The Cut‐Free Approach and the Admissibility‐Curry

Authors
Ulf Hlobil
Concordia University
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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