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On the relation between intuitionistic and classical modal logics. Algebra and Logic

By Frank Wolter and Michael Zakharyaschev


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. Atthe syntactical level, the Godel translation t embeds every intermediate logic L = Int+ into modal log-1 ics 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. Atthe 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

Year: 1996
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