Speaking about transitive frames in propositional languages
Reflexive and transitive Kripke frames can be talked about by both the classical modal and the intuitionistic non-modal (propositional) language, respectively, according to the standard semantic interpretations. Several results on the relationship between them have already been known; they have equivalent expressive power with respect to the axiomatizability of skeleton-closed classes, the lattice of intermediate logics is isomorphic to that of normal extensions of Grzegorczyk modal logic, and so on. In this paper the authors investigate such a relationship in the larger class of transitive Kripke frames without supposing reflexivity, where the basic logics turn out to be K4 and Visser's BPL instead of S4 and intuitionistic logic, respectively. With this generalization the non-modal language is shown to display some unusual features; its expressive power becomes weaker than that of classical modal language, the induced consequence relation does not enjoy the deduction theorem nor is it protoalgebraic. The consequence relation and its extensions, however, are analysed by making use of duality connecting algebraic and relational semantics. The expressive power of the non-modal language is restored as in the case of reflexive and transitive Kripke frames by adding to it one more implication.
scientific article; zbMATH DE number 1290125