Superintuitionistic Companions of Classical Modal Logics

This paper investigates partitions of lattices of modal logics based on superintuitionistic logics which are defined by forming, for each superintuitionistic logic L and classical modal logic Θ, the set L[Θ] of L-companions of Θ. Here L[Θ] consists of those modal logics whose non-modal fragments coincide with L and which axiomatize Θ if the law of excluded middle p V ¬p is added. Questions addressed are, for instance, whether there exist logics with the disjunction property in L[Θ], whether L[Θ] contains a smallest element, and whether L[Θ] contains lower covers of Θ. Positive solutions as concerns the last question show that there are (uncountably many) superclean modal logics based on intuitionistic logic in the sense of Vakarelov [28]. Thus a number of problems stated in [28] are solved. As a technical tool the paper develops the splitting technique for lattices of modal logics based on superintuitionistic logics and applies duality theory from [34]. © 1997 Kluwer Academic Publishers.