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A RECOVERY OPERATOR FOR NONTRANSITIVE APPROACHES

Published online by Cambridge University Press:  06 November 2018

EDUARDO ALEJANDRO BARRIO ,
FEDERICO PAILOS  and
DAMIAN SZMUC
EDUARDO ALEJANDRO BARRIO*
Affiliation:
University of Buenos Aires and IIF-SADAF (CONICET)
FEDERICO PAILOS*
Affiliation:
University of Buenos Aires and IIF-SADAF (CONICET)
DAMIAN SZMUC*
Affiliation:
University of Buenos Aires and IIF-SADAF (CONICET)
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BUENOS AIRES PUAN 480, C1420 BUENOS AIRES ARGENTINA and IIF-SADAF (CONICET) BULNES 642, C1176ABL BUENOS AIRES ARGENTINA E-mail: eabarrio@gmail.com
†DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BUENOS AIRES PUAN 480, C1420 BUENOS AIRES ARGENTINA and IIF-SADAF (CONICET) BULNES 642, C1176ABL BUENOS AIRES ARGENTINA E-mail: fpailos@hotmail.com
‡DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BUENOS AIRES PUAN 480, C1420 BUENOS AIRES ARGENTINA and IIF-SADAF (CONICET) BULNES 642, C1176ABL BUENOS AIRES ARGENTINA E-mail: szmucdamian@gmail.com
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Abstract

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities.

Keywords

03B0503B4703B53substructural logicscut rulerecovery operatorparadoxes
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 1 , March 2020 , pp. 80 - 104
Copyright
Copyright © Association for Symbolic Logic 2018 

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