1. Introduction, Motivation

Bulletin of the Section of Logic Volume 28/3 (1999), pp. 135–143

Vladimir V. Rybakov

AN EXPLICIT BASIS FOR RULES ADMISSIBLE IN MODAL SYSTEM S4

Abstract

We ﬁnd an explicit basis for all admissible rules of the modal logic S4. Our basis consists of an inﬁnite sequence of rules which have compact and easy readable form and depend on increasing set of variables. This gives a basis for all quasi- identities valid in the free modal algebra FS4(ω) of countable rank. KEY WORDS: inference rule, modal logic, free algebras, Kripke model, basis, admissible rule

1. Introduction, Motivation

The notion of admissible inference rules goes back to Lorenzen ([7], 1955). For any given logic the admissible rules are exactly those rules under which the logic is closed. And it is a well known fact that admissible rules are not always derivable (F. Harrop, [5], Mints [8]). How to describe all rules admissible in a given logic? One way is to give algorithmic criteria which allow to determine admissibility of rules. Such kind criteria were given for a number of non-standard modal and superintuitionistic logics and some classes of such logics (cf. [3], [4], [9], [12], [13], [14], [15], [16]). Another way is to describe bases for all admissible rules. Some research towards this aim has been already done. For instance, in Citkin [1] a basis for admissible quasi-characterizing rules of intuitionistic propositional logic -IPC- was suggested. But, in general, it was shown that basic 136 Vladimir V. Rybakov important modal and superintuitionistic logics do not have bases in finite number of variables (cf. Rybakov, – IPC – [10], modal logics – [11], [13]). Therefore the only bases for these logics which we could offer are those which appear from known algorithmic criteria. But these criteria are com- putationally difficult and there is no way to apply them to describe bases in an easy to handle form. Therefore we are interested to make an attempt to find bases which are precisely described and could be easily displayed, i.e. to give explicit bases. This question was investigated in R. Iemhoff [6] for the very case IPC and an explicit basis is found. Our paper is devoted to solve this task for modal logics. We focus our attention here on the basic transitive and reflexive modal logic S4 and we present a basis for all rules admissible in S4. The basis has a compact form, is easily visible and easy to handle. We use the technique of criteria for admissibility which constitute summarized in [16], the base for our approach.

2. Denotation, Preliminary Information

For common knowledge concerning modal logics and their semantics in- cluding Kripke models and modal algebras their connection and their in- teraction through Stone theory we refer to the literature (among modern one cf. [16] or [2]). For any frame F := hF,Ri, and any element a ∈ F , C(a) will denote the cluster containing a. For any subset X of F , X R := {a | ∃b ∈ X (bRa)}, i.e. X R is the upwards cone generated by X , and X R+ := {a | ∃b ∈ X (bRa)&∀c ∈ X (¬(aRc))}. For any antichain YS of clusters from F, a cluster C is a co- cover for Y if and only if CR+ = (CR ∪ C ); an element is a co-cover C1∈Y 1 1 if the cluster containing this element is a co-cover. For a pair of frames F1, F2, F1 v F2 is the abbreviation for F1 is an open subframe of F2. We are saying a frame F is a λ-frame for a logic λ if all theorems of λ are valid at F, and λ(F) is the logic generated by the frame F. A frame F is rooted, or sharp, if ∃a ∈ f such that ∀b ∈ F, aRb, then we say C(a) is the root of F. Sm(F) denotes the set of all elements of F of depth not exceeding m, and Slm(F) is the set of all elements of depth m from F - m-th slice. All preliminary information concerning inference rules and their admissibility can be found in [16], in particular, the construction of n-characterizing Kripke models Chλ(n) and criteria for recognizing admis- An explicit basis for rules admissible in modal system S4 137 sibility are presented there. They are the base for our research, therefore we have to recall the effective construction of Chλ(n). Given a modal logic λ extending S4 with finite model property, and a set Pn := {p1, ..., pn} of propositional letters. The first slice S1(Chλ(n)) consists of the collection of all clusters with all possible valuations V of letters from Pn which do not have doubling - elements with the same valuation, and no clusters which are identical as Kripke models. Assuming Sm(Chλ(m)) to be constructed, we put in Slm+1(Chλ(m)) the clusters as follows. We choose any antichain Y of clusters from Sm(Chλ(m)) having at least one cluster of depth m and put in Slm+1(Chλ(n)) any cluster C from S1(Chλ(n)) assuming C to be immediate predecessor for clusters from Y but only provided that (i) the frame CR is a λ-frame and (ii) if Y := {C1} then C is not a Kripke submodel of C1. Iterating this procedure we get as the result the model Chλ(n). We are saying a model M is n-characterizing for a logic λ if, for any formula

α, which is built up out of letters from Pn, α ∈ λ iff M α. We need for our research the following two well known facts. Theorem 2.1. (cf. [16] for instance) For any modal logic λ having fmp and extending S4, the model Chλ(n) is n-characterizing for λ. A subset X of given model M is definable if there is a formula α such that ∀x ∈ M[(x V α)⇔x ∈ X ]. And the valuation V is definable in M if, for any letter p from the domain of V , V (p) is definable. For a given frame F, a given valuation V and a given inference rule r := α1, ..., αn/β, we say r is valid in F (or at F) under V , and write F V r, if as soon as

∀x ∈ F, ∀i(x V αi) then we have ∀x ∈ F(x V β). A rule r is valid in a frame F if r is valid in F under any valuation, we write then F r. Theorem 2.2. (cf. [16] for instance) For any modal logic λ having fmp and extending S4 and for any inference rule r, r is admissible in λ iff r is valid in the frame of Chλ(n) for any n under any definable valuation. Definition 2.3. A logic λ extending S4 has weak co-cover property (WCCP for short) if, for any finite rooted λ-frame and any non-trivial anti-chain X of clusters from F, the frame F1 which is the result of adjoin- S R ing a single-element reflexive co-cover to the frame c∈X c is a λ-frame also. 138 Vladimir V. Rybakov

We need for our research the reducedW forms for modal inference rules. We say a rule r has reduced form if r := 1≤i≤t ϕi/2x0, where k(i, j, 1), k(i, j, 2) ∈ {0, 1}, s0 := s, s1 := ¬s, ^ ^ k(i,j,1) k(i,j,2) ϕi := xj ∧ (3xj ). 0≤j≤k 0≤j≤k

Theorem 2.4. (cf. [16]) For any modal rule r there is a rule rf(r) which has a reduced form and which is equivalent to r w.r.t. validness in modal algebras from V ar(S4) and reﬂexive and transitive Kripke frames; and r and rf(r) are mutually derivable in any modal logic over S4.

3. Main Results

We simply write out the rules which will form our basis. Let a couple of numbers n, m ∈ N be given n > 1, m ≥ 1 and let ^ An := 3pi; 1≤i≤n

^ ^ An,m := 2[ pi → ¬3qj]; 1≤i≤n 1≤j≤m

_ ^ ^ Bm := ( qi ∧ ¬3qi); D⊆{1,...,m} i∈D i6∈D,i∈{1,...,m}

And we introduce the following sequence of inference rules for n, m ∈ N, n > 1, m ≥ 1:

2(An,m ∧ ¬(An ∧ Bm)) ∨ 2z rn,m := 2¬An ∨ 2z

Lemma 3.1. Any inference rule rn,m is admissible in any modal logic λ having fmp, extending modal system S4 and having (i) the WCCP and (ii) the disjunction property. An explicit basis for rules admissible in modal system S4 139

Proof. Assume not. Since λ has the disjunction property, the rule 2(A ∧ ¬(A ∧ B )) R := n,m n m 2¬An is also not admissible for λ. Therefore by Theorem 3 there is a valuation V of variables from R in a certain constructive k-characterizing model Chλ(k) of λ disproving R. Therefore

Chλ(k) V 2(An,m ∧ ¬(An ∧ Bm)) and Chλ(k) V 2¬An. (1)

Consider b ∈ Chλ(k) such that a V 2¬An which exists by (1). Then there are elements b1, ..., bn ∈ Chλ(k) such that aRbi, 1 ≤ i ≤ n and b V pi. Since λ has WCCP, there is a reﬂexive element b ∈ Chλ(n) which is a co-cover for the set {bi | 1 ≤ i ≤ n}, that is [ R R {b} = {b} ∪ {bi} . 1≤i≤n

Using (1) it follows that b V 2An,m and b V An. Consider the following set D := {qi | b V qi}. Since b is the co-cover for b1, ..., bn it is easy to see that the disjunctive member of Bm corresponding to this D is valid at b under the valuation V . Thus b V An ∧ Bm which contradicts (1). 2 Given a modal algebra A := F +(X ) ∈ V ar(S4) generated by a set of subsets X in the wrapping modal algebra F + over a given reﬂexive and transitive frame F. Let rf(r) be a rule in reduced form. Lemma 3.2. If the rule rf(r) is admissible for S4, has k variables and is false in A then for some n, m, n > 1, m ≥ 1, where n + m ≤ k + 22k the rule rn,m is false in A also. W Proof. Our rule rf(r) has the form 1≤i≤t ϕi/2x0, where ^ ^ k(i,j,1) k(i,j,2) ϕi := xj ∧ (3xj ), 0≤j≤k 0≤j≤k and k(i, j, 1), k(i, j, 2) are each either 0 or 1, and w0 := w, w1 := ¬w. As soon as rf(r) is false in A, for a certain valuation V (xi) := Yi ∈ A, 1 ≤ i ≤ k, 140 Vladimir V. Rybakov

W F V 1≤i≤t ϕi and F V 2x0. (2)

Therefore, for certain elements a ∈ F, a 2x0. We choose among all these elements a some element b such that the set

ϕ(b) := {ϕj | ∃c ∈ V (ϕj)(bRc)} is maximal. Consider the wrapping modal algebra (bR)+ of the frame R R + b and its subalgebra B := (b ) (V (x0), ..., V (xk)) generated by the set of elements V (x0), ..., V (xk). Using (1) we get B 6|= rf(r). We will apply now Lemma 3.9.5 from [16] (see corresponding denotation there, in particular what θ1(ϕj), θ2(ϕj) mean). Consider the set Z consisting of all disjunctive members of the premise of the rule rf(r) which have in B under V non-zero value. It can be easily veriﬁed by direct computation that the properties (i) and (ii) of Lemma 3.9.5 from [16] hold for this Z. And since rf(r) is admissible for S4, by this lemma we conclude that there is a subset D of the set Z such that the property (iii) of Lemma 4.9.5 does not hold for Z, that is S ∀ϕ ∈ Z, θ (ϕ ) 6= θ (ϕ ) ∪ (θ (ϕ ) ∪ θ (ϕ )). (3) j 2 j 1 j ϕs∈D 2 s 1 s

Then, in particular, ||D|| > 1. Let PV be the set of all variables xi which Soccur in rf(r) and PT be the set of all variables which occur in the set (θ (ϕ ) ∪ θ (ϕ )). Let n := ||D||, m := ||P − P ||, note that ϕs∈D 2 s 1 s V T n > 1 and m > 0. Otherwise, if m = 0 we would have b V ϕj for some ϕj ∈ Z and this ϕj would disprove (3). Fix one-to-one correspondences: f - between p1, ..., pn and D and g - between q1, ..., qm and PV − PT . Then n + m ≤ 22k + k. Now we extend the valuation V in B from variables of rf(r) to variables of rn,m as follows:

V (pi) := V (f(pi)) and V (qj) := V (g(qj)). V Using the deﬁnition of B it is easy to see that b V 1≤i≤n 3pi because b occurs in the R-smallest cluster of the underlying frame of B and because our choice of D and Z. Thus

b V An. (4)

R To verify the validness of An,m in B under V assume c ∈ b and c V pi.

Then we have c V f(pi) for f(pi) ∈ D, and consequently, for any g(qi) ∈ An explicit basis for rules admissible in modal system S4 141

PV − PT , c V ¬3g(qi) because of choice of PT and since c V f(pi). Therefore we get

b V 2An,m. (5)

R Assume now that c ∈ b and c V An. Using (2) it follows c V ϕj for some j. And our assumption above yields c V 3ϕi for all ϕi ∈ D. There- fore PT ⊆ θ2(ϕj). Assume also that c V Bm. Suppose c V 3xi for xi ∈ [PV − PT ]. Then in accordance with c V Bm we get c V xi and xi ∈ θ1(ϕj). Thus [ θ2(ϕj) = θ1(ϕj) ∪ (θ2(ϕs) ∪ θ1(ϕs))

ϕs∈D and ϕj ∈ Z which contradicts (3). Thus in sum we get

B |=V 2(An,m ∧ ¬(An ∧ Bm)). (6) Now our aim is to choose a valuation V for the variable z of the rule rn,m in order to disprove rn,m in A.

Lemma 3.3. A 6|= rn,m.

Proof. By (2) there is a single formula ϕb,V from Z such that b V ϕb,V . We choose the valuation V for z in A as follows. ^ _ V (z) := V (¬[3ϕb,V ∧ 3ϕj ∧ 2( ϕj)]),

ϕj ∈ϕ(b) ϕj ∈Z where ϕ(b) was deﬁned above. Looking at our choice of ϕ(b) and Z we conclude that z is false under V at b. Thus by (4) the conclusion of the rule rn,m is false in b under V . We have to prove now that the premise of rn,m is valid in A under V . Indeed, assume that c ∈ F and c V 2z. Then by the choice of V for z we have V W ∃d ∈ V (ϕ )(cRd), and c 3ϕ ∧ 2( ϕ ). (7) b,V V ϕj ∈ϕ(b) j ϕj ∈Z j

Because b V 2x0 (see (2)), for some g, bRg and g V x0, e.i. g V ¬x0. By (2) g V ϕj for some j and ϕj ∈ Z. Therefore by (7), for some w, 142 Vladimir V. Rybakov cRw and w V ϕj, consequently w V ¬x0 and c V 2x0. Using that b was chosen as an element disproving 2x0 whose ϕ(b) is maximal and that (7) holds we conclude that ϕ(b) = ϕ(c). Therefore we derive that elements which are accessible from b and c, respectively, have in sum exactly the same tuple of valid under V formulas ϕj. Using this fact and c V 2x0 we can show that the premise of rn,m is valid in c under V using the same argument as showing (6). Thus A 6|= rn,m. 2 Immediately from Lemmas 3.1, 3.3 and Theorem 2.4 we derive

Theorem 3.4. Rules rn,m form a basis for all admissible rules of S4.

Evidently that transforming all inference rules rn,m into corresponding quasi-identities we get a basis for all quasi-identities which are valid in the free modal algebra FS4(ω) of countable rank.

References

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Department of Mathematics, Science faculty, Hacettepe University, Beytepe, Ankra, Turkey and Mathematical Department, Krasnoyarsk University pr. Svobodnyi 79, 660 062, Krasnoyarsk, Russia. e-mail: [email protected] [email protected]