Possible-Worlds Semantics for Modal Notions Conceived as Predicates

IF rectangle 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 rectangle 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 rectangle, we tackle both problems. Given a frame (W,R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret rectangle at every world in such a way that rectangle inverted perpendicularA inverted perpendicular holds at a world w epsilon W if and only if A holds at every world v epsilon W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Godel's Second Incompleteness Theorem, McGee's Theorem on the w-inconsistency of certain truth theories, etc.) show that many frame, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of rectangle 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.