Ancestral Kripke models and nonhereditary Kripke models for the Heyting propositional calculus

Abstract

This is a sequel to two previous papers, where it was shown that, for the Heyting propositional calculus H, we can give Kripke-style models whose accessibility relation R need not be a quasi-ordering relation, provided we have: (Formula Present) From left to right, this is the heredity condition of standard Kripke models for H, but, since R need not be reflexive, the converse is not automatically satisfied. These Kripke-style models for H were called “rudimentary Kripke models”. This paper introduces a kind of dual of rudimentary Kripke models, where the equivalence above is replaced by: (Formula Present). From right to left, this is again the usual heredity condition, but the converse, which is automatically satisfied if R is reflexive, yields a proper subclass of rudimentary Kripke models, whose members are called “ancestral Kripke models”. In all that, the semantic clauses for the connectives are as in standard Kripke models for H. The propositional calculus H is strongly sound and complete with respect to ancestral Kripke models. The remainder of the paper is devoted to Kripke-style models for H where the semantic clauses for the connectives are changed so that we need not assume any of the heredity conditions above. The resulting Kripke-style models are called “nonhereditary Kripke models”. These models are inspired by some particular embeddings of H into 54 and a somewhat weaker normal modal logic. With respect to a notion of quasi-ordered nonhereditary Kripke model, we can prove a certain form of strong soundness and completeness of H. With respect to another notion, where quasi-ordering is not assumed, we can only prove the ordinary soundness and completeness of H.