Notre Dame Journal of Formal Logic

Disarming a Paradox of Validity

Hartry Field

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Abstract

Any theory of truth must find a way around Curry’s paradox, and there are well-known ways to do so. This paper concerns an apparently analogous paradox, about validity rather than truth, which JC Beall and Julien Murzi (“Two flavors of Curry’s paradox”) call the v-Curry. They argue that there are reasons to want a common solution to it and the standard Curry paradox, and that this rules out the solutions to the latter offered by most “naive truth theorists.” To this end they recommend a radical solution to both paradoxes, involving a substructural logic, in particular, one without structural contraction.

In this paper I argue that substructuralism is unnecessary. Diagnosing the “v-Curry” is complicated because of a multiplicity of readings of the principles it relies on. But these principles are not analogous to the principles of naive truth, and taken together, there is no reading of them that should have much appeal to anyone who has absorbed the morals of both the ordinary Curry paradox and the second incompleteness theorem.

Article information

Source
Notre Dame J. Formal Logic, Volume 58, Number 1 (2017), 1-19.

Dates
Received: 6 December 2013
Accepted: 20 March 2014
First available in Project Euclid: 17 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1479351687

Digital Object Identifier
doi:10.1215/00294527-3699865

Mathematical Reviews number (MathSciNet)
MR3595339

Zentralblatt MATH identifier
1360.03028

Subjects
Primary: 03A99: None of the above, but in this section
Secondary: 03B99: None of the above, but in this section 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52} 03F45: Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25]

Keywords
Curry paradox substructural logic validity

Citation

Field, Hartry. Disarming a Paradox of Validity. Notre Dame J. Formal Logic 58 (2017), no. 1, 1--19. doi:10.1215/00294527-3699865. https://projecteuclid.org/euclid.ndjfl/1479351687


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References

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  • [2] Beall, J. C., and J. Murzi, “Two flavors of Curry’s paradox,” Journal of Philosophy, vol. 110 (2013), pp. 143–65.
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