The Final Cut

Abstract

In a series of works, Pablo Cobreros, Paul Égré, David Ripley and Robert van Rooij have proposed a nontransitive system (call it ‘K3LP’) as a basis for a solution to the semantic paradoxes. I critically consider that proposal at three levels. At the level of the background logic, I present a conception of classical logic on which K3LP fails to vindicate classical logic not only in terms of structural principles, but also in terms of operational ones. At the level of the theory of truth, I raise a cluster of philosophical difficulties for a K3LP-based system of naive truth, all variously related to the fact that such a system proves things that would seem already by themselves repugnant, even in the absence of transitivity. At the level of the theory of validity, I consider an extension of the K3LP-based system of naive validity that is supposed to certify that validity in that system does not fall short of naive validity, argue that such an extension is untenable in that its nontriviality depends on the inadmissibility of a certain irresistible instance of transitivity (whence the advertised “final cut”) and conclude on this basis that the K3LP-based system of naive validity cannot coherently be adopted either. At all these levels, a crucial role is played by certain metaentailments and by the extra strength they afford over the corresponding entailments: on the one hand, such strength derives from considerations that would seem just as compelling in a general nontransitive framework, but, on the other hand, such strength wreaks havoc in the particular setting of K3LP.