Modalities

Abstract We explain how propositional modal logics can be understood as subfragments of the two-variable fragment of first-order logic, in which the interpretation of a distinguished binary relation is subject to various semantic constraints, in particular, the properties of reflexivity, seriality, symmetry and transitivity. We introduce graded modal logic, which extends propositional modal logics (thus understood) with counting quantifiers. We determine the complexity of the satisfiability problems for modal logics and graded modal logics defined by all possible conjunctions of the semantic constraints just mentioned. We also characterize the expressive power of propositional modal logic and briefly consider some extensions contained within the two-variable fragment of first-order logic.