Faithfulness for naive validity
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Nontransitive responses to the validity Curry paradox face a dilemma that was recently formulated by Barrio, Rosenblatt and Tajer. It seems that, in the nontransitive logic ST enriched with a validity predicate, either you cannot prove that all derivable metarules preserve validity, or you can prove that instances of Cut that are not admissible in the logic preserve validity. I respond on behalf of the nontransitive approach. The paper argues, first, that we should reject the detachment principle for naive validity. Secondly, I show how to add a validity predicate to ST while avoiding the dilemma.
KeywordsNaive validity Nontransitive logic V-Curry paradox Substructural approaches to paradox
I am very grateful for invaluable conversations with and/or comments from Jaroslav Peregrin, Robert Brandom, Daniel Kaplan, Shawn Standefer, Ori Beck, David Ripley, Katharina Nieswandt, Stephen Mackereth, Shuhei Shimamura and two anonymous referees.
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