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Montague’s paradox, informal provability, and explicit modal logic
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Dean, Walter (2014) Montague’s paradox, informal provability, and explicit modal logic. Notre Dame Journal of Formal Logic, Volume 55 (Number 2). pp. 157-196. doi:10.1215/00294527-2420636 ISSN 0029-4527.
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Official URL: http://dx.doi.org/10.1215/00294527-2420636
Abstract
The goal of this paper is to explore the significance of Montague’s paradox—that is, any arithmetical theory T⊇Q over a language containing a predicate P(x) satisfying (T) P(┌φ┐)→φ and (Nec) T⊢φ ∴ T⊢P(┌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.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Divisions: | Faculty of Social Sciences > Philosophy | ||||
| Journal or Publication Title: | Notre Dame Journal of Formal Logic | ||||
| Publisher: | Duke University Press | ||||
| ISSN: | 0029-4527 | ||||
| Official Date: | 2014 | ||||
| Dates: |
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| Volume: | Volume 55 | ||||
| Number: | Number 2 | ||||
| Page Range: | pp. 157-196 | ||||
| DOI: | 10.1215/00294527-2420636 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access |
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