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ON THREE-VALUED PRESENTATIONS OF CLASSICAL LOGIC

Part of: General logic

Published online by Cambridge University Press:  11 May 2023

BRUNO DA RÉ Open the ORCID record for BRUNO DA RÉ [Opens in a new window] ,
DAMIAN SZMUC Open the ORCID record for DAMIAN SZMUC [Opens in a new window] ,
EMMANUEL CHEMLA Open the ORCID record for EMMANUEL CHEMLA [Opens in a new window]  and
PAUL ÉGRÉ Open the ORCID record for PAUL ÉGRÉ [Opens in a new window]
BRUNO DA RÉ*
Affiliation:
IIF (CONICET-SADAF) UNIVERSITY OF BUENOS AIRES BUENOS AIRES ARGENTINA E-mail: szmucdamian@gmail.com
DAMIAN SZMUC
Affiliation:
IIF (CONICET-SADAF) UNIVERSITY OF BUENOS AIRES BUENOS AIRES ARGENTINA E-mail: szmucdamian@gmail.com
EMMANUEL CHEMLA
Affiliation:
LABORATOIRE DE SCIENCES COGNITIVES ET DE PSYCHOLINGUISTIQUE DÉPARTEMENT D’ÉTUDES COGNITIVES CNRS, ENS, PSL UNIVERSITY, EHESS PARIS, FRANCE E-mail: emmanuel.chemla@ens.psl.eu
PAUL ÉGRÉ
Affiliation:
INSTITUT JEAN-NICOD DÉPARTEMENT D’ÉTUDES COGNITIVES & DÉPARTEMENT DE PHILOSOPHIE CNRS, ENS, PSL UNIVERSITY, EHESS PARIS, FRANCE E-mail: paul.egre@ens.psl.eu
*
E-mail: brunohoraciodare@gmail.com
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Abstract

Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations $st$, $ss$, $tt$, $ss\cap tt$, and $ts$, when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ the answer is trivial (no scheme works), and for $ss$ and $tt$ it is straightforward (they are the collapsible schemes, in which the middle value acts like one of the classical values). For $st$, the schemes in question are the Boolean normal schemes that are either monotonic or collapsible.

Keywords

classical logicstrict-tolerant logicsthree-valued logicslogical consequence

MSC classification

Primary: 03B05: Classical propositional logic 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 03B50: Many-valued logic
Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 23
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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