Proximity modal logics
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 of proximity relation between sets: the
univerce is a pseudo-metric space and the relation "P is near Q"
between subsets P,Q is fulfilled iff the distance between P and
Q is zero. Using filtration it is proved that PML has fmp with
respect to a class of non-standard models, which implies its decidability. Several
variations of PML are also introduced and studied. In the conclusion, an
application to the theory of generalized quantifiers is disscussed.
Dimiter Vakarelov