CHRIS MORTENSEN

TOPOLOGICAL SEPARATION PRINCIPLES AND LOGICALTHEORIES

ABSTRACT. This paper is dedicated to Newton da Costa, who, among his many achieve-ments, was the first to aim at dualising intuitionism in order to produce paraconsistentlogics, theC-systems. This paper similarly dualises intuitionism to a paraconsistent logic,but the dual is a different logic, namely closed set logic. We study the interaction betweenthe properties of topological spaces, particularly separation properties, and logical theorieson those spaces. The paper begins with a brief survey of what is known about the relationbetween topology and modal logic, intuitionist logic and paraconsistent logic in respectof the incompleteness and inconsistency of theories. Necessary and sufficient conditionswhich relate theT1-property to the properties of logical theories, are obtained. The resultis then extended to Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity.

1. PRELIMINARIES ON LOGIC AND TOPOLOGY

It is well known that there are significant connections between logic and to-pology. In the first section of this paper we survey these, in preparation forextending them later to take account of topological separation principles.Consider first the formal semantics of propositional modal logic. Modal lo-gic adds to the usual Boolean operators(& ,∨,∼) the unary propositionaloperator2, where2P is is interpreted as “It is necessary thatP ”. Addi-tional Boolean operators such as⊃ and≡ are defined in the usual way;and the modal operator3, interpreted as “it is possible that”, is defined as∼ 2 ∼. The possible worlds semantics for modal logic constructs modelsusing a setX of possible worlds. Propositions hold at some worlds and donot hold at others, so we can writePa for the statement “The propositionP holds at worlda”. For example, ifP is the proposition that snow iswhite, thenPa is the proposition that snow is white holds in worlda. Thisis given an algebraic setting by associating with each propositionP a set[P ] of members ofX, interpreted as the set of worlds at whichP holds.Thus we can definePa to meana ∈ [P ]. The simplest set of conditionsgoverning the behaviour of the operators(& ,∨,∼,2) is:

[P ] is a subset ofX, all propositionsP(1)

Synthese125: 169–178, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

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[∼ P ] = −[P ], the set complement of[P ](2)

[P&Q] = [P ] ∩ [Q](3)

[P ∨Q] = [P ] ∪ [Q](4)

[2P ] = X if [P ] = X(5)

else= ∅It is then simple to prove that:

[3P ] = X if [P ] not =∅(6)

else= ∅We can define adeducibility relation` in a model[ ] by: P ` Q in

[ ] iff [P ] ⊆ [Q]. Then define a proposition to be asemantic theoremjustin case it is true in all worlds of all models, i.e.,[P ] = X in all models.The set of semantic theorems so defined coincides exactly with the set ofprovable theorems of the logicS5. The simplicity of the semantics in thisresult has convinced the big majority of modal logicians thatS5 is thepreferred modal logic as a description of the properties of necessity andpossibility.

The idea thatX might have a topological structure(X,O) with opensetsO, allows a generalisation. Thus if we replace the semantic condition(5) for [2P ] with the condition:

[2P ] = int[P ], the interior of[P ](5.1)

we find instead that the semantic theorems coincide exactly with the prov-able theorems ofS4. Furthermore, it is apparent that we can readily recovertheS5 case with the additional stipulation that the topologyO onX shouldbe the indiscrete topology. Note also that this change implies that thesemantics of possibility, condition (6), changes to:

[3P ] = cl[P ], the closure of[P ].(6.1)

Modal logic is not the only place where logic connects with topology.One can link the behaviour of(& ,∨,∼), especially∼, directly with thetopological structure ofX. So instead one stipulates thatthe semantic valueof any proposition shall be an open set. This amounts to the stipulationthat(& ,∨,∼) shall be functional on the open sets. This is no problem for

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(& ,∨), since open sets are closed w.r.t. (finite) intersections and unions.However, for∼ to be an operator on open sets, it cannot generally be theset (Boolean) complement. Instead, one associates it with theopen com-plement: the largest open set contained in the set theoretic complement,which may be identical with the set theoretic complement but need notbe. Algebras of closed sets areHeyting algebras. Summarising, we take(1)–(4) but replace:

[P ] ∈ O, the open sets onX(1.1)

[∼ P ] = the open complement of[P ].(2.1)

There is also a natural implication operator⇒, which is not definable interms of(& ,∨,∼), but as:

[P ⇒ Q] = [P ] ⇒ [Q] = ∪{O : [P ] ∩O ⊆ [Q]}(7)

If as before a semantic theorem is any formula which holds at all pointsof X in all models, we have exactly the theorems ofintuitionist logic J .The above semantic features mean that intuitionist logic supportsincom-pletetheories, that is theories in which neitherP nor∼ P holds, for somepropositionP . To see this, consider a propositionP and leta be any pointon b[P ], the boundary of[P ]. [P ] is open, soa is not in [P ], hencePadoes not hold. But neither does∼ Pa hold, since that requiresa ∈ [∼ P ]= the open complement of[P ], which is disjoint fromb[P ]. Again, neitherP nor∼ P hold ata, so the theory consisting of the propositions whichhold ata, is incomplete. So one can describea, viewed as a possible world,as anincomplete world.

The topological duality between open and closed is mirrored in a dual-ity between intuitionist logic which supports incomplete theories, and (onevariant of) paraconsistentlogic, which supportsnontrivial inconsistenttheories. A theory isinconsistentif it contains at least one propositionPand its negation∼ P , andnontrivial if it does not contain every proposition(of its language). If we stipulate that all propositions hold onclosedsetsof points, then in order to have negation be a natural operation, we mustidentify the negation of a proposition with theclosed complementof thatproposition, which is the smallest closed set containing the set theoreticcomplement. That is, replace (1) and (2) instead by:

[P ] ∈ C, the closed sets onX(1.2)

[∼ P ] = the closed complement of[P ](2.2)

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We also have, instead of (7):

[P −Q] = [P ] − [Q] = ∩{C : [P ] ⊆ [Q] ∪ C}(7.1)

Algebras of closed sets areBrouwerianalgebras.This change requires rethinking the semantic condition determining

theoremhood. In moving from open sets to closed sets one is reversing theorder on the lattice. That is, the bijection which turns open sets into their setcomplements is contravariant w.r.t. subset inclusion, which is the order onthe set lattice. Now in any lattice-theoretic value space of more than twovalues there is in general a choice of more than one filter on the lattice,where membership of a given filter serves to determine membership of atheory. That is, a given Brouwerian lattice of closed sets, or for that matter aHeyting lattice of open sets, can support more than one theory for the samevalue function[ ], determined by different filters on the lattice. The mostnatural dual for the Heyting condition that a theorem be determined by theproperty of holding at every point, is the condition that a theorem be de-termined by the property of holding at some point. The set of propositionswhich hold at some point in all (closed set) models is closed set logic. Theproperties of the natural dual to intuitionist implication⇒, namely pseudo-difference−, prevent it from being a reasonable implication; for exampleP − P fails to be a theorem. This is not particularly paradoxical, sincedually J lacks a natural pseudo-difference operator, but which Booleanlogic possesses. Furthermore,⇒ is not the only reasonable implicationaround: there is always a reasonable implication on any lattice, namely theoperator which equalsX if [P ] ⊆ [Q] else equals∅.

Let [P ] be a closed set and leta be any point onb[P ]. Sincea ∈ [P ],Pa holds. But also sincea ∈ [∼ P ] = the closed complement of[P ],∼ Pa also holds. That is, the theory which is the propositions holdingat a, is inconsistent. The worlda can thus be described as aninconsistentworld. It is not in general trivial however, since many propositions may failto hold ata. In this sense closed set logic is paraconsistent, i.e., supportsnon-trivial inconsistent theories.

This completes our survey of what is known to date about semanticsusing open and closed sets. In the following sections, these ideas are ex-tended to a more general setting, in which value spaces include sets whichare open, closed or neither, but(& ,∨,∼) remain operators simultaneouslyon both open and closed sublattices, and where the logic addressed goesbeyond propositional logic to a fragment of quantifier logic. Results areobtained connecting specific topological properties such asT1, T2 with thelogical properties of the theories supported. Extension to the even moregeneral case of full first-order logic presents further complexities, requir-

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ing the use of product topologies, but is not necessary to make the pointhere.

2. INCONSISTENCY AND INCOMPLETENESS TOGETHER

We consider monadic predicatesF with a single free variablex. EachFis associated semantically with an extension[F ] which is a subset of atopological space(X,O,C) (usually shortened to(X,O)) with open setsO and closed setsC. Fa holds in a model[ ], or F holds ata in [ ], iffa ∈ [F ]. Now we stipulate that the operators(& ,∨,∼) shall be operatorson both the open and closed sublogics (and ignore intuitionist implicationand paraconsistent pseudo-difference). Again there are no problems with(& ,∨); but there is only one way to deal with negation, and that is to as-sociate it with thetopological complement, which is the open complementif [F ] is open, the closed complement if[F ] is closed, and the set comple-ment otherwise. Note that in the case where[F ] is clopen, i.e., both closedand open, then topological complement coincides with set complement.That is, we retain (1)–(4) except that we replace (2) with:

[∼ F ] = the topological complement of[F ](2.3)

An interesting consequence of this definition, pointed out by Greg Restall,is that we can have[F ] ⊆ [G] without [∼ G] ⊆ [∼ F ], for exampleif [F ] is open and[G] its closure. This is a kind of limited failure ofcontraposition which is significant because it occurs in the context of awell-motivated semantics.

The logic and associated theories can be further extended to sentenceswith a single, leading quantifier by adding:

(∀x)Fx holds iff Fa holds for alla(8)

(∃x)Fx holds iff Fa holds for somea(9)

If negated quantifiers are then defined by the Quantifier Equivalence Laws∼ ∀ =df ∃ ∼ and∼ ∃ =df ∀ ∼, these will have the consequences that:

∼ (∀x)Fx holds iff∼ Fa holds for somea(10)

∼ (∃x)Fx holds iff∼ Fa holds for alla(11)

Apart from being true in Boolean Logic, these are interesting consequencesbecause they permit the possibility of inconsistency and incompleteness

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among quantified sentences. A set of sentences is said to be atheory onatopological space(X,O) if it is the set of sentences which holds in somemodel[ ] where(X,O) is the codomain of[ ].PROPOSITION 1. If(X,O) has the discrete topology then every theoryon (X,O) is consistent and complete.

Proof.The discrete topology is that in which every subset is open. LetF

be any predicate of whatever logical complexity with a single free variablex. Then[F ] is both closed and open. Hence[∼ F ] is the set complementof F . Thus not bothFa and∼ Fa hold, for anya ∈ X. And if Fa does nothold, then a is not in[F ], so it is in[∼ F ], so that∼ Fa holds. That is, thesubtheory of quantifier-free sentences is consistent and complete. For thequantified case, if(∀x)Fx holds thenFa holds for alla. But [F ] is clopen,so a is not in[∼ F ], so∼ Fa holds for noa, so that∼ (∀x)Fx does nothold, which is consistency. If(∀x)Fx does not hold, then for somea, Fadoes not hold. Since[F ] is clopen,a ∈ [∼ F ]; and so∼ (∀x)Fx holds,which is completeness. The case of the existential quantifier is similar.2To find an appropriate converse, we need the following definition:

DEFINITION 2. A theory on a space is a 1-point theory iff for somenonlogical predicateF , [F ] = some singleton{a}.Clearly some theories are 1-point theories, e.g., any theory containing thepredicate “∈ {a}”; while some theories are not 1-point theories, e.g., anytheory with sole nonlogical axiom “a ∈ {a, b}” where “∼ a = b” alsoholds in the theory. Note that we exclude = from being such anF on thegrounds that it is a logical symbol. Of course = can be included in thetheory because “= a”, “= b”, “∼ = a” etc. can be; but allowing it to beall of F weakens later conclusions.

Now we have a proposition which further links the topological proper-ties of(X,O) with the logical properties of theories on it.

PROPOSITION 3. If(X,O) is a T1-space and every 1-point theory on(X,O) is consistent, then(X,O) has the discrete topology.

Proof. We use the formulation of aT1-space as: every singleton isclosed. (For definitions of this and other separation axioms see e.g., Sim-mons (1963, 130.) If(X,O) is not discrete then we show how to constructan inconsistent 1-point theory. Since it is not discrete, some singleton{a}is not open. But beingT1, {a} is closed. Consider any theory containingthe sentence “a ∈ {a}”. This is a 1-point theory w.r.t. the interpretation:[ ∈ {a}] = {a}. Clearly,a ∈ {a} holds. But{a} is closed and not open,

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so [∼ ∈ a] = X. This means that∼ a ∈ {a} also holds and the theory isinconsistent. However, for anyb different froma, b ∈ {a} does not hold,so the theory is non-trivial. 2PROPOSITION 4.(X,O) has the discrete topology iff(X,O) is T1 andevery theory on(X,O) is consistent.

Proof.L toR follows from Proposition 1 together with the fact that thediscrete topology isT1. R toL follows from Proposition 3. 2These propositions between them furnish us with examples of inconsistentand incomplete theories on topological spaces. For inconsistent theories,one cannot have the discrete topology. But any non-discreteT1 topologywill yield them. If {a} is any closed non-open singleton, the theories de-scribed in Proposition 3 suffice, i.e., any theory in whicha ∈ {a} holds,or more generally any theory in whichFa holds where[F ] = {a}. Thesetheories are in general non-trivial, for example if they contain a nameb

for any distinct elementb, thenFb does not hold. For incomplete theories,if we consider 1-point theories we must have a space which is notT1. Asimple case of a non-T1 space is the 3-member setX = {a, b, c}, withO = {X, {a, b}, {a, c}, {a},∅} andC = {X, {b, c}, {b}, {c},∅}. Clearly,b ∈ {a} does not hold. But also[∼ ∈ {a}] = ∅, so that∼ b ∈ {a}does not hold and the theory is incomplete. The theory is at the same timeinconsistent, since bothb ∈ {b} and∼ b ∈ {b} hold.

3. OTHER SEPARATION PRINCIPLES

To extend these results to other separation principles, we need furtherdefinitions. There are many theories on a given topological space(X,O).Here we want to define the theory of a subsetS of X. We also want tomodel identity. In the present setting of monadic predicates, modellingthe monadic predicates[ = a], one for eacha, as 1-point{a} producesin T1 spaces∼ a = a for every a, since all disidenties hold because[∼= a] = X. The simplest solution, which simplifies later results byconfining inconsistency and incompleteness to nonlogical predicates, is tomake identity classical:

[= a] = {a}, all a in X(12)

[∼= a] = −[a](13)

We briefly consider other models for identity in the next section.

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DEFINITION 5. If S is any subset ofX, we say thatthe theory ofS,Th(S), is the theory onX with names for all members ofX, a predicate“ ∈ 6” with [∈ 6] = S, and as well a possibly infinite number of identitypredicates “= a” and “a = ”, one for each name, satisfying (12) and (13)above.

Clearly, the theory ofS identifies all members of the space(X,O) andpicks out the members ofS. This is done consistently except at theboundary, where it is done incompletely or inconsistently.

DEFINITION 6. Theatomicsentences of a theory are those which lack thelogical connectives(& ,∨,∼, ∃,∀). Any two theories on a space(X,O)are atomically disjoint if the atomicnonlogical sentences (whose mainconnective is other than =) holding in each are disjoint; and one is anatomic extensionof the other if the set of atomic sentences of the firstis a superset of those of the second.

Now we have two propositions which characteriseT2 and normal spacesrespectively.

PROPOSITION 7. The space(X,O) is Hausdorff iff every pair of 1-pointtheories of subsets ofX have a pair of consistent atomic extensions whichare atomically disjoint.

Proof.Uses the definition of Hausdorff as: every distinct pair of pointscan be separated by open sets; and consists in noting that the bijectionbetween sets and their theories associates (a) singletons with disjoint 1-point theories which are complete in theT2 case; (b) open sets withconsistent (possibly incomplete) theories; (c) the subset relation with theatomic extension relation; and (d) disjoint sets with atomically disjointtheories. 2PROPOSITION 8. The space(X,O) is normal iff every pair of atomicallydisjoint complete theories of subsets ofX have a pair of consistent atomicextensions which are atomically disjoint.

Proof. Uses the definition of Normal as: every disjoint pair of closedsets can be separated by open sets; and consists in noting that in additionthe bijection associates closed sets with complete (possibly inconsistent)theories. 2The results of this section essentially involve a restatement of the topo-logical separation axioms in the language of theories. But the latter havea character of their own, since they involve the extendibility of pairs of

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complete and possibly inconsistent theories, to consistent theories whichseparate them.

4. OTHER IDENTITY CONDITIONS

There are other ways to model identity. Thus one can replace (12) by:

[= a] = [a =] = {a} ∩ S.(12.1)

This has the effect that only members ofS are self-identical, whilea = bdoes not hold for any distincta andb, nor even whena is b if it is outsideS. The pure theory of identity remains consistent and complete if we retain(13). However if (13) is changed to the topological counterpart of (12.1):

[∼ = a] = [∼ a = ] = ∼ [ = a](13.1)

then the theory of “= a" copies that of “∈ 6” when the latter is 1-point.Thus, whena is not inS, in addition toa = a not holding,∼ a = a holds.InsideS, in addition toa = a holding,∼ a = a holds if {a} is closedbut not open. That is,∼ [= a] is X if a is outsideS or {a} is closed andnot open, else∼ [= a] isX − {a}. As S contracts, the region in which noidentities hold expands, so this can be called the “scorched earth model”.

Alternatively, one can replace (12) by:

[ = a] = [a = ] = {a} ∩ (S∪ ∼ S).(12.2)

This has the effect thata = a holds for everya except those on theboundary ofS, if that is not included inS∪ ∼ S, i.e., if S is open andnot closed. No other identities hold. There are several options for[∼= a].For example, one can have∼ [ = a] which is (13.1); but there are also−[ = a],−{a} and∼ {a}.

There is also−{a} ∪ ({a} ∩ b(S) ∩ S). This has the effect that∼ b =a holds for everyb other thana, but also∼ a = a for everya on theboundary ofS as long as it is included inS, i.e., if S is closed and notopen. That is, ifS is closed and not open, it is surrounded by a boundary ofcontradictory self-identities, which can be called the “ring of fire” model.If howeverS is open and not closed, then∼ a = a does not hold on theboundary, so the boundary ofS is a fence where neither identities nor theirdenials hold.

Finally, there is:

[= a] = S if a ∈ S(12.3)

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else= {a}This has the effect that the members ofS are identified with one another.The methodology of identifying elements is ubiquitous in topology. Thismodel can be used to study the functional properties of inconsistent theor-ies determined by quotient topologies arising from an equivalence relation(see Mortensen (1995, chap. 9)).

5. CONCLUSION

It is seen that there are significant interactions between the topologicalproperties of spaces and the logical properties of theories on them. It isproposed to study these further in later papers.

REFERENCES

Mortensen, Chris: 1995,Inconsistent Mathematics, Kluwer, Dordrecht.Simmons, G. F.: 1963,Introduction to Topology and Modern Analysis, McGraw-Hill,

Kogakusha, Tokyo.

Department of PhilosophyThe University of AdelaideNorth Tce, Adelaide, SA 5005AustraliaE-mail: cmortens@arts.adelaide.edu.au