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On the Decidability of Certain Semi-Lattice Based Modal Logics

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10501))

Abstract

Sequent calculi are proof systems that are exceptionally suitable for proving the decidability of a logic. Several relevance logics were proved decidable using a technique attributable to Curry and Kripke. Further enhancements led to a proof of the decidability of implicational ticket entailment by Bimbó and Dunn in [12, 13]. This paper uses a different adaptation of the same core proof technique to prove a group of positive modal logics (with disjunction but no conjunction) decidable.

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Notes

  1. 1.

    See for example Dunn [20] and Kripke [27].

  2. 2.

    We may use other letters than \(\varphi \), from the latter part of the Greek alphabet, as variables for formulas.

  3. 3.

    In this paper, we only have use for finite multisets; thus, we use the term in a narrower sense than it is used elsewhere in the literature.

  4. 4.

    A quick approximation suggests that there are 89 logics that can be expected to be distinct.

  5. 5.

    \(\times \) excludes a pair of rules; shows that the rules are easily derivable, hence, it is better to omit them—for the sake of economy in proofs.

  6. 6.

    This analysis is fairly usual. For the ideas behind it and examples of it, we refer to [17] (and also to [9]).

  7. 7.

    This notion is an adaptation of a similar notion from Curry [17].

  8. 8.

    More details of a triple-inductive proof of the admissibility of the cut rule for a logic with no lattice operators may be found in [8]. Various enhancements of a more usual double-inductive proof of the cut theorem were introduced in [6, 7], where a goal was to accommodate constants like \(\textsf {Y},\textsf {y}\) and \(\varvec{t}\).

  9. 9.

    See, in chronological order, [28], [1], [22], [11], as well as [7] for motivations and logics that leave out the thinning rules from their sequent calculus formulations.

  10. 10.

    See Meyer [31] for a discussion of conceptual links that can be created between Dickson’s lemma and Kripke’s lemma.

  11. 11.

    The cut theorem is proved using the single cut in Lambek [28] and in display logics in Belnap [3] and Anderson et al. [2].

  12. 12.

    The so-called mix rule in [23] and the multicut rule explicitly stated, for example, in Dunn [18] are versions of the cut that were introduced specifically to facilitate the inductive proof of the cut theorem for the single cut. An early publication that exhibits a suitable version of cut in connection to a decidability proof using the Curry–Kripke method is [4], which is a precursor of the more readily available [5].

  13. 13.

    We defined heap numbers in a very liberal manner in order to make sure that all the necessary contractions are permitted. However, even if , for example, it may happen that in the \(L\mathfrak {X}_{3}^{\text {*}}\) logic no contraction will be applied to the formula, because it occurs on the left-hand side of the . (Similarly, but dually for .) This does not cause any problem in the proof search, because the heap number (like the notation) does not force contractions, rather, it places a limit on the number of potential applications of the contraction rules.

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Acknowledgments

I am grateful to the organizers of the TABLEAUX, FroCoS and ITP conferences for their invitation for me to speak at those conferences, which triggered the writing of this paper.

I would also like to thank the program committee for helpful comments on the first version of this paper.

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Authors and Affiliations

  1. Department of Philosophy, University of Alberta, Edmonton, AB, T6G 2E7, Canada

    Katalin Bimbó

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Correspondence to Katalin Bimbó .

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  1. University of Manchester, Manchester, United Kingdom

    Renate A. Schmidt

  2. University of Brasília, Brasília D.F., Brazil

    Cláudia Nalon

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Bimbó, K. (2017). On the Decidability of Certain Semi-Lattice Based Modal Logics. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_3

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