Glivenko Type Theorems for Intuitionistic Modal Logics

Transcription

1 Guram Bezhanishvili Glivenko Type Theorems for Intuitionistic Modal Logics Abstract. In this article we deal with Glivenko type theorems for intuitionistic modal logics over Prior s MIPC. We examine the problems which appear in proving Glivenko type theorems when passing from the intuitionistic propositional logic Int to MIPC. As a result we obtain two different versions of Glivenko s theorem for logics over MIPC. Since MIPC can be thought of as a one-variable fragment of the intuitionistic predicate logic Q-Int, one of the versions of Glivenko s theorem for logics over MIPC is closely related to that for intermediate predicate logics obtained by Umezawa [27] and Gabbay [15]. Another one is rather surprising. Key words: intuitionistic modal logics, Heyting algebras, monadic Heyting algebras, regular elements, dense elements, super-dense elements, completely modalized formulas, strongly modalized formulas, essentially negative formulas, essentially negative strongly modalized formulas. 1. Introduction In 1929 V. Glivenko [16] established a celebrated result that a propositional formula A is provable in the classical propositional logic Cl iff its double negation A is provable in the intuitionistic propositional logic Int. In 1952 S. Kleene [18] partly extended Glivenko s theorem to predicate logics by establishing that an -free predicate formula A is provable in the classical predicate logic Q-Cl iff its double negation A is provable in the intuitionistic predicate logic Q-Int. Unfortunatelly, this theorem can not be extended to all predicate formulas, see Kleene [18], Umezawa [27] 1 and Gabbay [15]. Indeed, it is obvious that the formula x(p (x) P(x)) is provable in Q-Cl. On the other hand, the formula x(p (x) P (x)) is not a theorem of Q-Int. Moreover, the formula x(p (x) P (x)) plays an essential role in proving a predicate version of Glivenko s theorem. According to Umezawa [27] and Gabbay [15], for a predicate logic L intermediate between Q-Int and Q-Cl we have (Q Cl A L A) iffl x(p (x) P (x)). In 1966 R. Bull [9] showed that the intuitionistic modal logic MIPC introduced by A. Prior [22] is closely related to Q-Int. Actually he proved that the set of theorems of MIPC coincides with the set of monadic theorems of Q-Int containing the one fixed individual variable. We have the same rela- 1 I would like to thank Nobu-Yuki Suzuki for drawing my attention to Umezawa s paper. Presented by Daniele Mundici; Received April 2, 1999; Final November 29, 1999 Studia Logica 67: , c 2001 Kluwer Academic Publishers. Printed in the Netherlands.

2 90 G. Bezhanishvili tion between the classical modal system S5 and Q-Cl. This fact was first established by M. Wajsberg [29] back in Therefore, it is natural to ask how to extend Glivenko s theorem to the logics which are intermediate between MIPC and S5, or more generally, how to extend Glivenko s theorem to the logics over MIPC? Here it should be noted that there exist consistent logics over MIPC which are not intermediate between MIPC and S5. For instance, all consistent proper extensions of S5 have this property. The aim of this article is to show how to prove Glivenko type theorems for logics over MIPC. In 2werecallbasicfactsfromtheduality theory for Heyting algebras and monadic Heyting algebras. In the same way as Heyting algebras can be thought of as algebraic models of intermediate propositional logics, monadic Heyting algebras can be thought of as algebraic models of logics over MIPC. Monadic Heyting algebras were first introduced by A. Monteiro and O. Varsavsky [20] as natural extensions of monadic Boolean algebras introduced by P. Halmos [17]. For the basic properties of monadic Heyting algebras and the duality theory for them the reader is refered to [1], [2], [3]. 3 has also an auxiliary purpose and in it we recall how one can use Heyting algebras and their dual spaces to obtain Glivenko s theorem. 4 is the central part of the paper. In it we prove two possible extensions of Glivenko s theorem to logics which are intermediate between MIPC and S5, and later show how to extend Glivenko type theorems to cover all logics over MIPC. Finally, in 5we show how to use Glivenko type theorems to obtain several general results about decidability and the finite model property for logics over MIPC. Here we would like to stress that all our arguments are purely semantical and mainly use algebraic models of logics over MIPC and their dual spaces. Some of our results can be obtained using purely proof-theoretical technique. However, it is not done here. Several results of this paper were reported at the annual meeting of mathematical logic in Japan, November Acknowledgements. Thanks are due to Leo Esakia, Hiroakira Ono, Hajime Ishihara, Nobu-Yuki Suzuki and Revaz Grigolia. I should also like to thank anonymous referees for valuable suggestions. 2. Preliminaries This section has an auxiliary purpose. Here we recall several basic facts from the duality theory for Heyting algebras and monadic Heyting algebras. All the needed proofs for Heyting algebras can be found in Esakia [13] and for monadic Heyting algebras in [1], [2].

3 Glivenko Type Theorems for Intuitionistic Modal Logics Heyting algebras Let us recall that a Heyting algebra (H,,,, 0) is a distributive lattice (H,,, 0) with an additional binary operation which satisfies the following condition: x a b iff a x b for any a, b H. We will denote the variety of all Heyting algebras by HA. For any intermediate propositional logic L Int, denote by K L the variety of Heyting algebras which corresponds to L. Recall that K L = {H HA : H L}, andthatl is complete with respect to K L. Let (X, R) be a partially ordered set. For any x X and A X, let R(x) = {y X : xry}, R(A) = x A R(x), R 1 (x) = {y X : yrx}, R 1 (A) = x A R 1 (x). A is said to be a R-cone of X if R(A) = A ( R 1 A = A). We call A a down R-cone if R 1 (A) =A ( R A = A). Let us also recall that a topological space (X, Ω) is called a Stone space if it is 0-dimensional, compact and Hausdorff. A X is said to be a clopen if A is simultaneously closed and open. For these and other elementary notions from general topology the reader is refered to Engelking [11]. Denote the set of all clopen subsets of X by CP(X). The set of all clopen R-cones of X will be denoted by CON R (X). A relation R is said to be point-closed if the set R(x) is closed for every x X. The dual spaces of Heyting algebras are the triples (X, Ω,R), where (X, Ω) is a Stone space and R is a point-closed partial order on X which satisfies the following condition: A CP(X) R 1 (A) CP(X). ( ) The dual spaces of Heyting algebras were first constructed by Esakia in [12]. Therefore, we call them Esakia spaces. Any point-closed quasi-order Q which satisfies the condition ( ) will be called an Esakia relation. Recall that every Esakia relation satisfies the following separation property: For a closed R-cone A and a closed down R-cone B, ifa B = ( ) then there exists U CON R (X) such that A U and B U =. Here we recall that the dual space (X, Ω,R) of a Heyting algebra H is constructed in the following way: X is the set of all prime filters of H, R is a set-theoretical inclusion, φ(a) denotes the set {x X : a x} for every a H, φ(h) denotes the set {φ(a)} a H and Ω is defined on X by announcing the Boolean closure B(φ(H)) of the set φ(h) as a base for topology. Also recall that (φ(h),,,, ), where A B = R 1 (A B) = {x X : y(xry & y A y B)} for every A, B φ(h), constitutes a Heyting algebra which is isomorphic to the initial H, andthatφ(h) = CON R (X).

4 92 G. Bezhanishvili In this way, many important algebraic results can be simply proved using the duality theory for Heyting algebras. For instance, filters of a Heyting algebra correspond to closed R-cones of its dual space. This correspondence can be obtained as follows. With any filter F H we associate the set C F = a F φ(a), which is obviously a closed R-cone of X. Now it is routine to check that this correspondence is one-to-one. For further results in this direction we refer to Esakia [13] Monadic Heyting algebras Let us recall that a monadic Heyting algebra is a triple (H,, ), where H is a Heyting algebra and, are unary operations on H (often called quantifiers ) which satisfy the following conditions: 1. a a a a; 2. (a b) = a b a b = (a b); 3. 1 =1 0= 0; 4. a = a a = a; 5. ( a b) = a b, for all a, b H. Denote the variety of all monadic Heyting algebras by MHA. Also recall that MIPC is the minimal set of formulas containing Int, the modal axioms (A1) p p p p (A2) ( p q) (p q) (p q) ( p q) (A3) p p p p (A4) (p q) ( p q) and closed under substitution, modus ponens and necessitation (A/ A). A set of formulas containing MIPC and closed under the above mentioned inference rules is called an intuitionistic modal logic over MIPC. The minimal intuitionistic modal logic containing a logic L MIPC and a set of formulas Γ is denoted by L Γ. Interpreting as and as we obtain an interpretation of a logic L over MIPC into a given monadic Heyting algebra. In this way, with every logic L MIPC is associated the variety K L = {(H,, ) MHA : (H,, ) L} of monadic Heyting algebras, and every logic L MIPC is complete with respect to K L (see [1] for details). For a given (H,, ), the set H 0 = { a : a H} forms a relatively complete Heyting subalgebra of H, which means that for every a H the sets {b H 0 : b a} and {b H 0 : a b} have the largest and the least

5 Glivenko Type Theorems for Intuitionistic Modal Logics 93 elements respectively. Moreover, every monadic Heyting algebra (H,, ) can be represented as a couple (H, H 0 ) of Heyting algebras, where H 0 is a relatively complete subalgebra of H (see [20] and [1] for details). AfilterF of (H,, ) is called monadic if a F a F. Recall that monadic filters correspond to congruences of (H,, ). Moreover, a filter F is monadic iff it is reconstructed from F H 0 (see [1]). The dual space of a monadic Heyting algebra (H,, ) is a quadruple (X, Ω,R,Q), where both R and Q are Esakia relations, R is a partial order, Q is a quasi order, R Q, Q = RE Q, i.e. xqy z X(xRz & xe Q y), where xe Q y Def (xqy & yqx), and in addition the following condition is satisfied: A CON R (X) Q(A) CON R (X). ( ) Roughly speaking, the dual spaces of monadic Heyting algebras are obtained by enriching the dual spaces of Heyting algebras with an additional quasi order Q which is an Esakia relation, extends R and can be reconstructed from R and E Q. The dual space of a given (H,, ) is constructed as follows: (X, Ω,R)is the dual space of H, andxqy iff a x a y for every a H. Notethat the last condition is equivalent to the following one: xqy iff a x a y for every a H. In this way, (φ(h),, ), where (B) = Q 1 (B) ={x X : y X(xQy y B)} and (B) =Q(B) ={x X : y B(yQx)} for every B φ(h), is isomorphic to the initial (H,, ), φ(h) =CON R (X) and φ(h 0 )=CON Q (X) (see [2] for details). Using this correspondence between monadic Heyting algebras and their dual spaces we can develop the duality theory for monadic Heyting algebras in the same way as it was done for Heyting algebras. For instance, monadic filters of a monadic Heyting algebra correspond to closed Q-cones of its dual space. For further results on the theory of duality for monadic Heyting algebras the reader is refered to [2]. 3. Glivenko s theorem In this section we recall Glivenko s celebrated result and show how Heyting algebras and their dual spaces can be used for obtaining Glivenko s theorem. We mainly skip all the proofs which are well known. Some of the proofs can be found in Rasiowa and Sikorski [23], the others belong to folklore. Let us, once again, recall one of the formulations of Glivenko s theorem: For any intermediate propositional logic L and any formula A, Cl A iff L A.

6 94 G. Bezhanishvili a Figure 1. b Note that one implication in Glivenko s theorem is trivial. Indeed, if L A, thensincel Cl, Cl A. NowsinceCl A A, weobtain Cl A. The problem is to show that if L A, thenalsocl A. Butwe know that if L A, then there exists a Heyting algebra H K L such that H A. If we could associate with H a Boolean algebra B H in such a way that B H A, then the proof of Glivenko s theorem would be completed. Let us call an element a H regular if a = a. Denote the set of all regular elements of H by R(H). It is a routine to check that R(H) = { a : a H} and that R(H) isa(,, 0)-subalgebra of H. However, in general R(H) is not a Heyting subalgebra of H. The easiest counterexample is shown in Fig. 1. (It is obvious that R(H) ={,a,b, } is not closed with respect to.) Although R(H) is not a Heyting subalgebra of H, (R(H),,,, 0) forms a Boolean algebra, where a b = (a b) for any a, b R(H). Indeed, since : H R(H) is a homomorphism from H onto (R(H),,,, 0), (R(H),,,, 0) is a Heyting algebra. Moreover, for any a R(H), a a = (a a) =, which means that (R(H),,,, 0) constitutes a Boolean algebra. Now it is easy to complete the proof of Glivenko s theorem. If Int A(p 1,...,p n ), then there exist a Heyting algebra H K L,a 1,...,a n H and a polynomial α(x 1,...,x n ), corresponding to A(p 1,...,p n ), such that α(a 1,...,a n ) in H. But then α(a 1,...,a n ) in R(H), and since is a homomorphism, α( a 1,..., a n ) in R(H). Now since a 1,..., a n R(H), we get that A(p 1,...,p n ) is not valid in R(H). Therefore, Cl A and the proof is completed. Note that since R(H) is a homomorphic image of H, there exists a filter F H such that R(H) H/ F. To construct F we need an additional notion: call an element a H dense if a =. It is a routine to check that

7 Glivenko Type Theorems for Intuitionistic Modal Logics 95 the set of all dense elements of H constitutes a (proper) filter of H. Letus denote it by F D.Sincea a is a dense element for every a H, itisobvious that H/ FD forms a Boolean algebra. Moreover, the natural morphism h : R(H) H/ FD, which is defined by putting h(a) =a/ FD for every a R(H), sets the needed isomorphism (see Rasiowa and Sikorski [23] for a proof). Since R(H) is a homomorphic image of H, the dual space of R(H) isa closed cone of the dual space of H. Let us denote the dual space of H by (X, Ω,R) and show exactly which closed cone of X corresponds to the dual space of R(H). Denote the set of all maximal points of X by max X. (Here we recall that a point x X is said to be maximal if xry implies x = y for every y X.) It is well known that the set max X is a non-empty closed subset of X, andthatforeveryy X there exists x max X such that yrx (see Esakia [13] for a proof). Now it is easy to check that an element a H is dense iff max X φ(a). Moreover, max X = a F D φ(a). Therefore, max X corresponds to the filter of all dense elements of H, and hence, the dual space of R(H) is isomorphic to max X. Thus, using the duality theory for Heyting algebras we can give an alternative proof of Glivenko s theorem: from every Esakia space X we can extract the set of all maximal elements of X. SincemaxX is a closed cone of X, maxx becomes an Esakia space. Moreover, since the restriction of R to max X is the identity relation, max X simply becomes a Stone space. Besides, for a given formula A, A is valid in X iff A is valid in max X, which completes the proof. Let us conclude this section by pointing out that actually regular elements of H have also a nice dual description. Indeed, for every R-cone A X let max A denote the set A max X. It is a routine to check that for every a H, x φ(a) iffmaxr(x) φ(a). Therefore, we obtain that a H is regular iff x X(max R(x) φ(a) x φ(a)). In other words, a is regular iff φ(a) is the greatest among the clopen cones which have the same maximum as φ(a). This description of regular elements makes it clear that R(H) H/ FD. Indeed, since for every regular element a H, φ(a) isthe greatest clopen R-cone among the clopen R-cones which have the same maximum as φ(a), there exists a natural isomorphism between {φ(a) :a R(H)} and clopens of max X which is set by putting h(φ(a)) = max φ(a). 4. Glivenko type theorems for MIPC This section is the heart of the paper and here we show several ways extending of Glivenko s theorem to the intuitionistic modal logics over MIPC.

8 96 G. Bezhanishvili x y Figure 2. Since the logics over MIPC are closely related to superintuitionistic predicate logics, several results of this section are related to those by Umezawa [27] and Gabbay [15]. An intuitionistic modal logic L over MIPC is called intermediate if MIPC L S5. In contrast to logics over Int, there exist logics over MIPC which are not intermediate. Witnesses are all proper extensions of S5 and many others. Therefore, we have different possibilities to formulate a modal version of Glivenko s theorem. One of the most natural formulations would be the following: For any intermediate logic L and any formula A, S5 A iff L A. However, as we will see below, this version of Glivenko s theorem never holds. Indeed, as in the case of logics over Int, it is trivial to show that if L A then S5 A. If we try to prove the converse, we need to show that L A implies S5 A. FromL A it follows that there exists a monadic Heyting algebra (H,, ) K L such that (H,, ) A. What we need now to do is to associate with (H,, ) a monadic Boolean algebra (B H, H, H ) in such a way that (B H, H, H ) A 2.Let(X, Ω,R,Q)denote the dual space of (H,, ). As in the case of Heyting algebras, the most natural way to get (B H, H, H )istoextractmaxx and try to show that the set of all clopens of max X is a homomorphic image of (H,, ), which by itself forms a monadic Boolean algebra. However, in general max X is not a Q-cone of X. The easiest witness is shown in Fig. 2. (It is obvious that max X = {x} is not a Q-cone of X.) Therefore, the set of all clopens of max X is not a homomorphic image of (H,, ). There are two possibilities to overcome this difficulty. The first one is to consider the set E Q (max X), which is the least Q-cone containing max X, and deal with the algebra of all clopen R-cones of E Q (max X), which is indeed a homomorphic image of (H,, ). The second one is to restrict ourselves only to those monadic Heyting algebras whose dual spaces have the property that max X is a Q-cone. In the first case we obtain a Glivenko 2 Recall that monadic Boolean algebras serve as algebraic models for S5.

9 Glivenko Type Theorems for Intuitionistic Modal Logics 97 a Figure 3. type correspondence between MIPC and the logic which is weaker than S5 but preserves several properties of S5, and in the second case we arrive at a Glivenko type correspondence between S5 and the logic which is a proper extension of MIPC. Let us investigate both cases in detail. First of all note that from algebraic point of view the difficulty we are facing is the following. In a monadic Heyting algebra (H,, ) the filter F D of all dense elements of (H,, ) is not a monadic filter. Therefore, either we should deal with a different kind of filter, or restrict ourselves to those monadic Heyting algebras in which F D is a monadic filter. As a possible substitution of F D we choose the filter of all super-dense elements. Let us call an element a H super-dense if a =. Denote the set of all super-dense elements of H by F SD. It is easy to check that F SD always forms a monadic filter of (H,, ). Since a a, every super-dense element is dense itself and hence F SD F D. Moreover, for every a H 0, a F SD iff a F D. However, in general F SD is a proper subset of F D. For instance, in the algebra shown in Fig. 3 it is obvious that a is dense but is not super-dense in H. It is worth noting that actually the frame (X, R, Q) shown in Fig. 2 is the dual space of (H,, ) shown in Fig. 3. Note that the inequality a a does not hold in the (H,, ) shown in Fig. 3. And actually this is the key why there is a difference between dense and super-dense elements. Indeed, if a a holds in a monadic Heyting algebra (H,, ), then a = a = a = and F SD = F D. Actually we can also prove the converse, that is if F SD = F D then a a holds in (H,, ). For this we need the following dual characterization of a a: Lemma 1. (see [3], Lemma 37) The formula a a holds in (H,, ) iff E Q (max X) =maxx. Having this lemma at hand we arrive at the following Theorem 2. F SD = F D iff a a holds in (H,, ). Proof. We already proved that if a a holds in (H,, ) then F SD = F D. Conversely, suppose a a does not hold in (H,, ).

10 98 G. Bezhanishvili Then from Lemma 1 it follows that there exist x max X and y X max X such that yrx and ye Q x.sincemaxxis a closed R-cone and y/ max X, there exists B CON R (X) such that max X B and y / B. From max X B it follows that B = X and B is dense in CON R (X). From y / B and ye Q x it follows that x / (B). Finally, from x max X and x/ (B) itfollowsthatx/ (B). Therefore, (B) X and B is not super-dense in CON R (X). Since a a is an important formula for us, we also underline the following Lemma 3. a a is equivalent to (a a) =. Proof can be carried over similarly to Umezawa [28], page 146. Corollary 4. MIPC ( p p) =MIPC (p p). Since p p is a modal analogue of Kuroda s formula K = x P (x) xp (x) we denote p p by K. Now let us return to super-dense elements of (H,, ) andshowthat actually the set E Q (max X) corresponds to the filter F SD. Indeed, we have the following Theorem 5. (1) a H is super-dense iff E Q (max X) φ(a). (2) E Q (max X) = a F SD φ(a). Proof. (1) a H is super-dense iff φ(a) =X iff max X φ(a) iff E Q (max X) φ(a). (2) From (1) it follows that E Q (max X) a F SD φ(a). Conversely, if x / E Q (max X), then since E Q (max X) is a closed Q-cone and Q is an Esakia relation, there exists B CON Q (X) such that E Q (max X) B and x/ B. FromE Q (max X) B it follows that B is a super-dense element in CON R (X). Therefore, from x/ B it follows that x/ a F SD φ(a). Corollary 6. The dual space of (H/ FSD, FSD, FSD ) is isomorphic to the space (E Q (max X), Ω EQ (max X),R EQ (max X),Q EQ (max X)). Note that in E Q (max X), Q coincides with E Q and hence Q becomes an equivalence relation in E Q (max X). Therefore, the set of all clopen Q- cones of E Q (max X) constitutes a Boolean algebra. Moreover, this Boolean algebra is isomorphic to H 0 / FSD H 0.NotethatF SD H 0 coincides with the

11 Glivenko Type Theorems for Intuitionistic Modal Logics 99 filter of all dense elements of H 0. Therefore, H 0 / FSD H 0 is isomorphic to R(H 0 ). Hence, we obtain that (H/ FSD, FSD, FSD ) (H/ FSD, R(H 0 )) and on the base of Theorem 23 of [1] we conclude that (H/ FSD, FSD, FSD ) MHA +( a = a). Let us also underline that although (H/ FSD ) 0 forms a Boolean algebra, H/ FSD in general is not a Boolean algebra. For instance, in the algebra shown in Fig. 3 the identity a = a holds, but still it is not a Boolean algebra. The variety MHA+( a = a) corresponds to the intuitionistic modal logic MIPC ( p p). Although MIPC ( p p) is properly contained in S5, there are several properties that MIPC ( p p) shares with S5. Definition 7. (1) A formula A is said to be completely modalized if every occurrence of a propositional variable in A appears in the scope of or. (2) A formula A is said to be strongly modalized if every occurrence of a propositional variable in A appears immidiately after or. Hence it follows that a formula A is strongly modalized iff A has the form B( p 1,..., p n ), where means either or, andb is -free (that is B contains only intuitionistic connectives). Examples of completely modalized formulas are p p and p p, while examples of strongly modalized formulas are p p and p p. It is obvious that every strongly modalized formula is also completely modalized, and that the converse does not hold. Hence the set of all strongly modalized formulas is properly contained in the set of all completely modalized formulas. Theorem 8. (1) MIPC ( p p) p p, p p. (2) The set of strongly modalized formulas of MIPC ( p p) coincides with the set of strongly modalized formulas of S5. Proof directly follows from Theorem 23 of [1], but in order to keep the paper self-contained, we give a (sketch of) proof: (1) p p p p p p. Hence p p p p. On the other hand, it is obvious that p p is equivalent to p p. (Actually, the converse, that is p p p p, holds as well, see Theorem 23 of [1]). (2) It is obvious that for every monadic Heyting algebra (H, H 0 ), strongly modalized formulas take their values in H 0. Hence a strongly modalized formula is valid in (H, H 0 ) iff it is valid in (H 0,H 0 ). Now since from (H, H 0 ) p p it follows that H 0 is a Boolean algebra, then a strongly modalized formula is MIPC ( p p)-valid iff it is S5- valid.

12 100 G. Bezhanishvili All these allow us to call the system MIPC ( p p) weak S5 and denote it by WS5. Remark 9. It is to be noted that the intuitionistic modal calculus L 4 (with one modal operator ) from Ono [21] has the same syntactical power as our WS5, and that actually WS5 coincides with IBM of Reyes and Zawadowski [24] 3, which gives another, category-theoretical motivation of WS5 (see [24] and the literature cited there). Hence the finite model property and decidability of WS5 follows from [21], [24] (as well as from a more general Theorem 42 of [1]). It is also worth to be mentioned that actually MAO and IBM C systems from [24] coincide with MIPC and WS5 (p p) (p p) 4 respectively. Hence the finite model property and decidability of MAO (which was stated as an open question in [24]) already follows from Bull [8]. (Though Bull s proof contained a gap. It was later filled independently by Ono [21] and Fisher Servi [14].) Note that, unlike strongly modalized formulas, the set of completely modalized formulas of WS5 does not coincide with the set of completely modalized formulas of S5. Indeed, the formula K is completely modalized and S5 K, while K is not valid in the algebra shown in Fig. 3 and hence WS5 K. It is also worth noting that actually there exists a continuum of logics in the interval [WS5, S5]. Indeed, for every two different propositional intermediate logics J 1,J 2 Int, we can prove that the logics WS5 J 1 and WS5 J 2 are also different, which implies that the cardinality of the interval [WS5, S5] is indeed that of continuum (see [1] and [7] for details). Now we are in a position to prove the first modal analogue of Glivenko s theorem. Theorem 10. (1) For any logic L intermediate between MIPC and WS5, WS5 A iff L A. (2) If A is completely modalized, then WS5 A iff L A, and WS5 A iff L A. 3 I should like to thank an anonymous referee for drawing my attention to the paper by Reyes and Zawadowski. 4 This has been shown in cooperation with R. Grigolia.

13 Glivenko Type Theorems for Intuitionistic Modal Logics 101 and (3) If A is strongly modalized, then S5 A iff L A, S5 A iff L A. Proof. (1) Since L WS5, WS5 A A and WS5 A A, then from L Ait follows directly that WS5 A. Conversely, if L A, then there exists a monadic Heyting algebra (H,, ) K L such that (H,, ) A. Consider the algebra (H/ FSD, FSD, FSD ). We already have mentioned that (H/ FSD, FSD, FSD ) WS5. Moreover, (H/ FSD, FSD, FSD ) A. Indeed, Corollary 6 implies that the dual space of (H/ FSD, FSD, FSD ) is isomorphic to E Q (max X), and hence contains max X. Therefore, B is valid in (H,, ) iff B is valid in (H/ FSD, FSD, FSD ), for any formula B. Now since (H/ FSD ) 0 R(H 0 ), (H/ FSD, FSD, FSD ) A, and hence (H/ FSD, FSD, FSD ) A. Butthen,WS5 A. 5 (2) For every completely modalized formula A, L A A. Moreover, L A A. Now it suffices to apply 1). (3) Directly follows from Theorem 8 (item 2) and (2). Note that this theorem covers only the logics intermediate between MIPC and WS5. Now we will slightly strengthen this theorem and cover all logics intermediate between MIPC and S5. Theorem 11. For any intermediate logic L [MIPC, S5], WS5 L A iff L A. Further, if A is completely modalized, then WS5 L A iff L A, and WS5 L A iff L A. Furthermore, if A is strongly modalized, then and S5 A iff L A, S5 A iff L A. Proof. Note that in the proof of Theorem 10, (H/ FSD, FSD, FSD ) WS5 L. The rest follows from Theorem In terms of the dual space (X, Ω,R,Q)of(H,, ) we have for an arbitrary Q-cone A that A is valid in X iff A is valid in E Q(max X), which implies the theorem.

14 102 G. Bezhanishvili Now let us turn to the second modal analogue of Glivenko s theorem. Note that in the first case we dealt with E Q (max X) which allowed us to associate with every monadic Heyting algebara the algebra which validates WS5. Now we will restrict ourselves to those monadic Heyting algebras whose dual spaces have the property that max X always constitutes a closed Q-cone. Note that since E Q (max X) istheleastq-cone containing max X, max X is a Q-cone iff max X = E Q (max X). But this equality holds only in dual spaces of algebras which validate the formula K. Therefore, we should deal with the system MIPC K. So, let us consider a monadic Heyting algebra which validates K.Then F SD = F D and hence H/ FSD R(H). Therefore, H/ FSD becomes a Boolean algebra. Besides, since a = a(= a) holds in every monadic Heyting algebra, R(H) becomes a (,,, )-subalgebra of (H,, ). Let us consider the algebra (R(H),,,,,, ), where a = a for every a R(H). It is obvious that is an onto homomorphism from (H,, ) to(r(h),, ), and that (R(H),, ) is a monadic Boolean algebra. Indeed, since K is valid in (H,, ), preserves. Moreover, since a = a = a holds in every monadic Heyting algebra, preserves and (R(H),, ) forms a monadic Boolean algebra. Remark 12. Consequently, the algebra (R(H), ) is always a homomorphic image of (H, ), and we arrive at the following result: For any -free formula A, S5 A iff MIPC A. This is closely related to Kleene s result (see our introduction). Therefore, for -free fragment of MIPC, which we denote by IntS5,wehavethe following fact: S5 A iff IntS5 A. Hence, -modal intuitionistic logics have an advantage over -modal or -modal intuitionistic logics in proving Glivenko type theorems. It is surprising, since as was shown in Wolter [30] and our [5], in general -modal intuitionistic logics are lacking almost all good logical properties which are preserved by -modal or -modal intuitionistic logics. The following may be considered as an explanation of this phenomenon. MIPC K is a conservative extension of IntS5 (see [6] for a proof and further results in this direction). Now returning to (R(H),, ) we obviously have that (R(H),, ) (H/ FD, FD, FD ), and that the dual space of (H/ FD, FD, FD ) is isomorphic to max X. Moreover, we arrive at the following

15 Glivenko Type Theorems for Intuitionistic Modal Logics 103 Theorem 13. (1) For any intermediate logic L, (S5 A L A) iff L K. (2) For any logic L [MIPC K, S5], S5 A iff L A. Proof. (1) Suppose L is intermediate and L K. Then obviously L A(p 1,...,p n ) implies S5 A(p 1,...,p n ). Conversely, suppose L A(p 1,...,p n ). Then there exist a monadic Heyting algebra (H,, ) K L, a 1,...,a n H and a monadic polynomial α(x 1,...,x n ), corresponding to A(p 1,...,p n ), such that α(a 1,...,a n ) in (H,, ). But then α(a 1,...,a n ) in (R(H),, ), and since is a monadic homomorphism, α( a 1,..., a n ) in (R(H),, ). Now since a 1,..., a n R(H), we get that A(p 1,...,p n ) is not valid in (R(H),, ). Therefore, S5 A. Conversely, if L K, then Corollary 4 implies that L (p p). On the other hand, S5 (p p). (2) Follows from (1). The last theorem can be regarded as a modal analogue of Umezawa- Gabbay theorem for intermediate predicate logics (see Umezawa [27] and Gabbay [15]). It is also worth noting that actually the logics WS5 and MIPC K are mutually incomparable, and that S5 is the join of these two logics (see [1] and [3]). In conclusion let us note that the two possible modal analogues of Glivenko s theorem which were offered here cover only the logics which are intermediate between MIPC and S5. However, as we mentioned before, there exist a lot of logics over MIPC which are not intermediate. Now we will show how to adopt Glivenko type theorems for them. Denote the lattice of all logics over MIPC by NExtMIPC and recall from [3] the structure of NExtMIPC. For this we will use the extensions of S5. As follows from Scroggs [26], S5 has only countably many extensions and they constitute a countable decreasing chain S5 1 S S5 n which converges to S5. Here S5 n = S5 B 2 n +1, where B n = i j, i,j n (p i p j ). Also recall that every S5 n is a tabular logic and is characterized by the (simple) algebra (2 n, 2), where 2 = {0, 1} denotes the two element Boolean algebra. The dual space of (2 n, 2) isthen-element cluster (X = {x 1,..., x n }, =,X X) shown in Fig. 4. This is the reason why the logics S5 n are often called n-cluster logics.

16 104 G. Bezhanishvili n Figure 4. A logic L MIPC is said to be an n-cluster logic, written L C n,if L S5 n and L S5 n+1. If L S5 n for every n ω, thenl is called a ω-cluster logic, written L C ω.nowwehavethat{c n } n ω is a partition of NExtMIPC,thatisC n C m = if n m, and n ω C n =NExtMIPC (see [3]). Moreover, every C n has the greatest element S5 n and the least element MIPC χ(2 n+1, 2), which is denoted by MIPC n.hereχ(2 n+1, 2) denotes the characteristic formula of the algebra (2 n+1, 2), that is χ(2 n+1, 2) isvaid in a monadic Heyting algebra (H,, ) iff(2 n+1, 2) is not a homomorphic image of a subalgebra of (H,, ). (See [1], [3], [4] for detailed investigation of characteristic formulas and other related results.) It is also obvious that the greatest element of C ω is S5 and the least element of C ω is MIPC. One can imagine the structure of NExtMIPC as it is shown in Fig. 5below (see [3] for details and other related results). The analogue of the logic WS5 in C n is the logic WS5 χ(2 n+1, 2) which will be denoted by WS5 n, and the analogue of the logic MIPC K in C n is the logic MIPC n K. Note that for every logic L NExtMIPC, if L is not intermediate, then there exists a natural number n such that L C n =[MIPC n, S5 n ]. Now we are in a position to formulate a modal analogue of Glivenko s theorem in the most general way. For the sake of uniformity of notations let us denote S5 by S5 ω, WS5 by WS5 ω and MIPC by MIPC ω. Theorem 14. (1) For every n ω and every logic L [MIPC n, WS5 n ], WS5 n A iff L A. Further, if A is completely modalized, then WS5 n A iff L A, and WS5 n A iff L A. Furthermore, if A is strongly modalized, then S5 n A iff L A, and S5 n A iff L A. (2) For every n ω and every logic L C n, WS5 n L A iff L A.

17 Glivenko Type Theorems for Intuitionistic Modal Logics 105 S5... S5 1 S MIPC 1 MIPC 2 MIPC Figure 5. Further, if A is completely modalized, then WS5 n L A iff L A, and WS5 n L A iff L A. Furthermore, if A is strongly modalized, then S5 n A iff L A, and S5 n A iff L A. (3) For every n ω and every logic L C n, (S5 n A L A) iff L K. Consequently, if L [MIPC n K, S5 n ],then S5 n A iff L A. 5. Essentially negative formulas In this section we will show how to benefit from Glivenko type theorems to obtain several results about decidability and the finite model property (FMP, for short). In particular, we will prove that adding an essentially

18 106 G. Bezhanishvili negative strongly modalized formula to a logic over MIPC preserves decidability and FMP, and that adding an essentially negative formulatoalogic over MIPC K also preserves decidability and FMP. These results extend analogous results for propositional intermediate logics from McKay [19] and Chagrov and Zakharyaschev [10] and are related to those by Rybakov [25]. AformulaA is said to be essentially negative if every occurence of a propositional variable in it is under the scope of (see McKay [19] or Chagrov and Zakharyaschev [10]). It is obvious that every essentially negative formula A has the form B( C 1,..., C n ) for some formulas B, C 1,..., C n. Call an essentially negative formula B( C 1,..., C n )anessentially negative strongly modalized formula if every C i is a strongly modalized formula. Now we are in a position to reduce the derivability problem in a logic over MIPC with an additional essentially negative completely modalized axiom to the derivability problem in MIPC. For this let us recall that the deduction theorem holds for logics over MIPC in the following form (see [7]): Γ,A L B iff Γ L A B, where L B means that B is derivable in L from with the help of modus ponens and necessitation. NowwehavethatA(p 1,...,p m ) MIPC B( C 1,..., C n )iffthereexists a derivation of A(p 1,...,p m )inmipc B( C 1,..., C n ) in which substitution instances of B( C 1,..., C n ) contain only the variables from the list p 1,..., p m. These substitution instances have the form B( C 1,..., C n) where every C i is a substitution instance of C i with the variables from the list p 1,..., p m. Since S5 is locally tabular, there exist only finitely many non-s5-equivalent such substitution instances of C i. Now in view of Theorem 10 (Item 3) there are only finitely many non-mipc-equivalent such substitution instances of C i. Therefore, there exist only finitely many non- MIPC-equivalent substitution instances of B( C 1,..., C n ) containing the variables p 1,..., p m. Denote them by B 1,..., B k(m). Then the deduction theorem and the fact that every B i is completely (even strongly) modalized imply that A MIPC B( C 1,..., C n )iffb 1 B k(m) A MIPC. Needless to say that in the last equivalence MIPC can be exchanged by any logic L over MIPC. Using the same arguments and Theorem 13 we can prove that for every essentially negative formula B( C 1,..., C n ) and every formula A(p 1,..., p m ), A MIPC K B( C 1,..., C n )iff (B 1 B l(m) ) A MIPC K.

19 Glivenko Type Theorems for Intuitionistic Modal Logics 107 Needless to say that in the last equivalence MIPC K can be exchanged by any logic L over MIPC K. As a result we arrive at the following Theorem 15. (1) For every decidable logic L NExtMIPC and every essentially negative strongly modalized formula B( C 1,..., C n ), the logic L B( C 1,..., C n ) is also decidable. (2) For every decidable logic L over MIPC K and every essentially negative formula B( C 1,..., C n ),thelogicl B( C 1,..., C n ) is also decidable. Let us close the paper by establishing that adding an essentially negative formula to a logic preserves FMP too. Theorem 16. (1) If L NExtMIPC enjoys FMP and B( C 1,..., C n ) is an essentially negative strongly modalized formula, then L B( C 1,..., C n ) also enjoys FMP. (2) If L is a logic over MIPC K, L enjoys FMP and B( C 1,..., C n ) is an essentially negative formula, then L B( C 1,..., C n ) also enjoys FMP. Proof. Suppose L B( C 1,..., C n ) A(p 1,...,p m ). Then L B 1 B k(m) A. But then there exists a finite monadic Heyting algebra (H,, ) K L with the generators a 1,..., a m such that interpreting every p i as a i we obtain B 1 B k(m) (a 1,...,a m ) A(a 1,...,a m ) in (H,, ). Consider the filter F =[B 1 B k(m) (a 1,...,a m )). (Here [a) denotes the filter generated by an element a.) Since every B i is completely (even strongly) modalized, B 1 B k(m) (a 1,...,a m ) H 0 and F is a monadic filter. Moreover, A(a 1,...,a m ) / F. Consider the algebra (H/ F, F, F ). Obviously A(a 1,...,a m ) in (H/ F, F, F ). It remains to show that (H/ F, F, F ) B 1 B k(m). Suppose otherwise, then there exist b 1,..., b m H such that interpreting every p i as b i we obtain B 1 B k(m) (b 1 / F,...,b m / F ) in (H/ F, F, F ). But then there exists i [1,k(m)] such that B i (b 1 / F,...,b m / F ) in (H/ F, F, F ), which means that B i (b 1,...,b m ) / F. Now recall that every B i has the form B( C 1,..., C n), and that every b k is a polynomial t k (a 1,...,a m ). Hence, B( C 1 (b 1,...,b m ),..., C n(b 1,...,b m )) has the form B( C 1 (t 1(a 1,..., a m ),..., t m (a 1,..., a m )),..., C n (t 1(a 1,..., a m ),..., t m (a 1,..., a m ))) = B( C 1 (a 1,..., a m ),..., C n (a 1,..., a m )), and B i (b 1,...,b m ) belongs to the list B 1 (a 1,...,a m ),..., B k(m) (a 1,...,a m ). Now since B i (b 1,...,b m ) / F,alsoB 1 B k(m) (a 1,...,a m ) / F,which

20 108 G. Bezhanishvili contradicts the construction of F. Therefore, (H/ F, F, F ) B 1 B k(m),and(h/ F, F, F ) is a finite member of K L B( C1,..., C n). Moreover, (H/ F, F, F ) A(p 1,...,p m ), which implies that L B( C 1,..., C n ) enjoys FMP. (2) can be proven analogously. The only difference is that in this case we should deal with the filter F =[ (B 1 B l(m) (a 1,...,a m ))). References [1] Bezhanishvili, G., Varieties of monadic Heyting algebras. Part I, Studia Logica 61 (1998), [2], Bezhanishvili, G., Varieties of monadic Heyting algebras. Part II: Duality theory, Studia Logica 62 (1999), [3] Bezhanishvili, G., Varieties of monadic Heyting algebras. Part III, Studia Logica 64 (2000), [4] Bezhanishvili, G., Splitting monadic Heyting algebras, 1997, Report IS-RR F, JAIST. [5] Bezhanishvili, G., Q-Heyting algebras as reducts of monadic Heyting algebras, Submitted. [6] Bezhanishvili, G., -free and -free reducts of monadic Heyting algebras, in preparation. [7] Bezhanishvili, G., and M. Zakharyaschev, Logics over MIPC, in Proceedings of Sequent Calculus and Kripke Semantics for Non-Classical Logics, RIMS Kokyuroku 1021, Kyoto University, 1997, pp [8] Bull, R. A., A modal extension of intuitionistic logic, Notre Dame Journal of Formal Logic 6 (1965), [9] Bull, R. A., MIPC as the formalization of an intuitionist concept of modality, Journal of Symbolic Logic 31 (1966), [10] Chagrov, A., and M. Zakharyaschev, Modal Logic, Oxford University Press, [11] Engelking, R., General Topology, Warszawa, [12] Esakia, L., Topological Kripke models (in Russian), Dokl. Akad. Nauk SSSR 214 (1974), [13] Esakia, L., Heyting Algebras I. Duality Theory (in Russian), Metsniereba Press, Tbilisi, [14] Fisher Servi, G., The finite model property for MIPQ and some consequences, Notre Dame Journal of Formal Logic 19 (1978), [15] Gabbay, D., Applications of trees to intermediate logics, Journal of Symbolic Logic 37 (1972),

21 Glivenko Type Theorems for Intuitionistic Modal Logics 109 [16] Glivenko, V., Sur quelques points de la logique de M. Brouwer, Bulletin de la Classe des Sciences de l Académie Royale de Belgique 15 (1929), [17] Halmos, P. R., Algebraic Logic, Chelsea Publishing Company, New York, [18] Kleene, S.,Introduction to Metamathematics, Van Nostrand, New York, North Holland, Amsterdam and Noordhoff, Groningen, [19] McKay, C., A class of decidable intermediate propositional logics, Journal of Symbolic Logic 36 (1971), [20] Monteiro, A., and O. Varsavsky, Algebras de Heyting monádicas, Actas de las X Jornadas de la Unión Matemática Argentina, Bahía Blanca, 1957, p [21] Ono, H., On some intuitionistic modal logics, Publications of Research Institute for Mathematical Sciences, Kyoto University 13 (1977), p [22] Prior, A., Time and Modality, Clarendon Press, Oxford, [23] Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, PWN, Warszawa, [24] Reyes, G., and M. Zawadowski, Formal systems for modal operators on locales, Studia Logica 52 (1993), [25] Rybakov, V., A modal analog for Glivenko s theorem and its applications, Notre Dame Journal of Formal Logic 33 (1992), [26] Scroggs, S. G., Extensions of the Lewis system S5, Journal of Symbolic Logic, 16 (1951), [27] Umezawa, T., On some properties of intermediate logics, Proceedings of the Japan Academy 35 (1959), [28] Umezawa, T., On logics intermediate between intuitionistic and classical predicate logics, Journal of Symbolic Logic 24 (1959), [29] Wajsberg, M., Ein erweiterter Klassenkalkül, Monatshefte für Mathematikund Physik 40 (1933), [30] Wolter, F., Superintuitionistic companions of classical modal logics, Studia Logica 58 (1997), Guram Bezhanishvili Department of Mathematical Sciences New Mexico State University Las Cruces, NM , USA gbezhani@nmsu.edu

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