Synthese

, Volume 176, Issue 2, pp 177–225 | Cite as

From the Knowability Paradox to the existence of proofs

Article

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-Löf 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\({\varphi}\)is true, then there exists a proof of\({\varphi}\). 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.

Keywords

Knowability Paradox Fitch Verificationism Intuitionistic logic BHK interpretation Existence predicate Logic of proofs Potential proof Bivalence 

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Graduate Center, City University of New YorkNew YorkUSA
  2. 2.Department of PhilosophyUniversity of WarwickCoventryUK

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