Disarming a paradox of validity.

*(English)*Zbl 1360.03028Summary: 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 J. Beall and J. Murzi [“Two flavors of Curry’s paradox”, J. Philos. 110, No. 3, 143–165 (2013; doi:10.5840/jphil2013110336 ] call the \(\mathrm{v}\)-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 “\(\mathrm{v}\)-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.

}In this paper I argue that substructuralism is unnecessary. Diagnosing the “\(\mathrm{v}\)-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.

##### MSC:

03A05 | Philosophical and critical aspects of logic and foundations |

03B47 | Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) |

03F45 | Provability logics and related algebras (e.g., diagonalizable algebras) |

03F52 | Proof-theoretic aspects of linear logic and other substructural logics |