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SUSZKO’S PROBLEM: MIXED CONSEQUENCE AND COMPOSITIONALITY

Published online by Cambridge University Press:  15 February 2019

EMMANUEL CHEMLA  and
PAUL ÉGRÉ
EMMANUEL CHEMLA*
Affiliation:
École normale supérieure, PSL University, EHESS, CNRS
PAUL ÉGRÉ*
Affiliation:
École normale supérieure, PSL University, EHESS, CNRS
*
*LABORATOIRE DE SCIENCES COGNITIVES ET PSYCHOLINGUISTIQUE DÉPARTEMENT D’ÉTUDES COGNITIVES ENS, PSL UNIVERSITY, EHESS, CNRS 75005 PARIS, FRANCE E-mail: chemla@ens.fr
†INSTITUT JEAN NICOD DÉPARTEMENT D’ÉTUDES COGNITIVES & DÉPARTEMENT DE PHILOSOPHIE ENS, PSL UNIVERSITY, EHESS, CNRS 75005 PARIS, FRANCE E-mail: paul.egre@ens.fr
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Abstract

Suszko’s problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski’s structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this article we give a more systematic perspective on Suszko’s problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that (intersective) mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those theorems to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos, and Wansing, for logics with various structural properties (reflexivity, transitivity, none, or both). We strengthen these results into exact rank results for nonpermeable logics (roughly, those which distinguish the role of premises and conclusions). We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties.

Keywords

03B4703B5003G27Suszko’s thesistruth valuelogical consequencemixed consequencecompositionalitytruth-functionalitymany-valued logicalgebraic logicsubstructural logicsregular connectives
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 4 , December 2019 , pp. 736 - 767
Copyright
Copyright © Association for Symbolic Logic 2019 

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