David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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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
|Keywords||modal logic necessity paradox possible worlds|
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References found in this work BETA
Alexander Chagrov (1997). Modal Logic. Oxford University Press.
George Bealer (1982). Quality and Concept. Oxford University Press.
Citations of this work BETA
Volker Halbach (2008). On a Side Effect of Solving Fitch's Paradox by Typing Knowledge. Analysis 68 (2):114 - 120.
Johannes Stern & Martin Fischer (2015). Paradoxes of Interaction? Journal of Philosophical Logic 44 (3):287-308.
James Kennedy Chase (2012). The Logic of Quinean Revisability. Synthese 184 (3):357-373.
Alexander Paseau (2009). How to Type: Reply to Halbach. Analysis 69 (2):280-286.
Johannes Stern (2014). Modality and Axiomatic Theories of Truth I: Friedman-Sheard. Review of Symbolic Logic 7 (2):273-298.
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