| Abstract: | 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  , the set L] of L-companions of . Here L] consists of those modal logics whose non-modal fragments coincide with L and which axiomatize 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], whether L] contains a smallest element, and whether L] contains lower covers of . 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 ap plies duality theory from 34]. |
