Notre Dame Journal of Formal Logic

Montague’s Paradox, Informal Provability, and Explicit Modal Logic

Walter Dean

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Abstract

The goal of this paper is to explore the significance of Montague’s paradox—that is, any arithmetical theory $T\supseteq Q$ over a language containing a predicate $P(x)$ satisfying (T) $P(\ensuremath {\ulcorner \varphi \urcorner })\rightarrow \varphi $ and (Nec) $T\vdash \varphi \,\therefore\,T\vdash P(\ensuremath {\ulcorner \varphi \urcorner })$ 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 $\mathcal {QLP}$ (the quantified logic of proofs) of Artemov’s logic of proofs ($\mathcal {LP}$). $\mathcal {QLP}$ contains both explicit modalities $t:\varphi $ (“$t$ is a proof of $\varphi $”) and also proof quantifiers $(\exists x)x:\varphi $ (“there exists a proof of $\varphi $”). 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.

Article information

Source
Notre Dame J. Formal Logic Volume 55, Number 2 (2014), 157-196.

Dates
First available in Project Euclid: 24 April 2014

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1398345779

Digital Object Identifier
doi:10.1215/00294527-2420636

Mathematical Reviews number (MathSciNet)
MR3201831

Zentralblatt MATH identifier
1352.03062

Subjects
Primary: 03F03: Proof theory, general
Secondary: 03F45: Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25] 03F50: Metamathematics of constructive systems

Keywords
Montague’s paradox Knower paradox constructive proof informal proof provability logic explicit modal logic justification logic

Citation

Dean, Walter. Montague’s Paradox, Informal Provability, and Explicit Modal Logic. Notre Dame J. Formal Logic 55 (2014), no. 2, 157--196. doi:10.1215/00294527-2420636. http://projecteuclid.org/euclid.ndjfl/1398345779.


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