On the relation between intuitionistic and classical modal logics

Intuitionistic propositional logic Int and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded just as fragments of classical modal logics containing S4. At the syntactical level, the Godel translation t embeds every intermediate logic L = Int+ into modal logics in the interval L = [ L = S4 t( ); L = Grz t( )]. Semantically this is re ected by the fact that Heyting algebras are precisely the algebras of open elements of topological Boolean algebras. From the lattice-theoretic standpoint the map is a homomorphism of the lattice of logics containing S4 onto the lattice of intermediate logics, while , according to the Blok{Esakia theorem, is an isomorphism of the latter onto the lattice of extensions of the Grzegorczyk system Grz. At the philosophical level the Godel translation provides a classical interpretation of the intuitionistic connectives. And from the technical point of view this embedding is a powerful tool for transferring various kinds of results from intermediate logics to modal ones and back via preservation theorems. (For details and references consult [6].) Both classical modal logic and the theory of intermediate logics have gained from this correspondence. The main aim of this paper is to construct a similar correspondence between intermediate logics enriched with modal operators|we call them intuitionistic modal logics|and classical polymodal logics. That the G odel translation can be extended to an embedding of at least a few particular intuitionistic modal systems into some classical polymodal logics was observed by several authors (cf. [13], [25], [26], [5]). Fischer Servi [13], [15] used a variant of the translation to de ne "true" intuitionistic analogues of a number of classical modal systems. In [27] we exploited the translation proposed by Shehtman [25] to embed intuitionistic modal logics with the single necessity operator 2 of K into bimodal logics