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Paradoxicality Without Paradox

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Abstract

It is not uncommon among theorists favoring a deviant logic on account of the semantic paradoxes to subscribe to an idea that has come to be known as ‘classical recapture’. The main thought underpinning it is that non-classical logicians are justified in endorsing many instances of the classically valid principles that they reject. Classical recapture promises to yield an appealing pair of views: one can attain naivety for semantic concepts while retaining classicality in ordinary domains such as mathematics. However, Julien Murzi and Lorenzo Rossi have recently suggested that revisionary approaches to truth breed revenge paradoxes when they are coupled with the thought that classical reasoning can be recaptured in certain circumstances. What’s novel about the paradoxes they put forward is that they cannot be dismissed so easily. The concepts used to generate these paradoxes—those of paradoxicality and unparadoxicality—are concepts that non-classical theorists need in order to offer a diagnosis of the truth-theoretic paradoxes. My goal in this paper is to argue that non-classical theorists can represent the concept of paradoxicality without falling prey to revenge paradoxes. In particular, I will show how to provide a formal fixed-point semantics for a language extended with a paradoxicality predicate that adequately expresses the non-classical logician’s notion of paradoxicality.
Erkenntnis (2023) 88:1347–1366
https://doi.org/10.1007/s10670-021-00405-w
ORIGINAL RESEARCH
Paradoxicality Without Paradox
Lucas Rosenblatt1
Received: 9 October 2020 / Accepted: 18 March 2021 / Published online: 30 May 2021
© The Author(s), under exclusive licence to Springer Nature B.V. 2021
Abstract
It is not uncommon among theorists favoring a deviant logic on account of the semantic
paradoxes to subscribe to an idea that has come to be known as ‘classical recap-
ture’. The main thought underpinning it is that non-classical logicians are justified in
endorsing many instances of the classically valid principles that they reject. Classi-
cal recapture promises to yield an appealing pair of views: one can attain naivety for
semantic concepts while retaining classicality in ordinary domains such as mathemat-
ics. However, Julien Murzi and Lorenzo Rossi have recently suggested that revisionary
approaches to truth breed revenge paradoxes when they are coupled with the thought
that classical reasoning can be recaptured in certain circumstances. What’s novel about
the paradoxes they put forward is that they cannot be dismissed so easily. The concepts
used to generate these paradoxes—those of paradoxicality and unparadoxicality—are
concepts that non-classical theorists need in order to offer a diagnosis of the truth-
theoretic paradoxes. My goal in this paper is to argue that non-classical theorists can
represent the concept of paradoxicality without falling prey to revenge paradoxes. In
particular, I will show how to provide a formal fixed-point semantics for a language
extended with a paradoxicality predicate that adequately expresses the non-classical
logician’s notion of paradoxicality.
A talk based on an earlier version of this paper was given in 2020 at the SeloI online seminar. I am
grateful to all its participants for a very stimulating discussion. Special thanks go to Eduardo Barrio, Jonas
Becker Arenhart, Pablo Cobreros, Bruno Da Ré, Pepe Martínez, Federico Pailos, and Damián Szmuc. I
also owe thanks to Camila Gallovich for many useful conversations about paradoxicality. Finally, I am
deeply indebted to Julien Murzi and Lorenzo Rossi not only for numerous comments and various
illuminating exchanges on the contents of the paper, but also for their encouragement and
open-mindedness throughout the process of writing it. Financial support for this work was provided by the
project “Logic and Substructurality” (FFI2017-84805-P), funded by the Spanish MINECO (Ministerio de
Economía, Industria y Competitividad).
BLucas Rosenblatt
l_rosenblatt@hotmail.com
1CONICET, Buenos Aires, Argentina
123
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Chapter
Is it possible to provide a theory of truth that is capable of distinguishing the semantic status of paradoxical sentences from that of other ungrounded sentences without bringing meta-linguistic resources into play? We explore an account that extends Kripke’s theory of truth with two primitive operators, one standing for the notion of paradoxicality and the other for the notion of hypodoxicality. Our results are mixed. While the paradoxicality operator behaves nicely, a number of restrictions need to be imposed to accommodate the hypodoxicality operator.
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Article
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