We investigate several intuitionistic modal logics (IMLs), mainly from semantical viewpoint. In these decades, in the area of type theory for programming language, various type systems corresponding (via Curry-Howard correspondence) to IMLs have been considered. However, IMLs arising from such a context are not extensively studied before. We try to understand the logical meaning of such IMLs, a...

In epistemic logic, Kripke structures are used to model the distribution of information in a multi-agent system. In this paper, we present an approach to quantifying how much information each particular agent in a system has, or how important the agent is, with respect to some fact represented as a goal formula. It is typically the case that the goal formula is distributed knowledge in the syst...

It is shown that every normal intuitionistic modal logic L over MIPC has the nite model property if there exists a universal rstorder sentence such that (1) L is characterized by the class of Kripke frames satisfying and (2) every Kripke frame that validates L satis es . Here, MIPC is a well-known intuitionistic modal logic introduced by Prior (1957).

The Gentzen-Kripke construction of the semantics for intermediate logic LQ, which is obtainable from the intuitionistic propositional logic H by adding the weak law of excluded middle -*A v —• —1>4, is presented. Our construction spans the Gentzen system and the Kripke semantics for LQ by providing the way from the cut-elimination theorem to model-theoretic results. The completeness and decidab...

In the paper we introduce logical calculi for representation of propositions with modal operators indexed by fuzzy values. There calculi are called Heyting-valued modal logics. We introduce the concept of a Heyting-valued Kripke model and consider a semantics of Heyting-valued modal logics at the class of Heyting-valued Kripke models.

The question as to whether the propositional logic of Heyting, which was a formalization of Brouwer's intuitionistic logic, is finitely many valued or not, was open for a while (the question was asked by Hahn). Kurt Gödel (1932) introduced an infinite decreasing chain of intermediate logics, which are known nowadays as Gödel logics, for showing that the intuitionistic logic is not finitely (man...

Kripke semantics plays a fundamental role for modal logics where it often provides modular semantics allowing a clear treatment of various logics related by the addition of axioms or connectives. Kripke semantics is also the “natural semantics” for classical modal logic in the sense that it is given by the representation theory for modal algebras. The algebraic structure corresponding to a trad...

We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order (or partial-order) with the modal accessibility relation generalise to the requirement that the relation and its inverse be lower semi-contin...

A Kripke Semantics is defined for a higher-order logic programming language with constraints, based on Church’s Theory of Types and a generic constraint formalism. Our syntactic formal system, hoHH(C) (higher-order hereditary Harrop formulas with constraints), which extends λProlog’s logic, is shown sound and complete. A Kripke semantics for equational reasoning in the simply typed lambda-calcu...