Possible worlds semantics for predicates

H Leitgeb, PD Welch, V Halbach

Research output: Chapter in Book/Report/Conference proceedingChapter in a book


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.
Translated title of the contributionPossible worlds semantics for predicates
Original languageEnglish
Title of host publicationIntensionality, Lecture Notes in Logic
EditorsR Kahle
PublisherAssociation for Symbolic Logic
Pages20 - 41
Number of pages22
ISBN (Print)156881268X
Publication statusPublished - 2005


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