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A Framework for Intuitionistic Grammar Logics

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Logic and Argumentation (CLAR 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13040))

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Abstract

We generalize intuitionistic tense logics to the multi-modal case by placing grammar logics on an intuitionistic footing. We provide axiomatizations for a class of base intuitionistic grammar logics as well as provide axiomatizations for extensions with combinations of seriality axioms and what we call intuitionistic path axioms. We show that each axiomatization is sound and complete with completeness being shown via a typical canonical model construction.

Work supported by the European Research Council (ERC) Consolidator Grant 771779 (DeciGUT).

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Notes

  1. 1.

    For a discussion of these conditions and their encompassing literature, see [25, Ch. 3].

References

  1. Bierman, G.M., de Paiva, V.C.V.: On an intuitionistic modal logic. Stud. Log.: Int. J. Symb. Log. 65(3), 383–416 (2000). http://www.jstor.org/stable/20016199

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

    Article  MathSciNet  Google Scholar 

  3. Brouwer, L.E.J., Heyting, A.: L.E.J. Brouwer: Collected Works, Volume 1: Philosophy and Foundations of Mathematics. North-Holland Publishing Company, American Elsevier Publishing Company, New York (1975)

    Google Scholar 

  4. del Cerro, L.F.n., Herzig, A.: Modal deduction with applications in epistemic and temporal logics. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming (Vol. 4): Epistemic and Temporal Reasoning, pp. 499–594. Oxford University Press Inc., USA (1995)

    Google Scholar 

  5. del Cerro, L.F., Penttonen, M.: Grammar logics. Log. Anal. 31(121/122), 123–134 (1988)

    MathSciNet  MATH  Google Scholar 

  6. van Dalen, D.: Logic and Structure. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-85108-0

    Book  MATH  Google Scholar 

  7. Davies, R., Pfenning, F.: A modal analysis of staged computation. J. ACM 48(3), 555–604 (2001). https://doi.org/10.1145/382780.382785

    Article  MathSciNet  MATH  Google Scholar 

  8. Demri, S., de Nivelle, H.: Deciding regular grammar logics with converse through first-order logic. J. Log. Lang. Inf. 14(3), 289–329 (2005). https://doi.org/10.1007/s10849-005-5788-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Došen, K.: Models for stronger normal intuitionistic modal logics. Stud. Log. 44(1), 39–70 (1985)

    Article  MathSciNet  Google Scholar 

  10. Ewald, W.B.: Intuitionistic tense and modal logic. J. Symb. Log. 51(1), 166–179 (1986). http://www.jstor.org/stable/2273953

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

    MATH  Google Scholar 

  12. Fairtlough, M., Mendler, M.: An intuitionistic modal logic with applications to the formal verification of hardware. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 354–368. Springer, Heidelberg (1995). https://doi.org/10.1007/BFb0022268

    Chapter  MATH  Google Scholar 

  13. Fischer Servi, G.: Axiomatizations for some intuitionistic modal logics. Rend. Sem. Mat. Univers. Politecn. Torino 42(3), 179–194 (1984)

    MathSciNet  MATH  Google Scholar 

  14. Fitch, F.B.: Intuitionistic modal logic with quantifiers. Portugaliae Math. 7(2), 113–118 (1948). http://eudml.org/doc/114664

  15. Glivenko, V.: Sur quelques points de la logique de m. brouwer. Bull. Classe Sci. 15(5), 183–188 (1929)

    MATH  Google Scholar 

  16. Heyting, A.: Die formalen regeln der intuitionistischen logik. Sitzungsbericht PreuBische Akad. Wissenschaften Berlin Phys.-Math. Klasse II, 42–56 (1930)

    MATH  Google Scholar 

  17. Horrocks, I., Sattler, U.: Decidability of \(\cal{SHIQ}\) with complex role inclusion axioms. Artif. Intell. 160(1–2), 79–104 (2004)

    Article  Google Scholar 

  18. Howard, W.A.: The formulae-as-types notion of construction. To HB Curry: Essays Combin. Log. Lambda Calculus Formalism 44, 479–490 (1980)

    Google Scholar 

  19. Kolmogorov, A.: On the principle of tertium non datur. Math. ussr sbornik 32, 646–667 (1925). In: Van Heijenoort, J. (ed.) From Frege to Gödel: a source book in mathematical logic, 1879–1931, vol. 9. Harvard University Press (1967)

    Google Scholar 

  20. Lyon, T.: Refining labelled systems for modal and constructive logics with applications. Ph.D. thesis, Technische Universität Wien (2021)

    Google Scholar 

  21. Orlov, I.E.: The calculus of compatibility of propositions. Mathe. USSR Sbornik 35, 263–286 (1928)

    Google Scholar 

  22. Osorio, M., Navarro, J.A., Arrazola, J.: Applications of intuitionistic logic in answer set programming. Theory Pract. Log. Program. 4(3), 325–354 (2004). https://doi.org/10.1017/S1471068403001881

    Article  MathSciNet  MATH  Google Scholar 

  23. Pitts, A.M.: Evaluation logic. In: Birtwistle, G. (ed.) IV Higher Order Workshop, Banff 1990, pp. 162–189. Springer, London (1991). https://doi.org/10.1007/978-1-4471-3182-3_11

    Chapter  Google Scholar 

  24. Plotkin, G., Stirling, C.: A framework for intuitionistic modal logics: extended abstract. In: Proceedings of the 1986 Conference on Theoretical Aspects of Reasoning about Knowledge, TARK 1986, pp. 399–406. Morgan Kaufmann Publishers Inc., San Francisco (1986)

    Google Scholar 

  25. Simpson, A.K.: The proof theory and semantics of intuitionistic modal logic. Ph.D. thesis, University of Edinburgh. College of Science and Engineering. School of Informatics (1994)

    Google Scholar 

  26. Straßburger, L.: Cut elimination in nested sequents for intuitionistic modal logics. In: Pfenning, F. (ed.) FoSSaCS 2013. LNCS, vol. 7794, pp. 209–224. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-37075-5_14

    Chapter  Google Scholar 

  27. Vakarelov, D.: Abstract characterization of some knowledge representation systems and the logic nil of nondeterministic information. In: Jorrand, P., Sgurev, V. (eds.) Artificial Intelligence II, pp. 255–260. North-Holland, Amsterdam (1987)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Computational Logic Group, Institute of Artificial Intelligence, Technische Universität Dresden, Dresden, Germany

    Tim S. Lyon

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  1. Tim S. Lyon
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Correspondence to Tim S. Lyon .

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Editors and Affiliations

  1. Department of Information Engineering, University of Brescia, Brescia, Italy

    Pietro Baroni

  2. Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany

    Christoph Benzmüller

  3. Department of Philosophy (Zhuhai), Sun Yat-sen University, Zhuhai, China

    Yὶ N. Wáng

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Lyon, T.S. (2021). A Framework for Intuitionistic Grammar Logics. In: Baroni, P., Benzmüller, C., Wáng, Y.N. (eds) Logic and Argumentation. CLAR 2021. Lecture Notes in Computer Science(), vol 13040. Springer, Cham. https://doi.org/10.1007/978-3-030-89391-0_29

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  • DOI: https://doi.org/10.1007/978-3-030-89391-0_29

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