive comprehension.

To handle such a contradictory set, the idea is as simple as it is naive: since, accto the law of excluded middle, the Russell set does belong and does not belong towe just take this for granted. Therefore one might intuitively think of a paraconsiset as an ordered pair of collections which cover the universe: the first part collthose objects which are supposed to belong to it, the second gathering those whsupposednot to belong to it, and where it is now agreed that these two parts may hnon-empty intersection. Thus the trick is just to consider membership and non-membas somewhat independent but symmetricalpositiveproperties.

The goal of this paper is mainly to illustrate how and to what extent this very simplecan be fruitful, showing at the end that there existnatural modelsfor a paraconsistent setheory. We shall review the techniques involved in the previous attempts so as to trastress the use of fixed-point arguments. Incidentally, we will point out and distinguisways of formalizing set theory, namelyabstractionandcomprehension. The distinctionespecially comes out in the manner in which the corresponding universe of sets is mo

We have aimed to make the paper as self-contained as possible, so that it be reaaccessible to a wide range of logicians. In any case, this should make the presentatthe comparison of the diverse approaches easier. As the reader has probably notihave also included a liberal selection of bibliographical references where he may findcomplete treatments and pursue some particular areas of interest to him.

To end this introduction, let us indicate how such a membership ambiguity canbe concocted within a classical context. Consider a universeU which consists of a collection U of objects together with a topology on it which might materialize some notioindiscernibility onU . Then, for anyx∈U andS ⊆ U , define{

x ∈U S ↔ x∈S,

x /∈US ↔ x∈U\S,where(·) is the closure operator onU.5

Note thatx∈S → x∈US, as well asx /∈S → x /∈US. Also, x∈UU\S ↔ x /∈US. But now itis allowed that bothx∈US andx /∈US, for somex∈U andS ⊆ U , and therefore some kinof relative inconsistency may be observed. Note that the non-contradictory subsetU

are nothing but the clopen subsets inU . It should also be stressed that here∈U and /∈Uare actually not independent. Anyhow, it is hopeless to define a model for a paraconset theory in this way, seeing that∈U and /∈U are subsets ofU × P(U), not of U × U .Nevertheless, we have a situation here in which a paraconsistent set can already be

4 By the way, a comprehensive bibliography on set theories with a universal set can be found at the foaddress:http://math.boisestate.edu/~holmes/holmes/setbiblio.html.

5 Throughout the paper we use a small ‘∈ ’ to denote and distinguish the membership relation in the metathfrom the big ‘∈’ in the language of models for some set theory.

http://math.boisestate.edu/~holmes/holmes/setbiblio.html


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et

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-

of as a covering pair ofclosedsubsets of the universe and, as we shall see, this is asuggestive way of looking at paraconsistent sets, indeed.

2. Preliminaries

In this section we give some basic definitions and specify some notations that wused throughout the paper. We also point out a key distinction in the formalizationtheory:comprehensionversusabstraction.

2.1. Structures for a paraconsistent set theory

For any setM , letPp(M) denote the set of ordered pairs of subsets which coverM , i.e.Pp(M) := {(X,Y ) | X ∪ Y = M}.

A structureM for a paraconsistent set theory is formally defined by a non-empty sM

together with a function[·]M from M into Pp(M), called theextension function, whichapplies anya∈M simultaneously to itspositiveextension[a]+M and itsnegativeextension[a]−M, i.e.

M :≡ ⟨M ; [·]M

⟩, where [·]M :M →Pp(M),

[·]M = ([·]+M, [·]−M).

Note that this conception of a paraconsistent set leads naturally to a related no(strong) extensionality which is the following:

for anya, b in M, [a]+M = [b]+M and[a]−M = [b]−M impliesa = b.6

In other words, such a structureM is said to bestrongly extensionalwhen the extensionfunction [·]M is injective; thenM can be identified with a subset ofPp(M), namely therange of[·]M.

Now, by setting{a ∈Mb iff a∈ [b]+Ma /∈Mb iff a∈ [b]−M

for anya, b∈M,

such a structureM is equally defined by means of two binary relations onM :

M :≡ 〈M ; ∈M, /∈M〉 where∈M ∪ /∈M = M × M.

This latter view is adopted in[22]7 and[24]. Basically,∈M and/∈M can be considered aweak negationof each other, since for somea∈M , [a]+M∩ [a]−M is possibly non-emptyAccordingly, ‘x ∈ y ’ can be interpreted as being both ‘true’ and ‘false’ for somex, y

in M . To formalize this, we define thetruth functionεM of the membership relation ‘∈’

6 Here ‘=’ is the meta-theoreticidentity; the interpretation of theequalityrelation within a structure is discussed in Section4.

7 Where∈M and /∈M are rather denoted by∈+ and∈− , respectively.
M M


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in M as follows:{t∈ εM(a, b) iff a ∈Mb

f∈ εM(a, b) iff a /∈Mbfor anya, b∈M.

In this way,εM(a, b) takes exactly one of the followingtruth degrees:

0 := {f}, 1 := {t} or i := {t,f}.The set of these truth degrees will be denoted byT := {0, i,1}, and in these terms, a struture for a paraconsistent set theory appears as

M :≡ 〈M ; εM〉 whereεM :M × M → T

or equivalently, by defining theextension function[·]M in this setting, as

M :≡ ⟨M ; [·]M

⟩where[·]M :M → T M,

y → (x → εM(x, y)

).

Again, such a structure is said to bestrongly extensionalexactly when the extension funtion [·]M thus defined is injective, so thatM can be identified with the range of[·]M,which is here a subset ofT M .

To sum up, a paraconsistent set can be seen, on the one hand, as an orderedsubsets which cover the universe, that is an element ofPp(M), and on the other hand, aa function on the universe which takes values inT , that is an element ofT M . It will beconvenient in the sequel to be able to exchange one view for the other. So, withoabuse, we shall write:{

[x]+M = [x]−1M {1, i},

[x]−M = [x]−1M {0, i}.

Remark 1. We could not end this description without mentioning the “dual” routethe alternative to Russell’s paradox: theparacompleteor partial case, where a setrather materialized by an ordered pair ofdisjoint parts of the universe. Of course, a(strongly extensional) paraconsistent structureM naturally gives rise to a (strongly extesional) paracomplete structureM∗ on the same universeM , and vice versa, by way o(·)∗ : (A,B) → (M\B,M\A).

In other terms:

{a ∈M∗b iff not(a /∈Mb)

a /∈M∗b iff not(a ∈Mb)for anya, b∈M.

Note that in the three-valued setting, the difference is not even perceptible unless theing of i is specified, namelyi := { } (neither ‘true’ nor ‘ false’) for the paracompletecase,instead ofi := {t,f} (both ‘true’ and ‘false’) for the paracompleteone. We shall furthediscuss this apparent duality in Section3.8

8 More on the duality between the paraconsistent and the paracomplete cases (and especially the natity between closed set and open set logics) can be found for instance in Mortensen’s book[27].


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2.2. Some notations

So far, only the truth degree of the atomic formula ‘x ∈ y ’ in a given structureMhas been defined, namely|a ∈ b|M := εM(a, b), for anya, b in M . More generally, thetruth degreeof any formulaϕ interpreted within a given structureM is denoted by|ϕ|M.Incidentally, whenever we write|ϕ|M, it is always assumed that an assignation has bgiven to the possible free variables ofϕ intoM so that the truth degree ofϕ in M is actuallycomputable. As usual the computation goes inductively as soon as thetruth functionsofthe logical connectives and quantifiers involved are defined. Section3 is intended to giveprecise definitions of these, here we just content ourselves with specifying some gnotations regarding the language of set theory.

In the sequel,L informally stands for the first-order language of set theory with∈’as unique primitive relational symbol and with all the logical symbols available. Wwe want to stress the use of further primitive symbols or specify the logical symbolare allowed in the formulae, we shall put them between ‘〈· · ·〉’ and write L〈· · ·〉 to de-note the corresponding language. Hopefully the context should make this conventionNow, relatively to any given structureM, LM〈· · ·〉 will be designating the languageL〈· · ·〉extended by constants naming the elements ofM . This is nothing but the usual labosaving device that will allow us to handle valuations conveniently. Incidentally, forϕ∈LM〈· · ·〉, ϕ(x) is equally used for denoting the formulaϕ provided its free variables aramongx := x1, . . . , xn, and then, ifτ := τ1, . . . , τn is any list of terms (e.g., variables, costants),ϕ(x|τ ), or most oftenϕ(τ ) when no confusion seems possible, will designateformula obtained fromϕ by substitutingτi for each free occurrence ofxi in ϕ. In order tohandle substitutions freely, we shall also conveniently assume that the terms and foof the language have been coded in the metatheory, in such a way as to identify the tformulae whose writings differ only in the name of their bound variables. For instancecan be achieved by using Bourbaki’s squares in the definition of the language and coterms and formulae as formal sequences of symbols in the metatheory. This is impthe definition of the so-calledterm modelsin Section5.

2.3. Comprehension and abstraction

We shall say that a structure as defined in Section2.1 is amodelfor some set theory iit fulfills some fragment of the comprehension scheme, in the following sense:

Definition 2. Let Σ ⊆ L be any given fragment of the language of set theory. We sayM fulfills Comp[Σ] if for each formulaϕ(x, y) in Σ , and for any list of parametersp inM , there existsb in M such that, for anya in M , |a ∈ b|M = |ϕ(a, p)|M.

Such a ‘b’ is usually denoted, at least in the metatheory, by theset abstract‘ {x | ϕ(x, p)}’, as it clearly depends onϕ and the list of parametersp. But it shouldbe stressed that such a ‘b’ is not necessarily unique, unlessextensionalityis required.

However, whether or not extensionality holds, one may also consider such a dtional devise in the language of set theory itself by making use of anabstraction operator


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‘ {· | −}’. As a characteristic feature, this step enables us to quantify and abstract oveables occurring free in set abstracts. A typical example of a term of the extended lanL〈{· |−}〉 is ‘{x | {t | x ∈ x} ∈ x}’.

This subtle yet important distinction has been rarely emphasized in the literature,we shall rather speak ofabstractionscheme instead ofcomprehensionto emphasize thaset abstracts may already appear astermsin the formulaϕ involved in the correspondininstance of that scheme, i.e.:

Definition 3. Let Σ ⊆ L〈{· |−}〉 be any given fragment of the language of set theory wan abstraction operator. We say thatM fulfills Abst[Σ] if for each formulaϕ(x, y) in Σ ,for any list of parametersp in M , and for anya in M , we have|a ∈ {x | ϕ(x, p)}|M =|ϕ(a, p)|M.

Of course it is tacitly assumed here that a suitable interpretation of the abstractiontor has been defined inM . Specifically, the so-calledterm modelsdescribed in Section5.2will provide us with typical examples of such structures. There we will discuss esionality together with the use of set abstracts in the object-language; then the diffbetween comprehension and abstraction should be clear in view of the results we arto present.

Finally, it is worth noting that both the comprehension and abstraction schemesrestated as follows: for each formulaϕ(x, y) in Σ , for any list of parametersp in M , thefunctionx → |ϕ(x, p)|M belongs to the range of[·]M. According to Cantor’s theorem, thrange of[·]M can never be equal toT M , so that our investigations might be summarizby the following simple question:

what can the range of[·]M be?

In a sense, this paper is devoted to showing how, by providingT (and M) with somesuitable structure, a satisfactory response to this question can be found.

3. The inner logic(s)

The three-valued interpretation described in Section2.1 leads to revisit the classicameaning of the usual logical operators in that context. It turns out that the deliberatependence of the interpretations in any structure ofϕ and¬ϕ, for any atomic formulaϕ,will result in the loss of expressiveness in the system.

As we shall see, this can be offset to some extent by introducing new connectivthus, depending on the choice of theprimitive ones (and on the definition of theconse-quencerelation), a variety of three-valuedlogics, each having a corresponding varietymerits and defects, comes up. Roughly, it might be said that all the difference lieschoice of the connective taken as officialimplication. We are not going to specify the primitive connectives here, so that we will not have to refer officially to some logic(s) knin the literature. In any case, a truth-functional characterization of the logical connewill be largely enough for our investigations.


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3.1. A consequence relation

In formal logic, the choice of an appropriate definition of the consequence relatoften guided by a suitable proof-theoretic characterization of the properties of thetives connectives. For such considerations, the reader is referred to[7]. Here we opted fothe common semantical definition.

We first specify thesatisfactionrelation ‘|=’ connected with our interpretation:

for any structureM and anyϕ in LM 〈· · ·〉, M |= ϕ :iff t∈|ϕ|M.

Now, for anyΣ ∪ {ϕ} ⊆ L〈· · ·〉, define

Σ �t ϕ :iff for any M such thatM |= ψ, for anyψ∈Σ, we haveM |= ϕ,

whereM is denoting avaluation, i.e. a structureM together with a specific assignatioin M of the variables occurring free in the formulae ofΣ ∪ {ϕ}. Thus defined, this consequence relation expresses nothing but thepersistence of‘true’ through valuations. Notethat in classical logic, the transmission of truth is obviously equivalent to the transmof non-falsity. This is not the case here, so that ‘�t’ actually says nothing about this latteWe will see below how this lack of symmetry interferes with the properties of somenectives in the object language. By the way, it should be said that other semantical vfor the definition of the consequence relation are possible (see[7]).

3.2. Monotonic connectives

From now on we strongly recommend the non-initiate reader to read and unde“t∈|ϕ|M” as “ϕ is true inM” and “f∈|ϕ|M” as “ϕ is false inM”. Thereby, a pleasanand natural way of introducing the truth functions of the basic connectives and quantisimply to translate theclassicalrules characterizing them. To distinguish the new logoperators so obtained from their corresponding metatheoretical ones, these latterexpressed in English.

The truth functions of ‘¬’, ‘ ∧’ and ‘∀’ are thus defined by the following rules:

t∈|¬ϕ|M iff f∈|ϕ|M,f∈|¬ϕ|M iff t∈|ϕ|M;

t∈|ϕ ∧ ψ |M iff t∈|ϕ|M andt∈|ψ |M,f∈|ϕ ∧ ψ |M iff f∈|ϕ|M or f∈|ψ |M;

t∈|∀xϕ|M iff for any a∈M, t∈|ϕ(x|a)|M,f∈|∀xϕ|M iff there existsa in M such thatf∈|ϕ(x|a)|M.

Those of ‘∨’ and ‘∃’ are obtained in the same way and so remainclassicallydefinablefrom these above. This also yields the following tables for the truth functions of theconnectives, where ‘⊃’ stands for thematerialconditional defined by ‘¬ϕ ∨ ψ ’, and ‘ ⊂⊃ ’


l con-

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re

s

es de-d that

e-

e

e can-eted asne, be-oth casesre-

is the related biconditional connective:9

¬1 0i i

0 1

∧ 1 i 0

1 1 i 0i i i 00 0 0 0

∨ 1 i 0

1 1 1 1i 1 i i

0 1 i 0

⊃ 1 i 0

1 1 i 0i 1 i i

0 1 1 1

⊂⊃ 1 i 0

1 1 i 0i i i i

0 0 i 1

‘Monotonic connectives.’

It is well known that these truth tables correspond to those of Kleene’s strong logicanectives, where then the middle degreei is rather interpreted as atruth value gap, i.e.i := { }, instead of atruth value glut, i := {t,f}. As a characteristic featurei can thus bethought of in both cases as animaginary truth valuesatisfying¬ i = i. In the paraconsistent case, we have|ϕ|M = i iff M |= ϕ ∧ ¬ϕ, namelyϕ is contradictoryin M, while inthe paracomplete case, we have|ϕ|M = i iff M �|= ϕ ∨ ¬ϕ, namelyϕ is undeterminedinM. Note that in both cases, the set of truth degreesT = {0, i,1} is naturally ordered bythe inclusion relation.10 These orderings, which are obviously dual of each other, wilreferred to in this paper as theinformation ordering�I and theknowledge ordering�K ,so that in the first casei might be considered as the token of aclash of information, andin the second case as the one of alack of knowledge. The ordered sets thus defined adenoted respectively byΛ andV :

Λ := 〈T ;�I 〉 ≡

i

/ \0 1

and V := 〈T ;�K 〉 ≡

0 1\ /

i

With regard to the consequence relation defined in Section3.1, a singular departure ithat i is a designated11 value in the paraconsistent case (sincet∈ i) whereas it is not inthe paracomplete one. Therefrom it is readily shown that, with the sole connectivfined hitherto, there are no paracomplete tautologies at all, while it can be provethe paraconsistent tautologies are still exactly the classical ones (see[28] or [7]). Anothermanifestation of that “asymmetry”, in connection with set theory, is the following.

It is proved in[29] that there existnon-trivial paraconsistent models of a full comprhension scheme, namely: there existsM, with M �|= ψ for someψ , such that for anyϕ(x, z)∈L〈∈;¬,∧,∀〉, M |= ∀ z∃y∀x(x ∈ y ⊂⊃ ϕ(x, z)). Particularly, in such a model wshould have|R ∈ R ⊂⊃ R /∈ R|M = i, for it is readily seen from the ‘⊂⊃ ’ table that this cannever be equal to 1, which incidentally shows that on the other hand the above schemnot be satisfied by any paracomplete structure. By no means should this be interpra defect of the paracomplete interpretation in comparison with the paraconsistent ocause the comprehension scheme as formulated here above does not express (in bactually) what it is intended to; in particular it does not fit in with our definition of comphension given in Section2.3, seeing thatt∈|ϕ ⊂⊃ ψ |M does not imply that|ϕ|M = |ψ |M.

9 Note that we are using the same notation for the connectives and their truth functions.10 If need were, we would remind the reader that the truth degrees are subsets of{t,f}.11 In any many-valued setting, a truth-value/degreev is said to bedesignatedif the satisfaction relation ‘|=’ is

defined in such a way thatM |= ϕ holds whenever|ϕ|M = v.


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In fact it is the use of ‘⊃’ as official “implication” connective which is irrelevant. Indeethis connective is not in any sense animplication for it is easily shown thatmodus ponenfails, namely{ϕ,ϕ ⊃ ψ} �t ψ . Though this holds for the paracomplete interpretation,⊃’cannot be considered either as an implication since then�t ϕ ⊃ ϕ. In the paraconsistencase, it is however worthy of note thatψ is actually derivable fromϕ andϕ ⊃ ψ as long asϕ is non-contradictory. But the problem is, precisely, that such a logic is unable to exthat one of its formulae is non-contradictory/determined. For if it was possible to dan unary connective, say ‘◦’, such thatM |= ◦ϕ iff |ϕ|M �= i, we should have◦ i = 0 and◦0 = 1 or i, and so, in any case,◦ i �K ◦0. Now it suffices to observe that all the connetives considered so far (and so anyone definable from these) aremonotonicwith respect tothe knowledge/information ordering.12 The quantifiers ‘∀’ and ‘∃’ thought of as generalized conjunction and disjunction are easily seen to be monotonic as well. By the wwill be convenient in the sequel to consider any truth value 0, 1,i as a propositional constant with the corresponding constant truth function, so that these are obviously montoo. However, none of these is definable from ‘¬’ and ‘∧’ alone. But note that in any setheoretical framework,i can be materialized by ‘R ∈ R’.

This limitative yet remarkable property has been largely explored in the literaturmight be considered as a kind ofsafety property13 with regard to the semantical and logicparadoxes. This will be illustrated and further discussed in Section5. By way of appetizerlet us just state here thatmonotonicityis actually a guarantee for the existence of afixedvalue of a truth function onV .14

Now the issue we have to face is how to definein a natural waynew logical connectivethat will increase the expressive power of the logic.

3.3. Non-monotonic connectives

Though the introduction of non-monotonic connectives can be carried out concurwe shall only focus on the paraconsistent case here. The point is that some signdifferences in the truth tables of the corresponding connectives will follow fromschizophrenic interpretation ofi, i.e. i = {} or i = {t,f} (whereas there is no differencat all in the tables as far as only monotonic connectives are considered). Again thisexplained by the fact thati is designated in one case and not in the other.

The need of a “true”implicationconnective is apparent from Section3.2. A natural wayof introducing such a one ‘→’ is as follows:

t∈|ϕ → ψ |M iff t∈|ϕ|M impliest∈|ψ |M,

f∈|ϕ → ψ |M iff t∈|ϕ|M andf∈|ψ |M.

12 Wheref :T n → T is said to bemonotonicif f (x1, . . . , xn) �K f (y1, . . . , yn) wheneverxi �K yi , fori = 1, . . . , n. [NB: this amounts to the same thing when�K is replaced by�I .]

13 In referring to the title of Avron’s evening lecture at the ESSLLI 2002, of which the WoPaLo was a pa14 To pursue a previous footnote, adding an imaginary truth valuei such that¬i = i yields a solution to any

reflexivepropositionalequationf (x) = x, just as adding an imaginary numberi such thati2 = −1 to the realssupplies any reflexivepolynomialequationf (x) = x with a solution.


nce

s,

f the.

mis-y

-e setulae

As this connective is just expressing thetransmission of truth, it is easily seen that ‘→’ hasthededuction propertyfor ‘�t’, stating that it is precisely the translation of the consequerelation at the object-language level:

Σ ∪ {ϕ} �t ψ iff Σ �t ϕ → ψ.

In fact, it can be shown that the{∧,∨,→,∀,∃}-fragment ofL equipped with ‘�t’ is ac-tually identical to the corresponding fragment of the two-valued classical logic (see[7]).That all the difference retires into the non-classical negation ‘¬’ is not a surprise. By theway, it is worth noticing that ‘¬’ fails to satisfy the substitutivity property (in other wordit is an intensionaloperator):

Φ ��t Ψ doesnot imply ¬Φ ��t ¬Ψ.15

Of course, this “intensionality” is nothing but the reflect at the object-language level oapparent and deliberate independence of the interpretations of∈ and/∈ within any structureA substitutableexternalnegation ‘∼’ can be obtained by setting∼ϕ :≡ ϕ → 0. It is seenfrom the table of ‘∼’ (below) that the unary connective ‘◦’ defined by◦ϕ :≡ ∼(ϕ ∧ ¬ϕ)

is then precisely the indicator connective requested in Section3.2.As mentioned in Section3.1, the consequence relation says nothing about the trans

sion of non-falsity. Consequently, ‘→’ is not self-contrapositive. To remedy this, we maintroduce another implication connective defined by:

ϕ ⇒ ψ :≡ (ϕ → ψ) ∧ (¬ψ → ¬ϕ).

It has no longer the deduction property but does still satisfymodus ponens. Note that allthe conditionals defined so far share a common “classical” negation, i.e.:

f∈|ϕ ⊃ ψ |M iff f∈|ϕ → ψ |M iff f∈|ϕ ⇒ ψ |M iff t∈|ϕ|Mandf∈|ψ |M.

Here are the truth tables of the connectives we have introduced in this section:

∼1 0i 00 1

◦1 1i 00 1

→ 1 i 0

1 1 i 0i 1 i 00 1 1 1

⇒ 1 i 0

1 1 0 0i 1 i 00 1 1 1

⇔ 1 i 0

1 1 0 0i 0 i 00 0 0 1

‘Non-monotonic connectives.’

It then can be confirmed that these arenot monotonic. As a token of this fact, observe that ‘∼’ has no fixed values. The consequence is that the existence of th‘ {x| ∼(x ∈ x)}’ is prohibited. Hence the presence of non-monotonic operators in formdefining sets can be devastating with regard to set theory.

By the way, a very characteristic property of ‘⇔’ is the following:

t∈|ϕ ⇔ ψ |M iff |ϕ|M = |ψ |M.

15 Here ‘Φ ��t Ψ ’ means ‘Φ �t Ψ andΨ �t Φ ’, or equivalently ‘�tΦ ↔ Ψ ’.


-

pearsulated

ssicaltuitive

an beobjectsuitive,

t

rop-

ll,

Thus the comprehension/abstraction schemes stated in Section2.3 can be entirely expressed at the object-language level now:

Comp[Σ] ≡ for anyϕ(x, y) in Σ ⊆ L,

∀ p ∃y ∀x(x ∈ y ⇔ ϕ(x, p)).

Abst[Σ] ≡ for anyϕ(x, y) in Σ ⊆ L〈{· |−}〉,∀ p ∀x(x ∈{x|ϕ(x, p)} ⇔ ϕ(x, p)).

Remark 4. Note that by using an abstractor operator, the existential quantifier disapin the formulation of the abstraction scheme, so that this latter may equally be reformby a “double two-ways rule”:{

x ∈ {x|ϕ(x, p)} ��t ϕ(x, p),

x /∈ {x|ϕ(x, p)} ��t ¬ϕ(x, p).

4. Equality and extensionality

Here it is shown that with the appropriate connectives in their formulations the claconcepts of strict and extensional equalities can be so defined as to maintain their inmeaning and classical properties.

4.1. The equality relation

In any set theory the role of the equality relation in the language, provided that it cused in formulae defining sets, is easily demonstrated. Indeed, to define such simpleas singletons might turn out to be ridiculously cumbersome, and certainly counter-intif the equality was not available.

In order that the binary relational symbol ‘=’ deserves theequalitystatus in the presencontext, we require that its truth function on any structureM, i.e.=M :M × M → T , bedefined in such a way that the following rules hold:{ �t x = x (reflexivity)

{x = y,ϕ} �t ϕ[x|y], for any formulaϕ16 (substitutivity).

Note that as soon as the negation can be used we havex = y �t ϕ(x) ⇔ ϕ[x|y], whichmeans that in any structureM, |ϕ(a)|M= |ϕ[b|a]|M whenevert∈|a = b|M.

Remark 5. The reader should easily convince himself that actually the substitutivity perty forL-formulae17 can be met by only prescribing ‘=’ to satisfyx = y �t x =◦ y∧x � y,

16 Whereϕ[x|y] is denoting the formula obtained fromϕ by substitutingy for some, but not necessarily afree occurrences ofx.

17 So in which ‘=’ does not occur.


pt the

n

f ‘

equal

that

preted‘

rule

y

where the following abbreviations are used:

x =◦ y :≡ ∀z(z ∈ x ⇔ z ∈ y) and x � y :≡ ∀z(x ∈ z ⇔ y ∈ z).

To single out a particular class of structures that play a central role, let us now adonext terminology:

Definition 6. We say that a structureM is normal if the equality relation is interpreted isuch a way that, for anya, b∈M , t∈|a = b|M iff a = b in M .18

It should be remarked that this does not completely determine the truth function o=’onM . All that is implied in the paraconsistent case is{

|a = b|M = 1 or i iff a = b in M,

|a = b|M = 0 iff a �= b in M.

Thus, the same object could be interpreted within a normal structure as being bothto and different from itself. If it is the case that|a = a|M = 1 for anya in M , then ‘=M’is called theclassical identityonM .

4.2. About extensionality

The extensionality principlestates that “two sets are equal as soon as it is shownthey share the same members”. Accordingly, we shall say that a structure isextensionalwhen it obeys this principle. The relation of “sharing the same members” can be interin any structureM by “having the same extension(s)”, which is exactly expressed by=◦ ’,so this latter is often referred to as theextensional equality. The extensionality principledoes precisely assert that this latter is indeed anequalityin the sense of Section4.1. Now,depending on whether the equality relation is admitted in the language, a structureM canbe said to be extensional in two ways essentially:

• If the use of ‘=’ is stipulated, the extensionality principle should appear as thex =◦ y �t x = y (and so we would have�t x =◦ y ↔ x = y). Then we say thatM isextensionalif M |= ∀x∀y(x =◦ y → x = y).

• If the use of ‘=’ is proscribed, according to the remark in Section4.1, ‘=◦ ’ could beconsidered as an equality provided that the rulex =◦ y �t x � y holds. Then we sathatM is weakly extensionalif M |= ∀x∀y(x =◦ y → x � y).

Remark 7. If for any a∈M , there isa ∈ M with M |= ∀x(x ∈ a ⇔ a ∈ x), i.e. a =‘{x|a ∈ x}’, then it is easily proved thatM |= ∀x∀y(x � y ⇒ x =◦ y). This should bethe case in any model for some version of naive set theory, showing that ‘=◦ ’ and ‘�’ areintimately related to each other, independently of extensionality.

18 This latter equality symbol is the meta-theoreticidentity; explicitly, ‘a = b in M ’ means that ‘a’ and ‘b’ aredesignating the same object inM .


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Any extensional structure is obviously weakly extensional. Conversely, it can bethat any weakly extensional structureM gives rise to an extensional one by defining=Mon M by |x = y|M := |x =◦ y|M. To see this, one has to prove that this interpretatio‘=’ fulfills reflexivity and substitutivity in Section4.1.

Hint. Remark thatx �=◦ y ��t δ(x, y), where this latter is theL-formula defined byδ(x, y) :≡ ∃z((z ∈ x ∧ z /∈ y) ∨ (z /∈ x ∧ z ∈ y)).

As in any case the extensionality principle asserts thatx = y ��t x =◦ y, it is interestingto note that within an extensional universe, by allowing ‘=’ in formulae defining sets, somnon-monotonic functions may sneak into them by the back door.19 This is going to have tobe explained in the last sections.

Finally, it should be recalled that, previously in Section2.1, a structureM was said tobestrongly extensionalif for any a, b in M , M |= a =◦ b impliesa = b in M .19 In fact thisnow coincides with the notion ofextensional normalstructure, i.e.:

strongly extensional≡ extensional+ normal.

It is worth noting that in any extensional normal structureM, for any distinct objectsa, b

in M , ast/∈|a = b|M, we must havet/∈|a =◦ b|M, and sof∈|a =◦ b|M. Therefore, if more-over, for anya in M such thatf∈|a = a|M, it was the case thatf∈|a =◦ a|M as well, thenwe would have thatM |= ∀x∀y(x =◦ y ⇒ x = y). Such a strongly extensional structuresaid to beperfect. The models which are described in Section5 fulfill this stronger axiom ofextensionality, considering that ‘=’ will be so defined as to satisfyx �= y ��t δ(x, y) (withδ(x, y) as above). Consequently, within such models the contradictory sets are ethose that are different from themselves, namelyx �= x ≡ δ(x, x) ≡ “x is contradictory”.

5. Kripke-style models: a tribute to monotonicity

Historically, the first “successful” attempts of building a universe of sets which iserned by some kind of full comprehension scheme in a non-classical logic were baa term modelconstruction (e.g.,[9,20]). Roughly, the universeM of such a model is simply made ofset abstracts, seen assyntacticalexpressions of the form{x|ϕ(x)}, say forsuitable formulaeϕ, and then, by afixed-point argument, the membership relation onM isdetermined in such a way that finally{x|ϕ(x)} be a solution to the corresponding instanof what is now called theabstraction scheme.

Basically, that technique initiated by Gilmore in[19], and then resumed by Brady[9], both for the paracomplete case, can be considered as the counterpart of thebut later work by Kripke on the liar paradox. This parallel has already been noticelargely investigated by Feferman in[16]. For a comprehensive treatment concerningliar paradox only, the reader is referred to Visser’s paper[30].

19 More precisely, by means of the ‘⇔’ involved in the definition of ‘=’.
◦


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In this section, we shall further explore the subject to point out the main defect ofattempts, namely the incompatibility of extensionality and abstraction with equality ilanguage.

5.1. A fixed-point theorem

We start with showing how/why a fixed-point argument comes into the picture.Let f (p) be any propositional function in one propositional variablep, and letτ :=

‘{x|f (x ∈ x)}’. If the existence ofτ was guaranteed in a structureM, we would have|τ ∈ τ |M = |f (τ ∈ τ)|M = |f |(|τ ∈ τ |M), where|f | is denoting the truth function off ,showing that this latter should have a fixed-point. Thus, roughly, sets defined by mof truth functions having fixed-points are most welcome. As furtively mentioned intion 3.2, monotonictruth functions onV/Λ are such ones; in that sense monotonicitysafetyproperty.

In order-theoretic terms, stating that any monotonic function onV/Λ has a fixed-poinamounts to saying thatV is adcpo:

Definition 8. We say that a partially ordered setD is directed complete, or in brief is adcpo, if any directed subset inD has a least upper bound; whereA ⊆ D is said to bedirected if for any a, b∈A, there existsc∈A with a, b �D c. [Note that∅ is directed, sothatD must have a least element (providedD �= ∅).]

The relevance of this notion lies in the following important result:

Theorem 9 (Knaster–Tarski). LetD be a dcpo andf :D → D be monotonic. Thenf hasa fixed point, i.e. there existsx in D suchf (x) = x. Moreover, if Fix(f ) denotes the set oits fixed points, then Fix(f ) is also a dcpo.

Using the machinery of ordinal numbers, the least fixed point can be reached induby iteratingf from the least element ofD, while the existence of maximal ones reliesZorn’s lemma (unlessD is finite, of course).

The original version of this statement due to Tarski was rather concerned with comlattices. This one involving dcpo’s is actually the best possible refinement in view ofollowing result (due to Markowski), of which the proof is by far much harder:

If D is an ordered set with a least element such that any monotonicf :D → D

has a fixed point, thenD is a dcpo.

The Knaster–Tarski fixed-point theorem, as many other fixed-point theorems achas shown itself to be a tremendous tool in many areas for solving so-calledreflexiveequations‘X � F(X)’. In a sense, any universe of sets might be conceived as a solutsuch a reflexive equation.

The framework of dcpo’s rather suggests consideringT with the knowledge ordering�K , even in the paraconsistent case, seeing thatΛ is not properly speaking a dcpo. AanywayV andΛ have the same class of monotonic functions, as long as only mono


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operators are involved in formulae defining sets, a result for the paracomplete cayield a corresponding result for the paraconsistent one and vice versa. The introducnon-monotonic operators (as the equality relation) may introduce a certain asymmthe duality. The attitude that consists in consideringT with the knowledge ordering whei = {t,f} can be summarized by the following slogan:

“a clash of information≡ a lack of knowledge” .

It is time now to proceed to some applications of this fixed-point theorem in connewith set theory.

5.2. A denotational universe

Let Σ be any given fragment ofL〈{· |−}〉 defined by means ofmonotonicconnectivesand quantifiers in such a way thatΣ be closed for substitutions, i.e. wheneverϕ(x) is inΣ andτ areΣ -terms, that are variables,Σ -constants or set abstracts{x|ψ} whereψ is inΣ , thenϕ(τ ) is also inΣ . In fact, this assumption guarantees that the abstraction opeis naturally interpretable on the set of allclosedΣ -terms, which is denoted here byM : theinterpretation of ‘{x|ϕ(x, y)}’, for y := τ in M , being nothing but{x|ϕ(x, τ )}.

We are going to show that among all the structures havingM as universe there do exisuch ones fulfillingAbst[Σ]. Since the universe is predetermined, we may identifystructureM with its membership truth functionεM∈V M×M . Now the “approximating”function(·)+ :V M×M → V M×M is defined by:

|a ∈ b|(M)+ := ∣∣ϕ(x|a)∣∣M for anya, b in M with b = {x|ϕ(x)}.

It is clear that the models ofAbst[Σ] are thus exactly the fixed points of(·)+. It then re-mains to show that such fixed points exist. To do that, observe thatV M×M is itself naturallyequipped with aknowledge/informationordering, defined byM1 �K M2 :iff for all x, y

in M , εM1(x, y) �K εM2(x, y). Clearly this ordering turnsV M×M into a dcpo as wellNow, as long as only monotonic connectives and quantifiers are involved inΣ , it can read-ily be proved that the approximating function(·)+ is monotonic onV M×M . Therefore theexistence of models ofAbst[Σ] is established by the fixed-point theorem in Section5.1.

It is apparent from the meaning of�K on V M×M that the most interesting modeshould be themaximal ones, namely those in which there is aminimumof undeter-mined/contradictory“memberships”. As a general rule, these models fail to obeyextensionality principle.

With regard to extensionality, a much more problematic aspect on which we shaelaborate is the use of the equality relation in formulae defining sets.

Note that it is actually possible to incorporate the equality relation, by defining=M inany structureM to be theclassical identityonM , namely:

|a = b|M :={

1 if a andb aresyntacticallyequal,0 otherwise.

Now, by setting|a = b|(M)+ := |a = b|M, the function(·)+ remains monotonic, and thethe consistency of an abstraction scheme with equality in formulae defining sets foll


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Theorem 10 (Gilmore). Abst[L〈∈,=, {· |−};0,¬,∧,∀〉] is consistent.

This result emerges from Gilmore’s work onpartial set theory(see[19,20]). The origi-nal proof makes use of partial predicates and it is based on the so-calledinductive method,later popularized by Kripke, so that the model considered is actually theminimal one. Itis not extensional, seeing that the syntactical interpretation of the equality is by nosuitable for the intended meaning of the elements ofM , as for instance it leads to diffeentiate{x|0 ∧ 0} from {x|0}, whereas these are clearly extensionally equal. Accordinthere would not be any extensional structure.

One might be tempted to remedy the situation by adopting a more suitable defiof the equality in the model itself. That is probably why, in a first report on his result[19]), Gilmore suggested that his model might be extensional. He however had to disit later in his paper[20]. Indeed, it is hopeless to combine extensionality and abstrawith the equality relation in formulae defining sets. To stress once more the asym(due here to the presence of the equality), it should be noted that the original proofresult, further explored and slightly simplified by Hinnion in[21], does not apply directlyto the paraconsistent case. In fact, and curiously enough, it is less sophisticatedlatter:

Theorem 11. There is no extensional paraconsistent model of Abst[L〈∈,=, {· |−};0〉].

Proof. Seta(x) := {u | x ∈ x} andτ := {x | a(x) = ∅}, where∅ := {u | 0}.Now letM be any paraconsistent model ofAbst[L〈∈,=, {· |−};0〉].We show thatM |= a(τ) =◦ ∅, whileM �|= a(τ) = ∅.To show thatM |= a(τ) =◦ ∅, supposet∈|u ∈ a(τ)|M. Hencet∈|τ ∈ τ |M, and thus

t∈|a(τ) = ∅|M. Then, by substitutivity, we would havet∈|u ∈ ∅|M, which is impossibleHence, for anyu, |u ∈ a(τ)|M = 0 = |u ∈ ∅|M, as desired.

On the other hand, we must have|a(τ) = ∅|M = 0; for if t∈|a(τ) = ∅|M, we wouldhavet∈|τ ∈ τ |M, by definition ofτ , and so, for anyu, t∈ |u ∈ a(τ)|M, which is impossi-ble as we have just seen.�

This observation is all the more interesting since Brady in[9] proved a complementarresult of Gilmore’s, even though he was not aware of that, namely that one can reextensionality if (and only if) one drops equality out of the language:20

Theorem 12 (Brady). Abst[L〈∈, {· |−};0,¬,∧,∀〉] is consistent with extensionality.

Brady’s ingenious proof consists in showing that the minimal model is weakly esional (see[9] for the paracomplete case and[12] for the paraconsistent one). It is actuabased on the fact that the minimal fixed point of a monotonic function on a dcpobe reachedinductively, so that extensionality can indeed be considered as a charact

20 It should be remarked that Brady in[9] (submitted in 1969) could not have been aware of a result of Gilmo[20] published in 1974, but even in subsequent works[10–12]this fact does not seem to have been noticed.


iom

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e-hemeningox by

property of theminimalfixed-point model. Incidentally, note that by its nature, the axof extensionality isnot monotonic since the non-monotonic connective ‘→’ is involved inits formulation. To testify to this fact, it is fairly easy to exhibitmaximalfixed point modelswhich arenot extensional:

Proof. Set τ := {x | x ∈ x} and τ ′ := {x | x ∈ x ∧ x ∈ x}. It is clear that, for anyM,M |= τ =◦ τ ′. It is easy to see as well that ifM0 is the minimal model, then we must ha|τ ∈ τ |M0 = i = |τ ′ ∈ τ ′|M0.

We now define a new structureM1 as follows:{|τ ∈ τ |M1 = 0 and|τ ′ ∈ τ ′|M1 = 1,

|a ∈ b|M1 = |a ∈ b|M0 otherwise.

Fact. M1 �K M+1 .

To see this, first observe that|τ ∈ τ |(M1)+ = |τ ∈ τ |M1 = 0 and that|τ ′ ∈ τ ′|(M1)

+ =|τ ′ ∈ τ ′ ∧ τ ′ ∈ τ ′|M1 = |τ ′ ∈ τ ′|M1 ∧ |τ ′ ∈ τ ′|M1 = 1.

Now, let a andb = {x | ϕ(x)} in M with a andb not both equal toτ or τ ′. Then,|a ∈b|M1 = |a ∈ b|M0 = |ϕ(x|a)|M0 �K |ϕ(x|a)|M1 = |a ∈ b|(M1)

+ , seeing thatM0 <K

M1 and thatϕ is supposed to be monotonic.Note that ifM∗

1 was any maximal(·)+-fixed point aboveM1, we would have|τ ∈τ |M∗

1= 0 and|τ ′ ∈ τ |M∗

1= |τ ′ ∈ τ ′|M∗

1= 1, and soM∗

1 �|= τ � τ ′. ThusM∗1 could not

be extensional. [Note that this already happens in the least fixed-point aboveM1.]To show that such a maximal fixed pointM∗

1 does exist is routine:Let D1 := {M |M1 �K M} and(·)+1 be the restriction of(·)+ to D1. We first observe

that(·)+1 is monotonic onD1. Indeed, ifM1 �K M, then we haveM1 �K M+1 �K M+

(using the fact), and this latter belongs toD1 as well. Therefore, asD1 is a dcpo, so isFix[(·)+1 ], and any maximal element suits.�Remark 13. The inductive method was further investigated by Brady, in[10] for theparaconsistent case and in[11] for the paracomplete one. Roughly, by subtly defininnon-truth-functional conditional ‘�’, he succeeded in proving the consistency of aabstraction scheme together with a corresponding weak form of extensionality, nam

Abst′ :≡ for anyϕ(x, y) in L〈∈, {· |−};0,¬,∧,∀,�〉,∀ p ∀x(x ∈{x|ϕ(x, p)} � ϕ(x, p)),

Ext′ :≡ ∣∣∀x∀y (∀z(z ∈ x � z ∈ y) → ∀z(x ∈ z � y ∈ z)).

In the paracomplete case, it is shown in[11] that the underlying logic is close to the infinitvalued Łukiasiewicz logic, for which besides it was proved that the full abstraction sc(without equality in the language either) is consistent. Come to this, it is worth mentiothat a similar technique to Brady’s has very recently been applied to the liar paradField in [15].21

21 It would seem that Brady has himself applied his technique to the Liar in his bookUniversal Logic, CSLIpubls, forthcoming (2003).


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5.3. The issue

In view of the results described above, it is rather apparent that suchdenotationaluni-verses cannot be considered as “natural models” for some set theory. Actually, theconsiderations raise the following question: by dropping set abstracts out of the lanbut keeping equality, can we get a theory with extensional models? In other words,

is Comp[L〈∈,=;0,¬,∧,∀〉] consistent with extensionality?

Yes, we can give a positive answer to this question for the paraconsistent case. Thewe are going to present were first introduced and manufactured in a different way in[22],and then further studied in[24]. It seems quite clear that the construction of such moshould involve other techniques. The remainder of this paper is devoted to showinone could have got these in a natural way.

It has already been argued by some authors (e.g.,[16]) that the sole use of monotonconnectives and quantifiers in formulae defining sets is not sufficient to be able to dclassical mathematics from a set theoretical point of view. To contrast with this, it shosaid the models we propose actually fulfill an extended comprehension scheme invsome non-monotonic functions (due to the presence of ‘=’) and then it turns out that thnatural paraconsistent set theory to which the properties of these models give riinterpret ZF.

6. Scott-style models: beyond monotonicity

The second technique of building models which is tackled here has its source in Swork on models for the untyped lambda calculus. Roughly, it was there the discovethe fixed-point theorem stated in the previous section is actually reflected within susubcategories of the corresponding objects, namely thedcpo’s, with the appropriate morphisms, the so-calledScott-continuousfunctions, which are specific monotonic functiohaving a computational meaning. Indeed, it has been proved there is a wide varfunctors ‘F(·)’ which do have fixed points within those categories, such fixed points bobtained byprojective limits. Since then, this technique has proved to be very useftheoretical computer science for producing semantics of programming languages.22

6.1. The limit of monotonicity

For our purposes it might therefore be tempting to solve a reflexive domain equatthe formX � F(X) whereF(X) ⊆ 〈X → V 〉, the set of all monotonic functions fromXinto V , for it is rather clear that such a fixed point could give rise to astrongly extensionamodel ofComp[L〈· · · ;0,¬,∧,∀〉].

Such an attempt for the paracomplete case is taken very seriously in[5], where it isfinally shown that the structure so constructed contains a model of “rough set th

22 We again rely on appropriate references and leave the interested reader to consult the abundant litethe subject, as[2] for instance.


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within its maximal elements (see also[6]). However attractive this idea is, it cannot soour question, at least as far as monotonicity alone is concerned. To see this, we firto point out a significant difference between this technique and the one describedprevious section: such a fixed-point modelM � F(M) would be obtained by projectivlimit within a subcategory of dcpo’s, so that the universeM of the final structure, whichis not fixed in advance here, carries its ownknowledge/informationordering�M that turnsit into a dcpo. Now, whatever the wayεM is actually defined onM , the existence othe sets ‘{x|x ∈ x}’, ‘ {x|x ∈ a}’ and ‘{x|a ∈ x}’, for any a in M , immediately compelsthe functionx → εM(x, x) and, for anya in M , the functionsx → εM(x, a) andx →εM(a, x), to belong toF(M) ⊆ 〈M → V 〉. Accordingly,εM :M × M → V should bemonotonic in both of its arguments with respect to�M . Of course, the same remark applto =M :M × M → V .

So, in order to be able to incorporate the equality relation in formulae defining setlatter should be monotonic. This is unfortunately not the case on a normal model,soon as there exist two objectsa, b∈M with a <M b, we would havet/∈|a = b|M whereast∈|a = a|M.

We shall thus abandon the category of dcpo’s for another one in which the idenwelcomed. It has often been shown in mathematics that a problem (an equation inular) can be solved by enlarging the realm of its potential solutions. What we are godo hereafter is another illustration of this.

6.2. From monotonicity to continuity

It is well known that monotonicity is a particular case ofcontinuity. Indeed, if anyordered set is equipped with the topology of which the closed subsets are simpdownwards-closed subsets, i.e.A is closediff x � a∈A impliesx∈A, then the monotonicmaps become precisely the continuous ones.

Applying this toV ≡ 〈T ;�K 〉, it is easy to see that

f :X → V is continuous iff f −1{0, i} andf −1{1, i} are both closed.

Now we would remind the reader that such a functionf defines a paraconsistent setx forwhich precisely[x]+X = f −1{1, i} and [x]−X = f −1{0, i}. So the next step is naturallylook at paraconsistent sets defined in terms ofclosedsubsets which cover the universe, bclosed in a broader topological sense, not only in connection with any given orderingleads to swap the set of monotonic functions for the set of all continuous functions ftopological spaceX into V , which then becomes identified with:

F(X) := {(A,B) | A,B closed inX andA ∪ B = X

} ⊆ Pp(X).

As we shall see, a solution toX � F(X) will come up to our expectations. To solve thisflexive equation, we shall work within a suitable category ofHausdorff topological spacesThis is a singular departure from the preceding attempts because the topology of anset isnot Hausdorff (unless the ordering is totally disconnected). Now, in spite of thethat the equality=M :M × M → V is not monotonic on a normal modelM, we will seethat it can however be continuous, but then=−1

M {1, i} = {(a, a) | a∈M} should be closedwhich exactly means thatM should be Hausdorff! So we are on the right track.


tthere

g that,ry in

in cer-

spletegram-

r someibe theoned,

mehownas

and a

by

gics is

6.3. Another fixed-point theorem to the rescue

In the light of Section6.2, the fixed-point theorem in Section5.1 simply states thaparticular continuous functions, namely monotonic ones, do have fixed points. Noware other well-known fixed-point theorems involving continuous maps.

One of the most famous (and much harder to prove by far) is Brouwer’s, statinany real continuous function on the cube[0,1]n (n � 1) has a fixed point. Interestinglythis theorem was invoked in the first attempts of building a model for naive set theothe infinite-valued Łukasiewicz logic.23

Another famous fixed-point theorem is Banach’s forcomplete metric spaces:

Definition 14. Let 〈X1, d1〉, 〈X2, d2〉 be metric spaces. We say thatf :X1 → X2 is con-tracting if there existsε < 1 such thatd2(f (x), f (y)) � ε · d1(x, y), for anyx, y in M .

Theorem 15 (Banach). Let〈X,d〉 be a complete metric space andf :X → X a contractingfunction. Thenf has a unique fixed point, i.e. there is a uniquex in X such thatf (x) = x.

Just as for dcpo’s, it has been shown that this fixed-point theorem is reflected withtain categories of corresponding objects, thecomplete metric spaces. Indeed, it is provedin [4] that there is a large class of functors, called “contracting”, that do have fixed pointwithin such a suitable category. It is then not surprising that the framework of commetric spaces too has shown itself to be a useful tool for producing semantics of proming languages.

6.4. Working with compact complete metric spaces

Roughly, using the techniques described in[4], a solution to our reflexive equationX �F(X) can be found. What we get then is acompactmetric spaceM together with anhomeomorphismh from M ontoF(M). As we shall see,compactnessis going to be anessential ingredient in order to show that such a structure gives rise to a model foparaconsistent set theory. Of course it is beyond the scope of this paper to descrconstruction of that structure. It will not be necessary anyway. As previously mentisuch a solution is already presented and manufactured in a different way in[22]. Moreover,it should be remarked that the results of[4] cannot be applied directly to our case. Sotechnical and precautionary checking is actually required. Particularly, it must be sthat, for any complete metric spaceX, F(X) admits a complete metric space structurewell. Here we shall content ourselves with proving this, introducing some notationspreparatory result.

We first recall that, for any metric space〈X,d〉, the set of its closed subsets, denotedPcl(X), is naturally equipped with a metric, the so-calledHausdorff distancedH which isdefined as follows:

23 The use of fixed-point theorems in connection with semantics for naive set theory in many-valued lodetailed in[25].


d

ce.

,

For anyx∈X andY ∈ Pcl(X), set

d∗(x,Y ) := infy∈Y

{d(x, y)

}and then, for anyA,B∈ Pcl(X), define

dH (A,B) := max{supa∈A

{d∗(a,B)}, supb∈B

{d∗(b,A)}}.In this way, it is proved that whenever〈X,d〉 is complete, so is〈Pcl(X), dH 〉.24 Now〈Pcl(X) ×Pcl(X), dmax〉 is also a complete metric space with

dmax((A1,B1), (A2,B2)

) := max{dH (A1,A2), dH (B1,B2)

}.

Thus, seeing thatF(X) ⊂ Pcl(X) × Pcl(X), it will be a complete metric space providewe show thatF(X) is closed:

Proof. Let (Ak,Bk) −→k→∞(A,B), with (Ak,Bk)∈ F(X) for all k∈N.

Suppose there existsx0∈X with x0/∈A∪B. Then we would have bothd∗(x0,A) � ε andd∗(x0,B) � ε, for someε > 0. Now, sinceAk −→

k→∞A andBk −→k→∞B, one could also find

n∈N, such that bothdH (An,A) < ε anddH (Bn,B) < ε, and it would follow therefromthatx0/∈An ∪ Bn, which is impossible. WhenceA ∪ B = X, and thus(A,B)∈ F(X). �Remark 16. It should be noted that this result is very characteristic of theparaconsis-tent interpretation. For instance, the “pseudo-dual” operatorF∗(·) defined byF∗(X) :={(A,B) | A,B closed inX andA ∩ B = ∅} may no longer yield a complete metric spaIndeed, takeX := [−1,1] with (Ak,Bk) := ([−1,−1

k], [1

k,1]), for eachk � 1. Then

(Ak,Bk) −→k→∞([−1,0], [0,1])/∈F∗(X). Anyhow, it is apparent from Section6.2 that a

paracompleteset should rather be considered as an ordered pair of disjointopensubsetsnot closed.

On the way, we conclude this section by proving a short preparatory result:

Lemma 17. Letf :X → Pcl(X) be continuous. Then{(x, y)∈X2 | x∈f (y)}∈ Pcl(X2).

Proof. SetF = {(x, y)∈X2 | x∈f (y)} and suppose(xk, yk) −→k→∞(x, y) with (xk, yk)∈F

for all k∈N. Then, sincef is continuous,f (yk) −→k→∞f (y). Hence, for anyε > 0, there

does existn such that bothd(xn, x) < ε2 anddH (f (yn), f (y)) < ε

2 . As xn∈f (yn), thisimpliesd∗(xn, f (y)) < ε

2 and then one can findz∈f (y) with d(xn, z) < ε2. Thus we have

d(x, z) � d(x, xn) + d(xn, z) < ε, showing thatx∈f (y) = f (y), and so(x, y)∈F . �24 See for instance Engelking’s book:General Topology, Polish Scientific, Warsaw, 1977.


of

-

iables

uan-

7. The model

It is remarkable that actually we do not even need to know the internal structureM

to define our model. Indeed, all we need is a compact metric spaceM together with anhomeomorphismh (not necessarily an isometry) ontoF(M):

Mh�F(M).

We begin with specifying the interpretation of the primitive symbols inM :

[·]M := h, i.e.: for anya, b in M,

{t∈|a ∈ b|M :iff a∈ (pr1 ◦ h)(b),

f∈|a ∈ b|M :iff a∈ (pr2 ◦ h)(b)(25).

As h is 1–1, this defines astrongly extensionalstructure (in the sense of Section2.1). Inconsequence, we may define the interpretation of ‘=’ as follows:

for anya, b in M,

{t∈|a = b|M :iff a = b in M,

f∈|a = b|M :iff t∈|δ(a, b)|M.

Thus we get an extensional normal structure that isperfect(as defined in Section4.2).We now move on to the analogue of monotonicity in this context:

Definition 18. We define thepositiveand negative extensionsof a LM 〈∈,=〉-formulaϕ(u(n)), with u = u1, . . . , un andn � 1, respectively as follows:{

〈ϕ(u)〉+M := {(a)∈Mn | t∈|ϕ(u|a)|M},〈ϕ(u)〉−M := {(a)∈Mn | f∈|ϕ(u|a)|M}.

Then we say that such a formulaϕ(u(n)) is continuousif its positive and negative extensions thus defined areclosedsubsets ofMn.

It is worth observing that this actually does not depend on the choice of the free varin ϕ, for it can be seen thatϕ(u(n)) is continuous iffϕ(v(m)) is,26 as well as ifϕ(u(k), v(l))

is continuous and(a)∈Ml , then so isϕ(v|a)(u(k)).Following the rules defining the truth functions of the monotonic connectives and q

tifiers, it is readily shown that:⟨(¬ϕ)(u)

⟩+M = ⟨

ϕ(u)⟩−M,⟨

(¬ϕ)(u)⟩−M = ⟨

ϕ(u)⟩+M;⟨

(ϕ ∧ ψ)(u)⟩+M = ⟨

ϕ(u)⟩+M ∩ ⟨

ψ(u)⟩+M,⟨

(ϕ ∧ ψ)(u)⟩−M = ⟨

ϕ(u)⟩−M ∪ ⟨

ψ(u)⟩−M;

25 pr1, pr2 denote respectively the projections on the first and on the second component.26 We would remind the reader that by writingϕ(u(n)) andϕ(v(m)) the actual free variables ofϕ are supposed

to be amongu = u1, . . . , un andv = v1, . . . , vm.


erved

ulae

rmula

t

⟨(∀xϕ)(u)

⟩+M =

⋂a∈M

⟨ϕ(x|a)(u)

⟩+M,

⟨(∀xϕ)(u)

⟩−M = pr2

⟨ϕ(x, u)

⟩−M.

In this latter we are rather considering〈ϕ(x, u(n))〉−M as a subset ofM × Mn, so thatpr2has to be understood as the projection on the second componentMn. Incidentally, we couldhave written⟨

(∀xϕ)(u)⟩−M =

⋃a∈M

⟨ϕ(x|a)(u)

⟩−M

but this would not have enabled us to prove the next lemma:

Lemma 19. Anyϕ(u(n))∈LM 〈∈,=;0,¬,∧,∀〉 is continuous and〈ϕ(u)〉+M ∪ 〈ϕ(u)〉−M =Mn.

Proof. The topological ingredients of the proof can be summarized by two facts:

(1) A spaceX is Hausdorff iffX := {(a, a) | a∈M} is closed inX2;(2) If X1,X2 are compact and Hausdorff spaces, then the projection mapspri :X1×X2 →

Xi (i = 1,2) areclosed, i.e. for any closed subsetF ⊆ X1 × X2, pri(F ) is closed inXi (i = 1,2).

From the rules described above, it is clear that the continuity of a formula is presunder the logical operators ‘¬’ and ‘∧’; because of the compactness ofM , the preservationunder ‘∀’ is now guaranteed by (2). Thus, it suffices to show that the atomic formare continuous. Using (1), (2) and the lemma in Section6.4, this easily follows from thefollowing observations:

〈0〉+M = ∅ ∣∣ 〈0〉−M = M

〈x ∈ y〉+M = {(x, y) | x∈ (pr1 ◦ h)(y)

} ∣∣ 〈x ∈ y〉−M = {(x, y) | x∈ (pr2 ◦ h)(y)

}〈x ∈ x〉+M = pr1

(〈x ∈ y〉+M ∩ M

) ∣∣ 〈x ∈ x〉−M = pr1(〈x ∈ y〉−M ∩ M

)〈x = y〉+M = M

∣∣ 〈x = y〉−M = ⟨δ(x, y)

⟩+M

〈x = x〉+M = M∣∣ 〈x = x〉−M = pr1

(〈x = y〉−M ∩ M

)[Recalling thatδ(x, y) :≡ ∃z((z ∈ x ∧ z /∈ y) ∨ (z /∈ x ∧ z ∈ y)) (see Section4.2).] Thesecond assertion of the lemma is easily proved by induction on the complexity of a foas well, seeing that it does hold for atomic formulae.�

Now we are ready to prove the expected result:

Theorem 20. M |= Comp[L〈∈,=;0,¬,∧,∀〉].Proof. Let ϕ(x)∈LM 〈∈,=;0,¬,∧,∀〉. By the preceding lemma,(〈ϕ(x)〉+M, 〈ϕ(x)〉−M) isa covering pair of closed subsets ofM . Therefore, by the surjectivity ofh, there does exisb in M such that[b]+ = 〈ϕ(x)〉+ and[b]− = 〈ϕ(x)〉− . �
M M M M


n ex-mee thets by

such

ence

l is a

ra

can

is

Hereby the equality relation makes its entrance in formulae defining sets, within atensional universe. According to Section4.2, some non-monotonic functions should coalong with it. In the model, this is certified by the next lemma that states that, besidequality relation, other non-monotonic functions are allowed in formulae defining semeans of restricted quantifications:

Lemma 21. If ϕ is continuous, so are‘∀x(x ∈ y → ϕ)’ and ‘∀x(x /∈ y → ϕ)’ .

Proof. Suppose thatϕ(x, y, z) is continuous and letψ(y, z) :≡ ∀x(x ∈ y → ϕ(x, y, z)).One the one hand, it must be shown thatF := 〈ψ(y, z)〉+M is closed.

So, let(yk, zk) −→k→∞(b, c) with (yk, zk)∈F , for anyk∈N.

Supposet∈|a ∈ b|M, for somea ∈ M . We have to show thatt∈|ϕ(a, b, c)|M.As (pr1 ◦ h)(·) = [·]+M is continuous, we havedH ([yk]+M), [b]+M) −→

k→∞ 0.

Now, asa∈ [b]+M, this impliesd∗(a, [yk]+M) −→k→∞ 0. Therefore, for anyn∈N0, one can

find xkn∈ [ykn ]+M such thatd(a, xkn) < 1/n (and such thatkn < kn+1). Thus, ast∈ |xkn ∈

ykn |M and (ykn, zkn)∈F , we havet∈|ϕ(xkn, ykn, zkn)|M, for any n∈N0. Now, since(xkn, ykn, zkn) −→

k→∞(a, b, c) and ϕ is continuous, it follows thatt∈|ϕ(a, b, c)|M, as ex-

pected. Hence〈ψ(y, z)〉+M is closed.On the other hand,〈ψ(y, z)〉−M = pr2(〈x ∈ y〉+M ∩ 〈ϕ(x, y, z)〉−M) is clearly closed.We thus have proved thatψ(y, z) is continuous.The proof is similar forψ ′(y, z) :≡ ∀x(x /∈ y → ϕ(x, y, z)). �We shall now review succinctly some properties of the model which show why

models can be considered asnatural modelsfor a paraconsistent set theory.Let us first adopt the following abbreviation: ‘x � y’ :≡ ‘∀z(z ∈ x ⇒ z ∈ y)’.It is easily observed thatM |= a � b iff [a]+M ⊆ [b]+M and[b]−M ⊆ [a]−M, and so:

M |= a = b iff M |= a � b ∧ b � a.

According to the last lemma, ‘�’ can be used in formulae defining sets, so that the existof a relevantpower-set operation, namelyP(y) := ‘{x|x � y}’, is actually derivable fromthe extended comprehension scheme the model fulfills.

From the algebraic point of view, let us mention that the universe of the modetypical example of what is called aparaconsistent boolean algebrain [13]. Incidentally,the underlying order of this algebra is nothing but ‘�’ above. Note also that the subalgebof classicalsets is exactly (up to isomorphism) the (classical) boolean algebra ofclopensubsets inM .

A very characteristic property of the model comes directly from its definition andbe expressed as follows:

Definition 22. For anya, b∈M , a is said to beless contradictorythanb, denoted bya � b,whenever[a]+M ⊆ [b]+M and [a]−M ⊆ [b]−M. [Note that in the three-valued setting, thamounts to saying that[a]M �I [b]M, namely[a]M(x) �I [b]M(x), for all x∈M , defin-ing the information/knowledge ordering at the set level.]


is ax-iom-trong

lity

ience,

nford,

J. Com-

(Ed.),

.), Lec-

276–

nford,

, Notre

acon-

ogic 24

utley,: Astron.

) (2001)

So, with this terminology, we have:∣∣∣∣ for each covering pair(A,B) of subsets ofM, there existsa � -minimalx0 in M such thatA ⊆ [x0]+M andB ⊆ [x0]−M.

Indeed, it is nothing but the (unique)x0 such that[x0]M = (A,B).As an immediate consequence, any collection definable by aL-formula ϕ(x) is opti-

mallyapproximated by a�-minimal setx0 defined by:

[x0]M = (〈ϕ(x)〉+M, 〈ϕ(x)〉−M).

The natural paraconsistent first-order theory to which all these properties give riseiomatized in[24]. There it was conjectured that, with a relevant formulation of an axof infinity, that theory could interpret ZF. Very recently, Esser[14] has proved the existence of models fulfilling such a relevant axiom of infinity. These are obtained from smodels of positive set theory calledhyperuniversesand described in[18].

As a concluding remark, it should be mentioned that the dual structureM∗ (as definedin Section2.1) is nota solution to the correspondingparacompleteproblem. AlthoughM∗is a strongly extensional paracomplete model ofComp[L〈∈;0,¬,∧,∀〉], it can be shownthat the “full” equality relation in formulae defining sets is still missing; actually, equabetween classical sets only is allowed. This should be discussed elsewhere.

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