Proximity Modal Logics

Abstract

A modal logic PML - Proximity Modal Logic, with a binary modality diamond(P,Q) with intuitive reading "P is near Q from certain point of view" is introduced. A possible world semantics for PML is given, which is based on the notion of proximity relation between sets, studied in the theory of proximity spaces [1]. An axiomatization and completeness theorem with respect to several classes of proximity spaces is proved, including one of the best examples: the universe is a pseudo-metric space and "P is near Q" iff the distance between P and Q is zero. Using filtration it is proved that PML has fmp with respect to a class of its models. which implies its decidability. In the conclusion, some extensions of PML are discussed and an application to the theory of generalized quantifiers is proposed.