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.
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I’ve often been asked why, since I do adopt a nontransitive system in order to solve the paradoxes of vagueness, I don’t adopt the nontransitive system to be discussed in this paper in order to solve the semantic paradoxes instead of adopting my favoured, noncontractive system (or why, since, again, I do adopt a nontransitive system in order to solve the paradoxes of vagueness, I don’t adopt that simpler nontransitive system in order to solve those paradoxes instead of adopting my favoured, more complex nontransitive system). This paper provides (part of) my answer to those questions. Earlier versions of the material in the paper were presented in 2020 at the BA Logic Group WIP Seminar (University of Buenos Aires); in 2021, at the Conference Formal Philosophy 2021 in Moscow (Higher School of Economics) and at the 10th SLMFCE 2021 Conference (University of Salamanca); in 2022, at the Current Debates in the Philosophy of Logic Seminar (University of Padua), at the 24th Valencian Philosophy Congress (Jaume I University) and at the 13th Panhellenic Logic Symposium in Volos (University of Thessaly). I’d like to thank all these audiences for very stimulating comments and discussions. Special thanks go to Eduardo Barrio, Carlos Benito, Colin Caret, Vitalij Dolgorukov, Elena Dragalina, Filippo Ferrari, Hartry Field, Michael Glanzberg, Anil Gupta, Volker Halbach, Ulf Hlobil, Antonis Kakas, Ben Martin, José Martínez, Sergi Oms, Federico Pailos, Simone Picenni, Lucas Rosenblatt, Yannis Stephanou, Damián Szmuc, Diego Tájer, Jordi Valor and several anonymous referees. None of them should be held responsible for any remaining mistake or long footnote in the paper. I’m also grateful to the guest editors Eduardo Barrio (again!) and Paul Égré for inviting me to contribute to this special issue and for their extraordinary support and patience throughout the process. I’m additionally grateful to Paul (again!) for his extremely perceptive and fair comments on several versions of what is for him not precisely a sympathetic paper. Work on the paper was supported by the Ramón y Cajal Research Fellowship RYC-2017-22883. Additionally, support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged. I also benefited from the FCT Project PTDC/FER-FIL/28442/2017 Companion to Analytic Philosophy 2, from the Project PID2019-105746GB-I00 of the Spanish Ministry of Science and Innovation Linguistic Relativity and Experimental Philosophy, from the Project 2019PIDPID-107667GB-I00 of the Spanish Ministry of Science and Innovation Worlds and Truth Values: Challenges to Formal Semantics, from the BBVA-Foundation Project Grants for Scientific-Research Projects 2021 Unstable Metaphysics and from the FCT Project 2022.03194. PTDC New Perspectives on the Objects and Grounds of Structural Rules.
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Zardini, E. The Final Cut. J Philos Logic 51, 1583–1611 (2022). https://doi.org/10.1007/s10992-022-09682-4
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DOI: https://doi.org/10.1007/s10992-022-09682-4
Keywords
- Classical logic
- Naive truth
- Naive validity
- Nontransitive logics