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International Workshop on Logic, Language, Information, and Computation

WoLLIC 2017: Logic, Language, Information, and Computation pp 349–357Cite as

Substructural Logics with a Reflexive Transitive Closure Modality

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10388))

Abstract

Reflexive transitive closure modalities represent a number of important notions, such as common knowledge in a group of agents or non-deterministic iteration of actions. Normal modal logics with such modalities are well-explored but weaker logics are not. We add a reflexive transitive closure box modality to the modal non-associative commutative full Lambek calculus with a simple negation. Decidability and weak completeness of the resulting system are established and extensions of the results to stronger substructural logics are discussed. As a special case, we obtain decidability and weak completeness for intuitionistic modal logic with the reflexive transitive closure box.

This work has been supported by the joint project of the German Science Foundation (DFG) and the Czech Science Foundation (GA ČR) number 16-07954J (SEGA: From shared evidence to group attitudes). The author would like to thank the anonymous reviewers for a number of suggestions, and to Adam Přenosil for reading a draft of the paper. A preliminary version of the paper was presented at the 8th International Workshop on Logic and Cognition in Guangzhou, China; the author is indebted to the audience for valuable feedback.

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Notes

  1. 1.

    Due to space limitations, we do not provide an introduction to substructural logics and their relational semantics. See [9], for example.

  2. 2.

    See [8], for example.

  3. 3.

    Stated more precisely, counterexamples to the K-axiom can be constructed if the frame property \(Syx \implies Rxxx\) fails. Similarly, counterexamples to distributivity of \(\Box \) over \(\otimes \) can be found if we have Swx and Ryzx but also \(Ry'z'w\) and \(Sy'u\) with \(y \not \le u\) for some u. Counterexamples to the converse implication can be found if a similar frame condition holds.

  4. 4.

    The reason is that if \(\Box ^{*}\varphi \in \varGamma \), then \(\varphi , \Box ^{n} \varphi \in \varGamma \) for all \(n \in \omega \) by \((*3 )\), but the converse implication cannot be established (our axiomatization is finitary).

  5. 5.

    Note that \((\varphi \otimes \psi ) \otimes \chi \rightarrow \varphi \otimes (\psi \otimes \chi )\) is valid in M if the model satisfies this frame condition. On the other hand, (5) entails the validity of \(\varphi \otimes \psi \rightarrow \psi \otimes \varphi \).

References

  1. Bílková, M., Majer, O., Peliš, M.: Epistemic logics for sceptical agents. J. Logic Comput. 26(6), 1815–1841 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buszkowski, W., Farulewski, M.: Nonassociative lambek calculus with additives and context-free languages. In: Grumberg, O., Kaminski, M., Katz, S., Wintner, S. (eds.) Languages: From Formal to Natural. LNCS, vol. 5533, pp. 45–58. Springer, Heidelberg (2009). doi:10.1007/978-3-642-01748-3_4

    Chapter  Google Scholar 

  3. Božić, M., Došen, K.: Models for normal intuitionistic modal logics. Stud. Logica. 43(3), 217–245 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. MIT Press, Cambridge (1995)

    MATH  Google Scholar 

  5. Fuhrmann, A.: Models for relevant modal logics. Stud. Logica. 49(4), 501–514 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press, Cambridge (2000)

    MATH  Google Scholar 

  7. Jäger, G., Marti, M.: Intuitionistic common knowledge or belief. J. Appl. Logic 18(C), 150–163 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mares, E.: Relevant Logic: A Philosophical Interpretation. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  9. Restall, G.: An Introduction to Substructural Logics. Routledge, London (2000)

    Book  MATH  Google Scholar 

  10. Sedlár, I.: Substructural epistemic logics. J. Appl. Non-Class. Logics 25(3), 256–285 (2015)

    Article  MathSciNet  Google Scholar 

  11. Sedlár, I.: Propositional dynamic logic with Belnapian truth values. In: Beklemishev, L., Demri, S., Máté, A. (eds.) Advances in Modal Logic 2016, pp. 503–519. College Publications, London (2016)

    Google Scholar 

  12. Sequoiah-Grayson, S.: Epistemic closure and commutative, nonassociative residuated structures. Synthese 190(1), 113–128 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Philosophy, Czech Academy of Sciences, Prague, Czech Republic

    Igor Sedlár

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  1. Igor Sedlár
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Correspondence to Igor Sedlár .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Juliette Kennedy

  2. Centro de Informática, Recife, Pernambuco, Brazil

    Ruy J.G.B. de Queiroz

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Sedlár, I. (2017). Substructural Logics with a Reflexive Transitive Closure Modality. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_25

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  • DOI: https://doi.org/10.1007/978-3-662-55386-2_25

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-55385-5

  • Online ISBN: 978-3-662-55386-2

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