View of A Neighbourhood Semantics for the Logic TK

A N

S

L

TK

C

A. M

H

A

F

Abstract. The logic TK was introduced as a propositional logic extending the classical propositional calculus with a new unary operator which interprets some conceptions of Tarski’s consequence operator. TK-algebras were introduced as models to TK. Thus, by using algebraic tools, the adequacy (soundness and completeness) of TKrelatively to the TK-algebras was proved. This work presents a neighbourhood semantics forTK, which turns out to be deductively equivalent to the non-normal modal logicEMT4.

Keywords: Consequence operator; TK algebra; TK logic; neighbourhood semantics.

Introduction

Considering algebraic aspects of the notion of Tarski’s consequence operator, Nasci- mento and Feitosa (2005) defined an algebra that rescues these conceptions in an algebraic context, the TK-algebra. So Feitosa, Grácio and Nascimento (2007) in- troduced a propositional logic which has as models exactly these TK-algebras. This logical system was presented in the Hilbert-style, with axioms and rules of inference, and the adequacy between the axiomatic system and the TK-models was proved. As the new operator was introduced to interpret the characteristics of the Tarski’s oper- ator, this propositional logic turns out to be a modal logic. In this paper, we present a neighbourhood semantics for this new logical system.

1. Tarski’s consequence operator

In what follows, we consider the concept of a consequence operator in a way a little more general than was introduced by Tarski, in 1935.

Definition 1.1. A consequence operator on S is a function C : P(S)^→ P(S) such that, for everyA, B⊆S:

(C₁) A⊆C(A);

(C₂) A⊆B⇒C(A)⊆C(B); (C₃) C(C(A))⊆C(A). Principia15(2): 287–302 (2011).

Of course, in view of (C₁) and (C₃), for everyA⊆S, the equalityC(C(A)) =C(A) holds.

Definition 1.2. A consequence operatorC onSisfinitarywhen, for everyA⊆S:

C(A) =∪{C(A0):A₀is a finite subset ofA}.

Definition 1.3. ATarski space(adeductive systemor aclosure space) is a pair(S,C) such thatSis a set andC is a consequence operator onS.

Definition 1.4. Let C be a consequence operator onS. The setAisclosedin (S,C) if C(A) =A; andAisopenin(S,C)if its complement relative toS, denoted byA⁰, is closed.

2. TK-algebras

The definition of a TK-algebra puts in the context of algebraic structures the notions of consequence operator.

Definition 2.1. A TK-algebra is a sextupleA = (A, 0, 1,∨,∼,•) such that (A, 0, 1,

∨,∼)is a Boolean algebra and•is a new operator, theTarski operator, such that:

(i) a∨ •a=•a;

(ii) •a∨ •(a∨b) =•(a∨b); (iii) •(•a) =•a.

Since we are working with a Boolean algebra, the item (i) of the above definition asserts that, for everya∈A,a≤ •a and we can define in a TK-algebra:

ašb=df∼a∨b.

Proposition 2.2. In any TK-algebra the following conditions are valid:

(i) ∼ •a≤ ∼a≤ • ∼a;

(ii) a≤ b⇒ •a≤ •b.

(iii) •(a∧b)≤ •a∧ •b;

(iv) •a∨ •b≤ •(a∨b).

(v) •(•a∧ •b) =•a∧ •b.

Naturally we can define a dual operation of•in any TK-algebra:

◦a=df∼ • ∼a.

Proposition 2.3. In any TK-algebra, the following conditions hold:

(i) ◦a≤a;

(ii) ◦(a∧b)≤ ◦a;

(iii) ◦a≤ ◦ ◦a;

(iv) a≤ b⇒ ◦a≤ ◦b.

An elementa∈ A isclosedwhen•a=aanda∈ A isopenwhen◦a=a.

Proposition 2.4.

(i) If a is open, then a≤b⇔a≤ ◦b;

(ii) If b is closed, then a≤ b⇔ •a≤b.

3. TK Logic

The propositional logicTKis the logical system associated to the TK-algebras. TKis determined over a propositional language L(¬,∨,→,‡,p₁,p₂,p₃, . . .)as follows:

Axiom Schemas:

(CPC) ϕ, ifϕis a tautology;

(TK₁) ϕ^→‡ϕ; (TK₂) ‡‡ϕ^→‡ϕ. Inference Rules:

(MP) ϕ^→ψ,ϕ / ψ; (RM^‡) ϕ^→ψ / ‡ϕ^→‡ψ.

As usual, we write`Sϕto indicate thatϕis a theorem of some axiomatic system S, and we drop the subscript if there is no danger of confusion.

Definition 3.1. LetΓ∪{ϕ}a set of formulas of some systemS. We say thatΓdeduces ϕ, what is denoted by Γ`Sϕ, if there is a finite subset ψ1, . . . ,ψn ofΓsuch that

`(ψ1∧. . .∧ψn)^→ϕ.

Notice that with the notion of syntactic consequence here presented the Deduc- tion Theorem holds; the inference rules are understood as rules of proof.

Proposition 3.2. `‡ϕ^→‡(ϕ^∨ψ).

Proposition 3.3. ϕ`‡ϕ.

As in the case of a TK-algebra, we can define the dual operator of ‡ in the following way:

„ϕ=df¬‡^¬ϕ. Proposition 3.4. ϕ^→ψ`„ϕ^→„ψ.

Corollary 3.5. ϕ^↔ψ`„ϕ<sup>↔</sup>„ψ. Proposition 3.6. `„ϕ^→ϕ. Proposition 3.7. `„ϕ<sup>→</sup>„„ϕ. Proposition 3.8. `„(ϕ^∧ψ)^→„ϕ. Corollary 3.9. `„(ϕ∧ψ)→(„ϕ∧„ψ).

We could, alternatively, consider the operator„as primitive and substitute the axioms TK₁and TK₂by the following ones:

(TK^∗₁) „ϕ^→ϕ, (TK^∗₂) „ϕ^→„„ϕ,

and the rule RM^‡by the rule RM^„: (RM^„) ϕ^→ψ /„ϕ^→„ψ.

Feitosa, Grácio and Nascimento (2007) showed the adequacy of TKrelative to TK-algebras.

4. A neighbourhood semantics for TK

In this section we introduce a new semantic for TKand prove, in later section, its adequacy.

We can show that TK is deductively equivalent to the classical modal system EMT4when considering the operators„and‡to be identical to the necessity and possibility operators ƒand◊. Takingƒas primitive, ◊can be defined in the usual way:

(Df◊) ◊ϕ=df¬ƒ^¬ϕ.

EMT4can be axiomatized by adding to the classical propositional calculus the following axiom schemes and rule of inference:

(M) ƒ(ϕ^∧ψ)^→(ƒϕ^∧ƒψ);

(T) ƒϕ^→ϕ; (4) ƒϕ^→ƒƒϕ;

(RE) ϕ^↔ψ /ƒϕ^↔ƒψ.

Proposition 4.1. Every theorem ofEMT4is a theorem ofTK.

Proof. It follows directly from the definition of „, TK^∗₁, TK^∗₂, and Corollaries 3.5 and 3.9.

Proposition 4.2. Every theorem ofTKis a theorem ofEMT4.

Proof. We only need to show thatEMT4provides RM^„.

1. ϕ^→ψ hypothesis

2. ϕ→(ϕ∧ψ) CPC in 1

3. (ϕ^∧ψ)^→ϕ CPC

4. ϕ↔(ϕ∧ψ) CPC in 2 and 3

5. ƒϕ^↔ƒ(ϕ^∧ψ) RE in 4

6. ƒ(ϕ∧ψ)→(ƒϕ∧ƒψ) M

7. ƒϕ^→(ƒϕ^∧ƒψ) CPC in 5 and 6

8. (ƒϕ^∧ƒψ)^→ƒψ CPC

9. ƒϕ^→ƒψ CPC in 7 and 8.

Definition 4.3. AframeforTKis a structureF=〈U,S〉 such thatU is a nonempty set of possible worlds and S is a function that associates to each x ∈ U a set of subsets ofU (that is,S(x)⊆ P(U)) that satisfies the following conditions:

(m) X∩Y ∈S(x)⇒X ∈S(x)andY ∈S(x); (t) X ∈S(x)⇒x ∈X;

(4) X ∈S(x)⇒ {y∈U:X∈S(y)} ∈S(x).

Definition 4.4. Avaluation V onU is a function from the set of atomic formulas to P(U).

Definition 4.5. LetF=〈U,S〉be a frame andV a valuation inU. AmodelforTKis a pairM=〈F,V〉or, equivalently, a tripleM=〈U,S,V〉.

Definition 4.6. LetM=〈U,S,V〉be a model andx an element ofU. A formulaϕ istruein the worldx, what is denoted by(M,x)ϕ, when:

(M,x)p_i ⇔x ∈V(pi), ifp_i is a propositional variable;

(M,x)^¬ϕ⇔(M,x)6ϕ;

(M,x)ϕ^→ψ⇔(M,x)6ϕor(M,x)ψ;

(M,x) ƒϕ⇔ kϕk^M∈S(x), withkϕk^M={x ∈U :(M,x)ϕ}.

Definition 4.7. The setkϕk^M from the above definition is called thetruth setofϕ inM.

When there is no risk of confusion, we will drop the superscript and write simply kϕk.

Definition 4.8. A formulaϕisvalid in a modelM=〈U,S,V〉when it is true in every x ∈U, and it is valid if it is true in any model M. We denote that a formulaϕ is valid in a modelMbyMϕ, and thatϕis valid byϕ.

IfΓis a set of formulas andM=〈U,S,V〉a model, then we writeMΓif and only ifMϕ, for eachϕ∈Γ. For every x ∈U, we say that(M,x)Γif and only if(M,x)ϕ, for eachϕ∈Γ.

Definition 4.9. LetΓ∪ {ϕ}be a set of formulas. We say that Γimpliesϕ, or that ϕ is alocal semantic consequence ofΓ, what is denoted by Ïϕ, when, for every modelM=〈U,S,V〉and every x∈U, we have: if(M,x)Γthen(M,x)ϕ.

5. Soundness

Since we have shown that TKandEMT4are the same logic, we will work, in what follows, with theEMT4axiomatization.

Lemma 5.1. LetM=〈U,S,V〉be a TK-model, andϕandψany formulas. Then:

(i) k^¬ϕk=−kϕk;

(ii) kϕ^∧ψk=kϕk ∩ kψk; (iii) kϕ∨ψk=kϕk ∪ kψk; (iv) kϕ→ψk=−kϕk ∪ kψk;

(v) kϕ^↔ψk= (−kϕk ∪ kψk)∩(−kψk ∪ kϕk); (vi) kƒϕk={x∈U:kϕk ∈S(x)}.

Proof. Items (i) to (v) are straightforward; we only show (vi). Now, for everyx ∈U, x ∈ {x ∈U :kϕk ∈S(x)}iff kϕk ∈S(x) iff x  ƒϕ iff x ∈ kƒϕk. It follows that {x∈U:kϕk ∈S(x)}=kƒϕk.

Theorem 5.2. If `ϕ, thenϕ.

Proof. By induction on theorems. LetM=〈U,S,V〉be a TK-model.

(A) Letϕbe an axiom. If it is a tautology, the proof is straightforward owing to the fact that every tautology is true in every state of a model, and thus in the model. So supposeϕis one of the modal axioms.

For M: Let x be an element ofU such that x  ƒ(ϕ^∧ψ). It follows thatkϕ^∧ψk ∈ S(x)and, sincekϕ ∧ψk=kϕk ∩ kψkby the preceding lemma, thenkϕk ∩ kψk ∈ S(x). Given that (m) holds in M, it follows that S(x) containskϕk andkψk. But then x ƒϕand x ƒψ, from what it follows that M holds.

For T: Let x be an element ofU such thatx  ƒϕ. By definition,S(x)containskϕk and thus, because (t) holds, x ∈ kϕk. But ifx belongs to the truth set ofϕ, we have that x ϕ, and it follows that T is valid.

For 4: Let x be an element of U such that x  ƒϕ. By definition, S(x) contains kϕkand thus, because (4) holds,{y ∈U:kϕk ∈S(y)} ∈S(x). By Lemma 5.1 (vi), {y ∈U:kϕk ∈S(y)}=kƒϕk. Thus,kƒϕk ∈S(x), sox  ƒƒϕ, and it follows that 4 is valid.

(B) Ifϕ was obtained by MP, the proof is immediate, sincemodus ponensis validity- preserving. So let us consider RE, and suppose`ϕ^↔ψ. Then (inductive hypothe- sis)ϕ^↔ψis valid. Soϕ andψare equivalent, hencekϕk=kψk. It follows that, for every x ∈U, kϕk ∈S(x)iffkψk ∈S(x). Thus x  ƒϕ iff x  ƒψ, from what it follows that x ƒϕ↔ƒψ. Hence RE preserves validity.

Corollary 5.3. If Γ`ϕ, thenÏϕ.

Proof. Suppose Γ `ϕ, and let Mbe some model, and x a world inM, such that (M,x)Γ. SinceΓ`ϕ, by definition there is a finite subsetψ1, . . . ,ψn ofΓsuch that`(ψ1 ∧. . .∧ψn)→ϕ. By the preceding theorem,(ψ1∧. . .∧ψn)→ϕ is valid and so true atx. Since(M,x)Γ, and everyψi ∈Γ, it follows that(M,x)ϕand, thus, thatÏϕ.

6. Completeness

Definition 6.1. A set of formulas∆ismaximal consistentif∆is consistent, and no proper extension of it is consistent.

Theorem 6.2(Lindenbaum). Every consistent set of formulasΓcan be extended to a maximally consistent set∆.

Proof. The proof is standard; see, for instance, Fitting and Mendelsohn 1998, p. 76.

Completeness will be proved using canonical models. Let S be the set of all TK-maximal consistent sets of formulas (TK-MCS).

Definition 6.3. Theproof setofϕis the set|ϕ|={Γ∈S:ϕ∈Γ}. Lemma 6.4. Letϕ andψany formulas. Then:

(i) |^¬ϕ|=−|ϕ|; (ii) |ϕ^∧ψ|=|ϕ| ∩ |ψ|; (iii) |ϕ^∨ψ|=|ϕ| ∪ |ψ|; (iv) |ϕ→ψ|=−|ϕ| ∪ |ψ|;

(v) |ϕ↔ψ|= (−|ϕ| ∪ |ψ|)∩(−|ψ| ∪ |ϕ|);

(vi) |ϕ| ⊆ |ψ| ⇔ `ϕ<sup>→</sup>ψ; (vii) |ϕ|=|ψ| ⇔ `ϕ^↔ψ.

Definition 6.5. M=〈U,S,V〉is acanonical modelforTKif it satisfies the following conditions:

(i) U=S;

(ii) |ϕ| ∈S(Γ)⇔ƒϕ∈Γ, for allΓ∈U;

(iii) V(pi) =|p_i|, for every propositional variablep_i.

Lemma 6.6. LetMbe a canonical model. Then, for every formulaϕ and everyΓ∈U:

(M,Γ)ϕ⇔ϕ∈Γ.

Proof. The proof proceeds by induction on formulas. LetΓbe some element ofU: (a) ϕ= p_i, for somei∈N. By definition,(M,Γ)p_i iff Γ∈V(p_i)iff Γ∈ |p_i|. By construction of|p_i|,Γis a set in|p_i|iff p_i∈Γ.

(b)ϕ=^¬ψ. (M,Γ)ϕiff(M,Γ)6ψiff (by induction hypothesis)ψ /∈Γiffϕ∈Γ. (c) ϕ = ψ ^→σ. By definition,Ïψ ^→σiff (M,Γ)6ψ or (M,x)σ. By the inductive hypothesis,(M,Γ)6ψiffψ /∈Γ, and(M,x)σiffσ∈Γ. Nowψ /∈Γor σ∈Γiffψ^→σ∈Γ. Thus(M,Γ)ψ^→σiffψ^→σ∈Γ.

(d)ϕ=ƒψ. By definition,(M,Γ) ƒψiffkψk ∈S(Γ). By the inductive hypothesis, for every ∆ ∈ U we have that (M,∆)  ψ iff ψ ∈ ∆, that is, kψk = |ψ|. So kψk ∈S(Γ)iff |ψ| ∈S(Γ). Now, by definition ofS, |ψ| ∈S(Γ) iff ƒψ∈Γ. Hence, (M,Γ) ƒψiffƒψ∈Γ.

It is well known with regard to monotonic logics that the smallest canonical model — that is, the model where, for everyΓ,S(Γ)contains only proof sets — does not satisfy condition (m). Fortunately we can show that there are other canonical models in which this condition holds.

Definition 6.7. ThesupplementationofMis the model M⁺ =〈U,S⁺,V〉such that for everyΓ∈U and everyX ⊆U:

X ∈S⁺(Γ)⇔Y ⊆X for someY ∈S(Γ).

It follows from this definition thatS⁺(Γ) ={X ⊆U :|ϕ| ⊆X for someƒϕ∈Γ}

and, obviously, for everyΓ∈U,S(Γ)⊆S⁺(Γ).

We need to prove thatM⁺is a canonical model forTK.

Lemma 6.8. M⁺=〈U,S⁺,V〉is a canonical model forTK.

Proof. It is enough to show that condition (ii) of the definition is satisfied, that is, for everyϕand everyΓ∈U:

|ϕ| ∈S⁺(Γ)⇔ƒϕ∈Γ.

(⇐) Ifƒϕ∈Γ, then|ϕ| ∈S(Γ)and sinceMis a canonical forTKso|ϕ| ∈S⁺(Γ). (⇒) Let|ϕ| ∈S⁺(Γ). Thus, for someY ⊆ |ϕ|, Y ∈S(Γ). SinceMis the smallest canonical model, this means thatY =|ψ|, for someψ. It follows that|ψ| ⊆ |ϕ|, and

ƒψ∈Γ. By Lemma 6.4 we have that`ψ →ϕ, and from RM that `ƒψ →ƒϕ.

Hence,ƒϕ∈Γ.

SoM⁺is a canonical model forTK.

Lemma 6.9. LetMbe the smallest canonical model forTK, andM⁺its supplementa- tion. Then the conditions (m), (t) and (4) hold inM⁺.

Proof. (a) For (m): We have from the previous lemma thatM⁺is a canonical model forTK. LetΓbe an element of U, andX andY be subsets of U such that X∩Y ∈ S⁺(Γ). By construction, there must be someZsuch thatZ⊆X∩Y andZ∈S(Γ). It follows thatZ⊆XandZ⊆Y and, again by construction,X ∈S⁺(Γ)andY ∈S⁺(Γ). (b) For (t): LetΓbe an element ofU, andX a subset ofU such thatX ∈S⁺(Γ). Suppose that X is a proof set, that is, X = |ϕ|, for some ϕ. By definition, we have that ƒϕ ∈ Γ. Since Γ is an MCS, and TK has T, it follows that ϕ ∈Γ. But then Γ∈ |ϕ|, and (t) holds. Suppose now thatX is not a proof set. By construction, for someϕ,|ϕ| ∈S(Γ),|ϕ| ⊆X. But if|ϕ| ∈S(Γ),ƒϕ ∈Γ,Γ`ϕ,Γ∈ |ϕ|,Γ∈X, and again (t) holds.

(c) For (4): LetΓbe an element ofU, andX a subset ofU such thatX ∈S⁺(Γ). We have to show that{∆∈U:X ∈S⁺(∆)} ∈S⁺(Γ).

Suppose first that X is a proof set, that is, X =|ϕ|, for someϕ. By definition, we have that ƒϕ ∈Γ. SinceΓis an MCS, and TKhas 4, it follows that Γ`ƒƒϕ and thatƒƒϕ∈Γ. By canonicity of the model,|ƒϕ| ∈S(Γ)and, by construction of M⁺,|ƒϕ| ∈S⁺(Γ). We must now show that|ƒϕ|={∆∈U :ƒϕ∈S⁺(∆)}. Now,

|ƒϕ|={∆∈U :ƒϕ∈∆}. Since the model is canonical,ƒϕ∈∆iff|ϕ| ∈S(∆)iff (by construction)|ϕ| ∈S⁺(∆). So|ƒϕ|={∆∈U :|ϕ| ∈S⁺(∆)}. It follows that {∆∈U:|ϕ| ∈S⁺(∆)} ∈S⁺(Γ), and (4) holds.

Suppose now that X is not a proof set. By construction of M⁺, however, there is some formula ϕ such that|ϕ| ⊆ X and|ϕ| ∈S(Γ). As above, we can show that

|ƒϕ| ∈S⁺(Γ), and that {∆∈U :|ϕ| ∈S⁺(∆)} ∈S⁺(Γ). Now, for every∆∈U, if ϕ ∈S⁺(∆), thenX ∈S⁺(∆). So{∆∈U :|ϕ| ∈S⁺(∆)} ⊆ {∆ ∈U : X ∈S⁺(∆)}. But {∆ ∈ U : |ϕ| ∈ S⁺(∆)} = |ƒϕ|, so it is a proof set. By construction of M⁺, {∆∈U:X∈S⁺(∆)} ∈S⁺(Γ), and (4) holds.

Theorem 6.10(Completeness). IfÏϕ, thenΓ`ϕ.

Proof. Suppose thatΓ0ϕ. Thus,Γ0^¬¬ϕ, and it follows thatΓ∪{^¬ϕ}is consistent.

By Lindenbaum’s Theorem, there exists anTK-MCS∆such thatΓ∪ {^¬ϕ} ⊆∆, that is,¬ϕ∈∆, andϕ /∈∆. Let nowMbe the smallest canonical model forTK, andM⁺ its supplementation. By the preceding lemma, conditions (m), (t) and (4) hold in M⁺, so it is a model forTK. Now∆is aTK-MCS andΓ⊆∆, so∆is a state inM⁺ such that, by Lemma 6.6,(M⁺,∆)Γand(M⁺,∆)2ϕ; henceΓ2ϕ.

7. Decidability

We show the decidability ofTKusing filtrations.

Definition 7.1. LetΓbe a set of formulas closed under subformulas, andMa model.

For any states x and y inM, we say that

x≡Γ y iff for everyϕ∈Γ,(M,x)ϕ iff(M,y)ϕ.

In other words, ifx ≡_Γ ythenx andyare equivalent with regard to the formulas inΓ. We can easily show that≡_Γis indeed an equivalence relation, partitioning the setU of states into disjoint equivalence classes.

Definition 7.2. Let Γ be a set of formulas closed under subformulas and M =

〈U,S,V〉a model. Then:

(i) if x∈U,[x]_Γ={y ∈U :x≡Γ y}; (ii) ifX ⊆U,[X]_Γ={[x]:x ∈X}.

Again, we will usually drop the subscript and write≡,[x], and[X].

Definition 7.3. Let M= 〈U,S,V〉 be some model, and Γa set of formulas closed under subformulas. Afiltration ofMthroughΓis any modelM^∗=〈U^∗,S^∗,V^∗〉such that:

(a) W^∗= [W];

(b) for every x ∈U, and every formulaƒϕ∈Γ, (i) kϕk^M∈S(x)⇔[kϕk^M]∈S^∗([x]); (ii) kƒϕk^M∈S(x)⇔[kƒϕk^M]∈S^∗([x]);

(c) V^∗(pi) = [V(pi)], for everyi∈Nsuch thatp_i∈Γ.

Notice that the above definition leaves room for different filtrations of a model.

Definition 7.4. A filtration is the finest filtration if, for every x ∈ U, S^∗([x]) con- tains only the sets [kϕk^M]and[kƒϕk^M]such that, respectively,kϕk^M∈S(x)and kƒϕk^M∈S(x), for everyƒϕ∈Γ.

This is what is needed for a model to be a filtration. Coarser filtrations will allow S^∗([x])to contain other sets besides the minimum required.

Theorem 7.5. LetM^∗=〈U^∗,S^∗,V^∗〉be aΓ-filtration of a modelM=〈U,S,V〉. Then, for everyϕ∈Γand every x∈U:

(M,x)ϕ⇔(M^∗,[x])ϕ, that is,[kϕk^M] =kϕk^M∗.

Proof. The proof is by induction on formulas. Let x be any state inM, andp some variable inΓ:

(M,x)p iff x ∈V(p) (by 4.6) iff [x]∈[V(p)] (by 7.2(ii)) iff [x]∈V^∗(p) (by 7.3(c)) iff (M^∗,[x])p (by 4.6).

The boolean cases are straightforward; we show only the case in whichϕ=ƒψ, for someψ. Let againx be any state inM:

(M,x) ƒψ iff kψk^M∈S(x) (by 4.6) iff [kψk^M]∈S^∗([x]) (by 7.3(b))

iff kψk^M∗ ∈S^∗([x]) (inductive hypothesis) iff (M^∗,[x]) ƒψ (by 4.6).

Corollary 7.6. LetM^∗be aΓ-filtration of a modelM. ThenMandM^∗are equivalent moduloΓ, that is, for everyϕ∈Γ,Mϕiff M^∗ϕ.

A well-known result says that if a logic is axiomatizable and has the finite model property—that is, every nontheorem fails in some finite model—, then it is decidable.

TK is axiomatizable, as we have shown before. All we need to show is thatTK is determined by the class of finite models satisfying conditions (m), (t) and (4).

Lemma 7.7. LetMbe a model,Γa set of formulas closed under subformulas, andM^∗ aΓ-filtration of M. Then, for everyϕ∈Γ:

(i) kƒϕk^M={x∈U:kϕk^M∈S(x)};

(ii) [kƒϕk^M] ={[x]∈U^∗:[kϕk^M]∈S^∗([x])}. Proof.

(i)x ∈ kƒϕk^M iff x  ƒϕ;[Def. truth-set] iff kϕk^M∈S(x);[Def. 4.6] iff x ∈ {x ∈U :kϕk^M∈S(x)}. (ii)[x]∈[kƒϕk^M] iff [x]∈ kƒϕk^M∗;

iff [x] ƒϕ; iff kϕk^M∗ ∈S^∗([x]); iff [kϕk^M]∈S^∗([x]);

iff [x]∈ {[x]∈U^∗:[kϕk^M]∈S^∗([x])}.

Theorem 7.8. LetMbe a model satifying conditions (m), (t) and (4), and, for some set of formulasΓclosed under subformulas, letM^∗be the finestΓ-filtration ofM. Then its supplementation,M^∗+, is aΓ-filtration of Mand satisfies (m), (t) and (4).

Proof. We first show thatM^∗+is aΓ-filtration ofM. That is, we must show that, for every x∈U and everyƒϕ∈Γ,

(i) kϕk^M∈S(x)⇔[kϕk^M]∈S^∗+([x]), and (ii) kƒϕk^M∈S(x)⇔[kƒϕk^M]∈S^∗+([x]).

(i). Suppose first that kϕk^M ∈ S(x). Then [kϕk^M] ∈ S^∗([x]), since M^∗ is a Γ- filtration ofM. Thus[kϕk^M]∈S^∗+([x])by supplementation.

Suppose now that[kϕk^M]∈S^∗+([x]). By the definition of a supplementation, there must be someψ(eventuallyψ=ϕ) such thatƒψ∈Γ,[kψk^M]∈S^∗([x])and [kψk^M]⊆[kϕk^M]. It follows thatkψk^M ∈S(x), sinceM^∗is the finestΓ-filtration of M. Now M satisfies condition (m), which means that every superset ofkψk^M belongs toS(x). We just need to show thatkψk^M⊆ kϕk^M.

Now,[kψk^M]⊆[kϕk^M]means that{[x]:x ∈ kψk^M} ⊆ {[x]: x∈ kϕk^M}. Let y ∈ kψk^M. Hence[y]∈ {[x]:x ∈ kψk^M}, and[y]∈ {[x]:x ∈ kϕk^M}. It follows that y∈ kϕk^M. Sokψk^M⊆ kϕk^M, and from this we have thatkϕk^M∈S(x). (ii). Ifkƒϕk^M∈S(x), then[kƒϕk^M]∈S^∗([x])becauseM^∗is a Γ-filtration ofM, and, by the definition of a supplementation,[kƒϕk^M]∈S^∗+([x]).

Suppose now that[kƒϕk^M]∈S^∗+([x]). By the definition of a supplementation, there must be someψsuch thatƒψ∈Γand

(a) [kψk^M]∈S^∗([x])and[kψk^M]⊆[kƒϕk^M], or (b) [kƒψk^M]∈S^∗([x])and[kƒψk^M]⊆[kƒϕk^M].

(a). Since M^∗ is a Γ-filtration of M, we havekψk^M ∈S([x]). Suppose now y ∈ kψk^M. Hence [y] ∈ {[x] : x ∈ kψk^M}, and [y] ∈ {[x] : x ∈ kƒϕk^M} by the second part of (a) above. But then y ∈ kƒϕk^M. Thuskψk^M⊆ kƒϕk^M. SinceMis supplemented,kƒϕk^M∈S([x]).

(b). Since M^∗ is a Γ-filtration of M, we have kƒψk^M ∈ S([x]). Suppose now y ∈ kƒψk^M. Hence[y]∈ {[x]:x ∈ kƒψk^M}, and[y]∈ {[x]:x ∈ kƒϕk^M}by the second part of (b) above. But then y ∈ kƒϕk^M. Thuskƒψk^M ⊆ kƒϕk^M. SinceM is supplemented,kƒϕk^M∈S([x]).

Thus, M^∗+ is a Γ-filtration of M. Does it satisfy conditions (m), (t) and (4)?

SinceM^∗+is a supplementation, it automatically satisfies (m).

Let us consider (t). We need to show, for any[x]∈U^∗and any subsetX ofU^∗, that[x]∈X, ifX ∈S^∗+([x]). Suppose first thatX ∈S^∗([x]). SinceM^∗is the finest filtration, X = [kϕk^M], for some formulaϕ such thatƒϕ ∈Γ andkϕk^M ∈S(x). Now condition (t) holds inM, sox ∈ kϕk^M, and[x]∈[kϕk^M] =X.

If nowX ∈/S^∗([x])then, by the definition of a supplementation,X is a superset of some [kϕk^M] such that ƒϕ ∈ Γ and kϕk^M ∈ S(x). As above, it follows that [x]∈[kϕk^M], and, thus, that[x]∈X. Thus (t) holds inM^∗+.

With regard to condition (4), we show first that M^∗ has (4). Let [x] be an element ofU^∗, andX a subset ofU^∗such thatX ∈S^∗([x]). We have to show that

{[y]∈U^∗:X ∈S^∗([y])} ∈S^∗([x]).

SinceM^∗is the finest filtration, there must be some formulaƒϕ∈Γsuch that (i) X = [kϕk^M]andkϕk^M∈S(x), or

(ii) X = [kƒϕk^M]andkƒϕk^M∈S(x).

Suppose it is (i). Since[kƒϕk^M] ={[y]∈U^∗:[kϕk^M]∈S^∗([y])}(Lemma 7.7), what we have to show is that[kƒϕk^M]∈S^∗([x]).

Now condition (4) holds inM, so{y ∈U :kϕk^M∈S(y)} ∈S(x). But (Lemma 7.7) kƒϕk^M={y ∈U :kϕk^M ∈S(y)}; thuskƒϕk^M ∈S(x). By condition (b.ii) of the definiton of filtration, we immediately have[kƒϕk^M]∈S^∗([x]).

Now consider (ii). We now have to show that

{[y]∈U^∗:[kƒϕk^M]∈S^∗([y])} ∈S^∗([x]).

By Lemma 7.7,[kƒƒϕk^M] ={[y]∈U^∗:[kƒϕk^M]∈S^∗([y])}. So what we have to show is that[kƒƒϕk^M]∈S^∗([x]).

Since condition (4) holds inM,{y∈U :kƒϕk^M∈S(y)} ∈S(x). By Lemma 7.7, kƒƒϕk^M={y∈U:kƒϕk^M∈S(y)}; thuskƒƒϕk^M∈S(x).

Now`ƒϕ^↔ƒƒϕ (it follows from T and 4), so ƒϕ ^↔ƒƒϕ. But then, for every x∈U, x ƒϕiff x  ƒƒϕ. Thuskƒϕk^M=kƒƒϕk^M.

From this it follows thatkƒϕk^M∈S(x), and also that[kƒϕk^M] = [kƒƒϕk^M]. By conditon b.ii of the definition of filtration, we immediately have [kƒϕk^M] ∈ S^∗([x]), and[kƒƒϕk^M]∈S^∗([x]).

It follows from (i) and (ii) above thatM^∗has (4). We now show thatM^∗+has (4).

Let[x]be an element of U^∗, and X a subset of U^∗such that X ∈S^∗+([x]). We thus have to show that{[y]∈U^∗:X ∈S^∗+([y])} ∈S^∗+([x]).

SinceM^∗+ is the supplementation ofM^∗, there must be some formula ϕ such thatƒϕ∈Γ,[kϕk^M]∈S^∗(x)and[kϕk^M]⊆X (eventually[kϕk^M] =X, of course).

But as we have shown above, M^∗ has (4), so {[y] ∈ U^∗ : [kϕk^M] ∈ S^∗([y])} ∈ S^∗([x]). Now, for every [y] ∈ U^∗, if [kϕk^M] ∈ S^∗([y]) then X ∈ S^∗+([y]) by supplementation. So we have:

{[y]∈U^∗:[kϕk^M]∈S^∗([y])} ⊆ {[y]∈U^∗:X ∈S^∗+([y])}.

Finally, since{[y]∈U^∗ :[kϕk^M]∈S^∗([y])} belongs toS^∗([x]), {[y]∈U^∗: X ∈ S^∗+([y])} ∈S^∗+([x])by supplementation and we are done.

Theorem 7.9. TKis determined by the class of finite models satisfying conditions (m), (t) and (4).

Proof. If ϕ is a theorem of TK, then it is valid in the class of all models satisfying conditions (m), (t) and (4); in particular, in the class of finite models satisfying these conditions.

For the other direction, supposeϕ is not a theorem ofTK. Thenϕ fails in some world x of some modelMforTK. LetΓ be any finite set of formulas closed under subformulas such that ϕ ∈ Γ, and let M^∗+ be the supplementation of a finest Γ- filtrationM^∗ ofM. By Theorem 7.8,M^∗+, is aΓ-filtration ofMand satisfies (m),

(t) and (4). SinceΓis a finite set,M^∗+is a finite model. By Theorem 7.5,x and[x] agree on every formula inΓ; thus(M^∗+,[x])2ϕ.

In view of the preceding result, every nontheorem of TK fails in some finite model, from what it follows that TKhas the finite model property. Since it is also axiomatizable,TKis decidable.

Acknowledgements

This work has been sponsored by FAPESP through the Projects 2004/14107-2. The article was written while Hércules Feitosa was at UFSC doing post-doctoral research.

References

Bell, J. L. & Machover, M. 1977.A course in mathematical logic. Amsterdam: North-Holland.

Blackburn, P.; Rijke, M.; Venema, Y. 2001. Modal logic. Cambridge: Cambridge University Press.

Carnielli, W. A.; Pizzi, C. 2001.Modalità e multimodalità. Milano: Franco Angeli.

Chagrov, A. & Zakharyaschev, M. 1997.Modal logic. Oxford: Clarendon Press.

Chellas, B. F. 1980.Modal Logic: an introduction. Cambridge University Press.

Ebbinghaus, H. D.; Flum, J.; Thomas, W. 1984. Mathematical logic. New York: Springer- Verlag.

Feitosa, H. A.; Grácio, M. C. C.; Nascimento, M. C. 2007. A propositional logic for Tarski’s consequence operator. Campinas: CLE E-prints, p.1–13.

Fitting, M. & Mendelsohn, R. L. 1998.First-order modal logic. Dordrecht: Kluwer.

Hamilton, A. G. 1978.Logic for mathematicians. Cambridge: Cambridge University Press.

Mendelson, E. 1987.Introduction to mathematical logic. 3. ed. Monterey, CA: Wadsworth &

Brooks/Cole Advanced Books & Software.

Miraglia, F. 1987. Cálculo proposicional: uma interação da álgebra e da lógica. Campinas:

UNICAMP/CLE. (Coleção CLE, v.1)

Nascimento, M. C. & Feitosa, H. A. 2005. As álgebras dos operadores de conseqência. São Paulo:Revista de Matemática e Estatística23(1): 19–30.

Rasiowa, H. 1974.An algebraic approach to non-classical logics. Amsterdam: North-Holland.

Rasiowa, H. & Sikorski, R. 1968.The mathematics of metamathematics. 2. ed. Waszawa: PWN – Polish Scientific Publishers.

Vickers, S. 1990.Topology via logic. Cambridge: Cambridge University Press.

Wojcicki, R. 1988. Theory of logical calculi: basic theory of consequence operations. Dor- drecht: Kluwer. (Synthese Library, v.199)

CEZARA. MORTARI

Departamento de Filosofia Universidade Federal de Santa Catarina Campus Universitário – Trindade 88040-010 Florianópolis, SC

BRASIL

cmortari@cfh.ufsc.br H^{ÉRCULES DE}A^RAÚJOF^EITOSA Departamento de Matemática Universidade Estadual Paulista (UNESP) Campus de Bauru 17033-360 Bauru, SP BRASIL

haf@fc.unesp.br Resumo. A lógicaTKfoi introduzida como uma lógica proposicional estendendo o cálculo proposicional clássico com um novo operador unário que interpreta algumas concepções do operador de consequência de Tarski. TK-álgebras foram introduzidas como modelos para TK. Assim, usando ferramentas algébricas, foi demonstrada a adequação (correção e com- pletude) de TK relativamente às TK-álgebras. Este trabalho apresenta uma semântica de vizinhanças paraTK, lógica que resulta ser dedutivamente equivalente à lógica modal não normalEMT4.

Palavras-chave:Operador de consequência; álgebra TK; lógica TK; semântica de vizinhan- ças.

Notes

1 Of course, one can also define deductionlocally: we say thatΓ` ϕ if there is a finite subsetψ1, . . . ,ψnofΓsuch that`(ψ1∧. . .∧ψn)→ϕ. With this definition, obviously, the Deduction Theorem holds.