Rudimentary Kripke Models for the Intuitionistic Propositional Calculus

DoSen, K., Rudimentary Kripe models for the intuitionistic propositional calculus, Annals of Pure and Applied Logic 62 (1993) 21-49. It is shown that the intuitionistic propositional calculus is sound and complete with respect to Kripke-style models that are not quasi-ordered. These models, called rudimentary Kripke models, differ from the ordinary intuitionistic Kripke models by making fewer assumptions about the underlying frames, but have the same conditions for valuations. However, since accessibility between points in the frames need not be reflexive, 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 rudimentary Kripke models, particular attention is paid to those that guarantee that the assumption of heredity and converse heredity for propositional variables implies heredity and converse heredity for all propositional formulae. These frames need to be neither reflexive nor transitive. The intuitionistic propositional calculus, which we call H (after Heyting), is sound and complete with respect to Kripke models based on quasi-ordered frames. Besides this class of Kripke models there are many smaller classes of Kripke models with respect to which H is sound and complete. For example, we may require from the frames of Kripke models that in addition to being quasi-ordered they satisfy one or more of the following: the frame is partially ordered (i.e., we have added antisymmetry), the frame is generated (i.e., there is a point which is lesser than or equal to every point), the frame is a tree, the frame is a Jaskowski tree, the frame is finite. 0168-0072/93/$06.00