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General characterization results on intuitionistic modal propositional logics. (English) Zbl 0598.03015

Systems of modal logic based on classical logic can be characterized either by Boolean algebras or by Kripke frames, and it is straightforward to pass from a frame to an algebra and vice-versa, using Stone representation. Similarly when the underlying non-modal logic is intuitionistic, except that Heyting algebras replace Boolean ones. Say that C(F), a condition on a frame for modal intuitionistic logic, characterizes C’(A), a condition on a normal modal Heyting algebra, if (a) whenever A is constructed from F and C(F) holds, so does C’(A); and (b) if F is the reduct of the Stone representation frame for an algebra A, then if C’(A) holds, so does C(F). In this paper, the author establishes a series of very general characterization theorems of somewhat complex form, extending those of Lemmon and Scott, and Salqvist. These theorems enable one to derive conditions on algebras from conditions on frames.
Reviewer: G.Forbes

MSC:

03B45 Modal logic (including the logic of norms)
03G10 Logical aspects of lattices and related structures
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