Weak logics with strict implication

Abstract

Kripkean semantics for intuitionistic logic and in general for intermediate logics contains a principle of 'truth-preservation' to the effect that if a sentence letter p is true at a world w, then p is true at every world related or accessible from w. If we consider frames whose accessibility relation is transitive, then this principle holds for arbitrary formulae. In this paper we investigate those sublogics of intuitionistic propositional logic, to which we shall refer simply as weak logics, which are characterized (valid and complete) by classes of Kripkean models in which the truth conditions are the standard ones for intuitionistic logic and in which no assumption of truth-preservation is made.