Possible-worlds semantics for modal notions conceived as predicates

Journal of Philosophical Logic 32 (2):179-223 (2003)


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 (W, R) 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 □ $\ulcorner A \ulcorner$ holds at a world ᵆ ∊ W if and only if A holds at every world $\upsilon$ ∊ W such that ᵆR $\upsilon$ . The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, 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 □ 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

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Author Profiles

Hannes Leitgeb
Ludwig Maximilians Universität, München
Volker Halbach
Oxford University

References found in this work

Outline of a Theory of Truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
Quality and Concept.George Bealer - 1982 - Oxford, England: Oxford University Press.

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Citations of this work

Validity as a Primitive.J. Ketland - 2012 - Analysis 72 (3):421-430.
Ungroundedness in Tarskian Languages.Saul Kripke - 2019 - Journal of Philosophical Logic 48 (3):603-609.
Advanced Temporalising.Daniel Deasy - 2021 - In Karen Bennett & Dean Zimmerman (eds.), Oxford Studies in Metaphysics Vol.12. Oxford, UK: Oxford University Press.

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