Abstract
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.
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Notes
See Pogorzelski (1964) and Avron (1991) for more on the deduction theorem in relation to Łukasiewicz’s three-valued conditional. Pogorzelski shows that Łukasiewicz’s conditional satisfies a more complex form of the deduction theorem relative to consequence defined as the preservation of the value 1. Avron shows that it can satisfy the deduction theorem in standard form if the definition of logical consequence is modified in a way that rules out the above counterexample, by giving up structural contraction. We note that relative to the mixed consequence relation st (introduced below), Łukasiewicz’s conditional would satisfy the deduction theorem (but not what we call premise Gentzen-regularity, unlike with Avron’s consequence).
A recent exception is Wintein (2016) looking at 3-valued and 4-valued mixed consequence relations, but not at intersective mixed relations.
Computer-aided investigations of this kind still seem quite rare, which is striking considering that some pioneers such as Foxley (1962) had bravely started deploying them for very related tasks, when much more ingenuity was needed to compensate for the lower power of computers.
One may entertain other ways to extend an order on truth values (possibly with more properties, such as the systematic presence of infimums and/or supremums) onto a truth-relation between subsets of truth values. Below are some examples, close to descriptions in Chemla et al. (2017), but which we will not attend to specifically here: \(\gamma |\!\!\equiv \delta \) iff \(\inf (\gamma )\le \sup (\delta )\), or \(\gamma |\!\!\equiv \delta \) iff \(\exists d\in \delta : \inf (\gamma )\le d\).
Given an ordering \(\le \), an upset is a set that is closed under \(\le \), namely such that y belongs to the set whenever x belongs and \(x\le y\).
Gentzen originally stated only the right-to-left direction of those rules, but it is natural to use invertible rules.
This is not to imply that a regular connective may satisfy only one regularity rule. For instance, in reflexive logics, as in classical logic, adding a conjunct of the form \(\Gamma , A \vdash A, \Delta \) to a regularity rule produces a new rule, but it is essentially the same rule and certainly it is satisfied by the same connectives.
This conditional corresponds to a three-valued version of the so-called Gödel implication, see Hájek (1998).
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Acknowledgements
We are very grateful to Benjamin Spector for providing inspiration and support to this project. We thank two anonymous referees for detailed and helpful comments. We also thank Johan van Benthem, Denis Bonnay, Keny Chatain, Roy Cook, Christian Fermüller, João Marcos, Hitoshi Omori, Francesco Paoli, David Ripley, Lorenzo Rossi, Hans Rott, Philippe Schlenker, Jan Sprenger, Shane Steinert-Threlkeld, Heinrich Wansing for helpful conversations, as well as audiences in Regensburg (workshop “New Perspectives on Conditionals and Reasoning”, organized by H. Rott and C. Michel), Beijing (Tsinghua workshop “Logical in Theoretical Philosophy”, organized by A. Jiang, F. Liu, M. Stokhof), Bochum (Logic in Bochum IV, organized by H. Omori and H. Wansing), Dagstuhl (Dagstuhl Seminar 19032 “Conditional logics and conditional reasoning”, organized by G. Aucher, P. Egré, G. Kern-Isberner, F. Poggiolesi), and Buenos Aires (VIII Workshop on Philosophical Logic, organized by E. Barrio). The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 313610, and from the ANR program “Trivalence and Natural Language Meaning” (ANR-14-CE30-0010). We also thank the Ministerio de Economía, Industria y Competitividad, Gobierno de Espana, as part of the project “Logic and substructurality” (Grant FFI2017-84805-P), as well as grant FrontCog, ANR-17-EURE-0017 for research conducted in the Department of Cognitive Studies at ENS.
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We dedicate this paper to the memory of Carolina Blasio.
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Chemla, E., Egré, P. From many-valued consequence to many-valued connectives. Synthese 198 (Suppl 22), 5315–5352 (2021). https://doi.org/10.1007/s11229-019-02344-0
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DOI: https://doi.org/10.1007/s11229-019-02344-0