Incompleteness of semantics for intermediate predicate logics, I. Kripke's Semantics
Hiroakira Ono · Proceedings of the Japan Academy Series A Mathematical Sciences · 1973
Kripke models have been introduced in [2] for the intuitionistic logic, but in [3] we have studied their basic properties as models for intermediate propositional logics.We presented there the following problem. Has every intermediate propositional logic a characteristic Kripke model?Though it is important, e.g. in connection with the finite model property ([4]), it remains unsolved.Instead of this, we will show in the present paper that there exists an intermediate predicate logic having no characteristic Kripke models.In this sense, we can say that Kripke's semantics for intermediate predicate logics is incomplete.Similar incompleteness results for algebraic semantics will appear in the subsequent paper.In order to abbreviate definitions, we use some of the terminology in Church [1].We identify the word predicate logics with the word pure functional calculi of first order.