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Basing Sequent Systems on Exclusive-Or

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12842))

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Abstract

In the standard Gentzen-type systems for classical logic, the right hand side of a sequent is interpreted as the inclusive-or of its elements. In this paper we investigate what happens if the exclusive-or \(\oplus \) is used instead. We provide corresponding analytic systems, and some of the decision procedures that are based on them. The latter are particularly efficient for the negation-equivalence fragment of classical logic.

This research was supported by The Israel Science Foundation (grant no. 550/19).

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Notes

  1. 1.

    For being definite, we may assume here association to the right. In practice, this would not matter, since the interpretations of \(+\) are almost always associative.

  2. 2.

    In multiplicative linear logic and in many other logics, the empty sequent is interpreted by a special propositional constant \(\bot \), such that \(\Rightarrow \) and \(\Rightarrow \bot \) are derivable from each other. In the logics we consider in this paper (as well as in plenty of others), the role of \(\bot \) can be played by any formula of the form \(\lnot \tau \), where \(\tau \) is a valid formula of the logic. In particular: we may take \(\tau =\lnot \varphi +\varphi \), since \(\lnot \varphi +\varphi \) is necessarily valid for every general consequence relation (in the sense of [2]) for which \(+\) and \(\lnot \) are internal disjunction and negation, respectively.

  3. 3.

    Recall that a rule is analytic if every formula which occurs in one of its premises is a subformula of some formula in its conclusion.

  4. 4.

    Nevertheless, for reasons that will become clear in Note 8, we prefer to leave it as one of the primitive rules of \(GCL_{\oplus }\).

  5. 5.

    In particular: to use this procedure in order to check whether a formula \(\psi \) in the language of \(\{\lnot ,\leftrightarrow \}\) is classically valid, we should first find a clause which is obtained from \(\Rightarrow \psi \) by applying backward the logical rules of the system described in Note 5. Then we should check whether that clause satisfies the two conditions given in Lemma 5. It is not hard to see that this happens iff \(\psi \) satisfies the criterion of McKinsey and Mihailescu for validity of such formulas ([10], or [4], Corollary 1).

  6. 6.

    An exact characterization of the expressive power of \(\mathcal {L}_{\oplus }\) is given in [4].

  7. 7.

    Hypersequents were independently introduced by Mints in [11], Pottinger in [12], and the author in [1] and [3]. Among other applications, they now provide the main framework for the proof theory of fuzzy logics ([9]).

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Authors and Affiliations

  1. School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Arnon Avron

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  1. Arnon Avron
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Correspondence to Arnon Avron .

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Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Anupam Das

  2. University of Genoa, Genoa, Italy

    Sara Negri

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Avron, A. (2021). Basing Sequent Systems on Exclusive-Or. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_7

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  • DOI: https://doi.org/10.1007/978-3-030-86059-2_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-86058-5

  • Online ISBN: 978-3-030-86059-2

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