Rudimentary Beth Models and Conditionally Rudimentary Kripke Models for the Heyting Propositional Calculus

This paper continues the investigation of rudimentary Kripke models, i.e. non-quasiordered Kripke-style models for the Heyting propositional calculus H. Three topics, which make three companion pieces of [3], are pursued. The first topic is Beth-style models for H called rudimentary Beth models. Rudimentary Kripke models may be conceived as a particular type of these models, which are analogous to rudimentary Kripke models in not assuming quasi-ordering for the underlying frames. The second topic is a first step into the correspondence theory for rudimentary Kripke models. It is shown what conditions on frames are now defined by the characteristic schemata of Dummett's logic, the logic of weak excluded middle and classical propositional logic. The third topic is a generalization of rudimentary Kripke models that yields models called conditionally rudimentary Kripke models. It is shown that, if we don't want to change the usual semantic clauses for the connectives, conditionally rudimentary Kripke models make the largest class of Kripkestyle models with respect to which we can demonstrate the ordinary soundness and completeness of H.