On the Semantics of Graded Modalities

We enrich propositional modal logic with operators ◇ (n ∈ N) which are interpreted on Kripke structures as “there are more than n accessible worlds for which ...”, thus obtaining a basic graded modal logic GrK. We show how some familiar concepts (such as subframes, p-morphisms, disjoint unions and filtrations) and techniques from modal model theory can be used to obtain results about expressiveness (like graded modal equivalence, correspondence and definability) for this language. On the basis of the class of linear frames we demonstrate that the expressive power of the language is considerably stronger than that of classical modal logic. We give a class of formulas for which a first-order equivalent can be systematically obtained, but also show that the set of formulas for which such an equivalence exists is in some sense a proper subset of the set of so called Sahlqvist formulas, a syntactically defined set of modal formulas for which a corresponding formula is guaranteed to exist. Finally we show how, combining the technique of ‘filtration’ with a notion of ‘copying worlds’ — in view of the “more than n” interpretation, one cannot simply collapse worlds — for some graded modal logics (GrK, GrT, . . . ), the finite model property (and also decidability) is obtained.