Intuitionistic Modal Logics as Fragments of Classical Bimodal Logics

Godel's translation of intuitionistic formulas into modal ones provides the well-known embedding of intermediate logics into extensions of Lewis' system S4, which re ects and sometimes preserves such properties as decidability, Kripke completeness, the nite model property. In this paper we establish a similar relationship between intuitionistic modal logics and classical bimodal logics. We also obtain some general results on the nite model property of intuitionistic modal logics rst by proving them for bimodal logics and then using the preservation theorem. The aim of this paper is to show how well known results and technique developed in the eld of intermediate and classical (monoand poly-) modal logics can be used for studying various intuitionistic modal systems. The current state of knowledge in intuitionistic modal logic resembles (to some extent) that in classical modal logic about a quarter of a century ago, when the discipline was just an extensive collection of individual systems and the best method of proving decidability was the art of ltration. The reason why intuitionistic modal logic lags far behind its classical counterpart is quite apparent. Intuitionistic modal logics are much more closely related to classical bimodal logics than to usual monomodal ones, and only few recent years have brought fairly general results in polymodal logic (e.g. [9], [13], [20]). This explains also the strange fact that the embedding of intuitionistic modal logics into classical bimodal ones (via the evident generalization of G odel's translation The work of the second authorwas supportedby the Alexandervon Humboldt Foundation.