Recent studies in semantics of modal and superintuitionistic predicate logics provided many examples of incompleteness, especially for Kripke semantics. So there is a problem: to find an appropriate possible- world semantics which is equivalent to Kripke semantics at the propositional level and which is strong enough to prove general completeness results. The present paper introduces a new semantics of Kripke metaframes' generalizing some earlier notions. The main innovation is in considering "n"-tuples of individuals as abstract "n"-dimensional vectors', together with some (...) transformations of these vectors. Soundness of the semantics is proved to be equivalent to some non- logical properties of metaframes; and thus we describe the maximal semantics of Kripke- type. (shrink)

An example of finite tree Mo is presented such that its predicate logic (i.e. the intermediate predicate logic characterized by the class of all predicate Kripke frames based on Mo) is not finitely axiomatizable. Hence it is shown that the predicate analogue of de Jongh - McKay - Hosoi's theorem on the finite axiomatizability of every finite intermediate propositional logic is not true.

A method for constructing continua of logics squeezed between some intermediate predicate logics, developed by Suzuki [8], is modified and applied to intervals of the form [L, L+ ¬¬S], where Lis a predicate logic, Sis a closed predicate formula. This solves one of the problems from Suzuki's paper.

This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...) to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia’s work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues. (shrink)

The problem of completeness for predicate modal logics is still under investigation, although some results have been obtained in the last few years . As far as we know, the case of multimodal logics has not been addressed at all. In this paper, we study the combination of modal logics in terms of combining their semantics. We demonstrate by a simple example that in this sense predicate modal logics are not so easily manipulated as propositional ones: mixing two Kripke-complete predicate (...) modal logics results in a Kripke-incomplete system. (shrink)

Some properties of Kripke-sheaf semantics for super-intuitionistic predicate logics are shown. The concept ofp-morphisms between Kripke sheaves is introduced. It is shown that if there exists ap-morphism from a Kripke sheaf 1 into 2 then the logic characterized by 1 is contained in the logic characterized by 2. Examples of Kripke-sheaf complete and finitely axiomatizable super-intuitionistic (and intermediate) predicate logics each of which is Kripke-frame incomplete are given. A correction to the author's previous paper Kripke bundles for intermediate predicate logics (...) and Kripke frames for intuitionistic modal logics (Studia Logica, 49(1990), pp. 289–306 ) is stated. (shrink)

This article builds on Humberstone’s idea of defining models of propositional modal logic where total possible worlds are replaced by partial possibilities. We follow a suggestion of Humberstone by introducing possibility models for quantified modal logic. We show that a simple quantified modal logic is sound and complete for our semantics. Although Holliday showed that for many propositional modal logics, it is possible to give a completeness proof using a canonical model construction where every possibility consists of finitely many formulas, (...) we show that this is impossible to do in the first-order case. However, one can still construct a canonical model where every possibility consists of a computable set of formulas and thus still of finitely much information. (shrink)

We propose a new, rather simple and short proof of Kripke-completeness for the predicate variant of Dummett's logic. Also a family of Kripke-incomplete extensions of this logic that are complete w.r.t. Kripke frames with equality (or equivalently, w.r.t. Kripke sheaves [8]), is described.

In [Ono 1987] H. Ono put the question about axiomatizing the intermediate predicate logicLFin characterized by the class of all finite Kripke frames. It was established in [ Skvortsov 1988] thatLFin is not recursively axiomatizable. One can easily show that for any finite posetM, the predicate logic characterized byM is recursively axiomatizable, and its axiomatization can be constructed effectively fromM. Namely, the set of formulas belonging to this logic is recursively enumerable, since it is embeddable in the two-sorted classical predicate (...) calculusCPC 2. Thus the logicLFin is II 2 0 -arithmetical.Here we give a more explicit II 2 0 -description ofLFin: it is presented as the intersection of a denumerable sequence of finitely axiomatizable Kripke-complete logics. Namely, we give an axiomatization of the logicLB n P m + characterized by the class of all posets of the finite height m and the finite branching n. A finite axiomatization of the predicate logicLP m + characterized by the class of all posets of the height m is known from [Yokota 1989]. We prove thatLB n P m + =,B n being the propositional axiom of the branching n. (shrink)

We generalize the incompleteness proof of the modal predicate logic Q-S4+ p p + BF described in Hughes-Cresswell [6]. As a corollary, we show that, for every subframe logic Lcontaining S4, Kripke completeness of Q-L+ BF implies the finite embedding property of L.

Kripke bundle [3] and C-set semantics [1] [2] are known as semantics which generalize standard Kripke semantics. In [3] and in [1], [2] it is shown that Kripke bundle and C-set semantics are stronger than standard Kripke semantics. Also it is true that C-set semantics for superintuitionistic logics is stronger than Kripke bundle semantics [5].In this paper, we show that Q-S4.1 is not Kripke bundle complete via C-set models. As a corollary we can give a simple proof showing that C-set (...) semantics for modal logics are stronger than Kripke bundle semantics. (shrink)

The completeness w.r.t. Kripke frames with equality (or, equivalently, w.r.t. Kripke sheaves, [ 8 ] or [4, Sect. 3.6]) is established for three superintuitionistic predicate logics: ( Q - H + D *), ( Q - H + D *&K), ( Q - H + D *& K & J ). Here Q - H is intuitionistic predicate logic, J is the principle of the weak excluded middle, K is Kuroda’s axiom, and D * (cf. [ 12 ]) is a (...) weakened version of the well-known constant domains principle D . Namely, the formula D states that any individual has ancestors in earlier worlds, and D * states that any individual has $${\neg\neg}$$ -ancestors (i.e., ancestors up to $${\neg\neg}$$ -equality) in earlier worlds. In particular, the logic ( Q - H + D *& K & J ) is the Kripke sheaf completion of ( Q - H + E & K & J ), where E is a version of Markov’s principle (cf. [ 12 ]). On the other hand, we show that the logic ( Q - H + D *& J ) is incomplete w.r.t. Kripke sheaves. (shrink)

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic (...) Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Gödel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic. (shrink)

The Kripke-completeness and incompleteness of some intermediate predicate logics is established. In particular, we obtain a Kripke-incomplete logic (H* +A+D+K) where H* is the intuitionistic predicate calculus, A is a disjunction-free propositional formula, D = x(P(x) V Q) xP(x) V Q, K = ¬¬x(P(x) V ¬P(x)) (the negative answer to a question of T. Shimura).

The quantified extension of a canonical prepositional intermediate logic is complete with respect to the generalization of Kripke semantics taking into consideration set-valued functors defined on a category.

By means of models in toposes of C-sets (where C is a small category), necessary conditions are found for the minimum quantified extension of a propositional (intermediate, modal) logic to be complete with respect to Kripke semantics; in particular, many well-known systems turn out to be incomplete.