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2014 Montague’s Paradox, Informal Provability, and Explicit Modal Logic
Walter Dean
Notre Dame J. Formal Logic 55(2): 157-196 (2014). DOI: 10.1215/00294527-2420636

Abstract

The goal of this paper is to explore the significance of Montague’s paradox—that is, any arithmetical theory TQ over a language containing a predicate P(x) satisfying (T) P(φ)φ and (Nec) TφTP(φ) is inconsistent—as a limitative result pertaining to the notions of formal, informal, and constructive provability, in their respective historical contexts. To this end, the paradox is reconstructed in a quantified extension QLP (the quantified logic of proofs) of Artemov’s logic of proofs (LP). QLP contains both explicit modalities t:φ (“t is a proof of φ”) and also proof quantifiers (x)x:φ (“there exists a proof of φ”). In this system, the basis for the rule NEC is decomposed into a number of distinct principles governing how various modes of reasoning about proofs and provability can be internalized within the system itself. A conceptually motivated resolution to the paradox is proposed in the form of an argument for rejecting the unrestricted rule NEC on the basis of its subsumption of an intuitively invalid principle pertaining to the interaction of proof quantifiers and the proof-theorem relation expressed by explicit modalities.

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Walter Dean. "Montague’s Paradox, Informal Provability, and Explicit Modal Logic." Notre Dame J. Formal Logic 55 (2) 157 - 196, 2014. https://doi.org/10.1215/00294527-2420636

Information

Published: 2014
First available in Project Euclid: 24 April 2014

zbMATH: 1352.03062
MathSciNet: MR3201831
Digital Object Identifier: 10.1215/00294527-2420636

Subjects:
Primary: 03F03
Secondary: 03F45, 03F50

Rights: Copyright © 2014 University of Notre Dame

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Vol.55 • No. 2 • 2014
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