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First-Order Intuitionistic Epistemic Logic

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Logic, Rationality, and Interaction (LORI 2019)

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Abstract

Intuitionistic epistemic logic (IEL), introduced by Artemov and Protopopescu (2016), accepts the co-reflection axiom: “\(A \supset KA\)” in terms of Brouwer-Heyting-Kolmogorov interpretation. There are two variants for IEL, one of which has the axiom “\(KA \supset \lnot \lnot A\)”, while the other does not have it. The aim of this paper is to study the first-order expansions of these two IELs. Hilbert systems and sequent calculi of the first-order expansion of these two intuitionistic epistemic logic are provided to be proved sound and complete for the intended semantics. We also prove the cut-elimination theorems for both systems. Furthermore, the Craig interpolation theorems of both systems are established by Maehara’s method as consequences of cut-elimination theorems.

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References

  1. Artemov, S., Protopopescu, T.: Intuitionistic epistemic logic. Rev. Symbolic Logic 9, 266–298 (2016). https://doi.org/10.1017/S1755020315000374

    Article  MathSciNet  MATH  Google Scholar 

  2. Fitch, F.: A logical analysis of some value concepts. J. Symbolic Logic 28(2), 135–142 (1963). https://doi.org/10.2307/2271594

    Article  MathSciNet  MATH  Google Scholar 

  3. Hermant, O.: Semantic cut elimination in the intuitionistic sequent calculus. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 221–233. Springer, Heidelberg (2005). https://doi.org/10.1007/11417170_17

    Chapter  Google Scholar 

  4. Hintikka, J.: Knowledge and Belief. College Publications, London (2005)

    MATH  Google Scholar 

  5. Jäger, G., Marti, M.: A canonical model construction for intuitionistic distributed knowledge. In: Advances in Modal Logic, pp. 420–434 (2016)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Kashima, R.: Mathematical logic. Asakura Publishing Co. Ltd., Tokyo (2009). (in Japanese)

    Google Scholar 

  8. Krupski, V.N., Yatmanov, A.: Sequent calculus for intuitionistic epistemic logic IEL. In: Artemov, S., Nerode, A. (eds.) LFCS 2016. LNCS, vol. 9537, pp. 187–201. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-27683-0_14

    Chapter  Google Scholar 

  9. Ma, M., Palmigiano, A., Sadrzadeh, M.: Algebraic semantics and model completeness for intuitionistic public announcement logic. Ann. Pure Appl. Logic 165(4), 963–995 (2014)

    Article  MathSciNet  Google Scholar 

  10. Ono, H., Komori, Y.: Logics without the contraction rule. J. Symbolic Logic 50(1), 169–201 (1985)

    Article  MathSciNet  Google Scholar 

  11. Proietti, C.: Intuitionistic epistemic logic, Kripke models and Fitch’s paradox. J. Philos. Logic 41, 877–900 (2012). https://doi.org/10.1007/s10992-011-9207-1

    Article  MathSciNet  MATH  Google Scholar 

  12. Protopopescu, T.: Intuitionistic epistemology and modal logics of verification. In: van der Hoek, W., Holliday, W.H., Wang, W. (eds.) LORI 2015. LNCS, vol. 9394, pp. 295–307. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48561-3_24

    Chapter  MATH  Google Scholar 

  13. Su, Y., Sano, K.: Double negation translation in logics for knowability paradox. In: Proceedings of the 4th Asian Workshop on Philosophical Logic, pp. 3–15. Tsinghua University, Beijing, 20 October 2018

    Google Scholar 

  14. Troelstra, A., van Dalen, D.: Constructivism in Mathematics, Studies in Logic and the Foundations of Mathematics, vol. 1. Elsevier Science, Amsterdam (1988)

    Google Scholar 

  15. Williamson, T.: On intuitionistic modal epistemic logic. J. Philos. Logic 21, 63–89 (1992)

    MathSciNet  MATH  Google Scholar 

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Acknowledgement

We would like to thank the reviewers for their helpful comments. The work of all authors was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) Grant Number 19K12113. The work of the second author was partially supported also by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) Grant Number 17H02258 and JSPS Core-to-Core Program (A. Advanced Research Networks).

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Authors and Affiliations

  1. Graduate School of Letters, Hokkaido University, Sapporo, Japan

    Youan Su

  2. Faculty of Humanities and Human Sciences, Hokkaido University, Sapporo, Japan

    Katsuhiko Sano

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  1. Youan Su
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  2. Katsuhiko Sano
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Correspondence to Youan Su .

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Editors and Affiliations

  1. Roskilde University, Roskilde, Denmark

    Patrick Blackburn

  2. Université Paul Sabatier, IRIT-CNRS, Toulouse, France

    Emiliano Lorini

  3. Southwest University, Chongqing, China

    Meiyun Guo

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Su, Y., Sano, K. (2019). First-Order Intuitionistic Epistemic Logic. In: Blackburn, P., Lorini, E., Guo, M. (eds) Logic, Rationality, and Interaction. LORI 2019. Lecture Notes in Computer Science(), vol 11813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60292-8_24

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  • DOI: https://doi.org/10.1007/978-3-662-60292-8_24

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  • Print ISBN: 978-3-662-60291-1

  • Online ISBN: 978-3-662-60292-8

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