Models for Normal Intuitionistic Modal Logics

Kripke-style models with two accessibility relations, one intuitionistio and the other modal, are given for analogues of the modal system K based on Heyting's propositional logic. I t is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for an investigation of intuitionistio analogues of systems stronger than K. A brief survey is given of the existing literature on intuitionistic modal logic. w