Possible-Worlds Semantics for Modal Notions Conceived as Predicates

If is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate , we tackle both problems. Given a frame hW;Ri consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret at every world in such a way that pAq holds at a world w 2 W if and only if A holds at every world v 2 W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several `paradoxes' (like Montague's Theorem, G\u7fodel's Second Incompleteness Theorem, McGee's Theorem on the !-inconsistency of certain truth theories etc.) show that many frames, e.g., reexive frames, do not allow for such an interpretation. We present sucient and necessary conditions for the existence of a suitable interpretation of at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic