Possible worlds semantics: a research program that cannot fail? (Q1062670)

There exist incomplete modal logics, which fail to coincide with the modal theory of any class of possible worlds frames. Here it is pointed out that, in some of these cases, not even a free choice of frame class plus truth definition for the modal operator will model the incomplete logic. On this theme of varying truth definitions, a model-theoretic preservation theorem is proven for first-order definable filter quantifiers (satisfying the axioms of the minimal modal logic): these are essentially just syntactic variants on the well-known Kripke truth definition.