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)

We prove that the positive fragment of first-order intuitionistic logic in the language with two individual variables and a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals \ and \, where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser’s basic and formal logics, (...) respectively. We also show that, for most “natural” first-order modal logics, the two-variable fragment with a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz—among them, QK, QT, QKB, QD, QK4, and QS4. (shrink)

We obtain an effective embedding of the classical predicate logic into the logic of partial quasiary predicates. The embedding has the property that an image of a non-theorem of the classical logic is refutable in a model of the logic of partial quasiary predicates that has the same cardinality as the classical countermodel of the non-theorem. Therefore, we also obtain an embedding of the classical predicate logic of finite models into the logic of partial quasiary predicates over finite structures. As (...) a consequence, we prove that the logic of partial quasiary predicates is undecidable—more precisely, $\varSigma ^0_1$-complete—over arbitrary structures and not recursively enumerable—more precisely, $\varPi ^0_1$-complete—over finite structures. (shrink)

In this paper, the predicate counterparts, defined both axiomatically and semantically by means of Kripke frames, of the modal propositional logics $\textbf {GL}$, $\textbf {Grz}$, $\textbf {wGrz}$ and their extensions are considered. It is proved that the set of semantical consequences on Kripke frames of every logic between $\textbf {QwGrz}$ and $\textbf {QGL.3}$ or between $\textbf {QwGrz}$ and $\textbf {QGrz.3}$ is $\Pi ^1_1$-hard even in languages with three (sometimes, two) individual variables, two (sometimes, one) unary predicate letters, and a single (...) proposition letter. As a corollary, it is proved that infinite families of modal predicate axiomatic systems, based on the classical first-order logic and the modal propositional logics $\textbf {GL}$, $\textbf {Grz}$, $\textbf {wGrz}$ are not Kripke complete. Both $\Pi ^1_1$-hardness and Kripke incompleteness results of the paper do not depend on whether the logics contain the Barcan formula. (shrink)

We show that—unlike products of ‘transitive’ modal logics which are usually undecidable—their ‘expanding domain’ relativisations can be decidable, though not in primitive recursive time. In particular, we prove the decidability and the finite expanding product model property of bimodal logics interpreted in two-dimensional structures where one component—call it the ‘flow of time’—is • a finite linear order or a finite transitive tree and the other is composed of structures like • transitive trees/partial orders/quasi-orders/linear orders or only finite such structures expanding (...) over time. The decidability proof is based on Kruskal’s tree theorem, and the proof of non-primitive recursiveness is by reduction of the reachability problem for lossy channel systems. The result is used to show that the dynamic topological logic interpreted in topological spaces with continuous functions is decidable if the number of function iterations is assumed to be finite. (shrink)

This volume presents mathematical game theory as an interface between logic and philosophy.