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From many-valued consequence to many-valued connectives.

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for...

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Published in:Synthese Vol. 198; pp. 5315 - 5353
Main Authors: Chemla, Emmanuel, Egré, Paul
Format: Article
Published: Springer Nature Oct2021 Supplement 22
Subjects:
Many-valued logic
Algebraic logic
Online Access:View this record in EBSCOhost
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        10.1007/s11229-019-02344-0
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        atl: From many-valued consequence to many-valued connectives.
      aug:
        au:
          Chemla, Emmanuel
          Egré, Paul
        affil:
          Laboratoire de Sciences Cognitives et Psycholinguistique, Département d'études cognitives, ENS, EHESS, CNRS, PSL University, 75005, Paris, France
          Institut Jean Nicod, Département d'études cognitives & Département de philosophie, ENS, EHESS, CNRS, PSL University, 75005, Paris, France
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        Many-valued logic
        Algebraic logic
      sug:
        subj:
          Many-valued logic
          Algebraic logic
      keyword:
        Conditionals
        Connectives
        Deduction theorem
        Logical consequence
        Mixed consequence
        Sequent calculus
        Strict-Tolerant logic
        Substructural logic
        Truth value
        Truth-functionality
      ab: Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but also on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.
      pubtype: Academic Journal
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    language: English
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