An investigation of various Kripke models for intuitionistic logic

Abstract

Kripke models are semantic models for intuitionistic logic and other intermediate logics. This class of models is sound and complete with respect to the intuitionistic propositional calculus H. However, there are other classes of models that are sound and complete for H. These models are based on frames that are not partially ordered like the frames of standard Kripke models. It is relevant to examine other classes of models for an important logic like H, since different semantics create different points of view. First, the standard Kripke models will be discussed. Second, quasi-ordered Kripke models will be examined. This class of models is based on quasi-ordered frames where the relation is required to be reflexive and transitive, although not necessarily antisymmetric. The third class of models is also sound and complete with respect to H; the class of rudimentary Kripke models. These models consist of frames with pseudo-valuations that satisfy certain rules of heredity. Finally, the class of pseudo Kripke models is discussed. Pseudo Kripke models consist of frames with pseudo-valuations that do not satisfy the rules of heredity and the class of pseudo Kripke models is therefore not sound and complete for H. In this paper the various Kripke models will be discussed and the mutual relations will be clarified.