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From the knowability paradox to the existence of proofs
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Dean, W. and Kurokawa, H.. (2010) From the knowability paradox to the existence of proofs. Synthese, Vol.176 (No.2). pp. 177-225. ISSN 0039-7857
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Official URL: http://dx.doi.org/10.1007/s11229-009-9490-3
Abstract
The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Lof using the sort of propositional modal operators which are employed in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if phi true, then there exists a proof of phi. Building on the work of Artemov (Bull Symb Log 7(1): 1-36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science | ||||
| Divisions: | Faculty of Social Sciences > Philosophy | ||||
| Journal or Publication Title: | Synthese | ||||
| Publisher: | Springer | ||||
| ISSN: | 0039-7857 | ||||
| Official Date: | September 2010 | ||||
| Dates: |
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| Volume: | Vol.176 | ||||
| Number: | No.2 | ||||
| Number of Pages: | 49 | ||||
| Page Range: | pp. 177-225 | ||||
| Identification Number: | 10.1007/s11229-009-9490-3 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access | ||||
| URI: | http://wrap.warwick.ac.uk/id/eprint/5203 |
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