The Cut‐Free Approach and the Admissibility‐Curry

Abstract
The perhaps most important criticism of the nontransitive approach to semantic paradoxes is that it cannot truthfully express exactly which metarules preserve validity. I argue that this criticism overlooks that the admissibility of metarules cannot be expressed in any logic that allows us to formulate validity-Curry sentences and that is formulated in a classical metalanguage. Hence, the criticism applies to all approaches that do their metatheory in classical logic. If we do the metatheory of nontransitive logics in a nontransitive logic, however, there is no reason to think that the argument behind the criticism goes through. In general, asking a logic to express its own admissible metarules may not be a good idea.
Keywords naive validity  nontransitive logic  paradox  substructural approaches to paradox  v-Curry paradox
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DOI 10.1002/tht3.267
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Two Flavors of Curry's Paradox.Jc Beall & Julien Murzi - 2013 - Journal of Philosophy 110 (3):143-165.
Paradoxes and Failures of Cut.David Ripley - 2013 - Australasian Journal of Philosophy 91 (1):139 - 164.
Tolerant, Classical, Strict.Pablo Cobreros, Paul Egré, David Ripley & Robert van Rooij - 2012 - Journal of Philosophical Logic 41 (2):347-385.

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