Maximal Kripke-type semantics for modal and superintuitionistic predicate logics

Recent studies in semantics of modal and superintuitionistic predicate logics provided many examples of incompleteness, especially for Kripke semantics. So there is a problem: to find an appropriate possible-world semantics which is equivalent to Kripke semantics at the propositional level and which is strong enough to prove general completeness results. The present paper introduces a new semantics of ‘Kripke metaframes’ generalizing some earlier notions (such as Kripke bundles [12] or functor semantics [5]). The main innovation is in considering n-tuples of individuals as ‘abstract n-dimensional vectors’, together with some transformations of these vectors (‘abstract permutations and projections’). Soundness of the semantics is proved to be equivalent to some non-logical properties of metaframes; and thus we describe the maximal semantics of Kripke-type. The construction of canonical metaframes using n-types allows us to prove completeness theorems rather easily. Representation theorems in the last section show that in some cases metaframes are equivalent to ‘Cartesian’ ones in which all the vectors are n-tuples.