RUDIMENTARY KRIPKE MODELS FOR THE INTUITIONISTIC PROPOSITIONAL CALCULUS

It is shown that the intuitionistic propositional calculus is sound an d complete with respect to Kripke-style models that are not quasi-orde red. These models, called rudimentary Kripke models, differ from the o rdinary intuitionistic Kripke models by making fewer assumptions about the underlying frames, but have the same conditions for valuations. H owever, since accessibility between points in the frames need not be r eflexive, we have to assume, besides the usual intuitionistic heredity , the converse of heredity, which says that if a formula holds in all points accessible to a point x, then it holds in x. Among frames of ru dimentary Kripke models, particular attention is paid to those that gu arantee that the assumption of heredity and converse heredity for prop ositional variables implies heredity and converse heredity for all pro positional formulae. These frames need to be neither reflexive nor tra nsitive.