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Axiomatizing Epistemic Logic of Friendship via Tree Sequent Calculus

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Logic, Rationality, and Interaction (LORI 2017)

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Abstract

This paper positively solves an open problem if it is possible to provide a Hilbert system to Epistemic Logic of Friendship (EFL) by Seligman, Girard and Liu. To find a Hilbert system, we first introduce a sound, complete and cut-free tree (or nested) sequent calculus for EFL, which is an integrated combination of Seligman’s sequent calculus for basic hybrid logic and a tree sequent calculus for modal logic. Then we translate a tree sequent into an ordinary formula to specify a Hilbert system of EFL and finally show that our Hilbert system is sound and complete for an intended two-dimensional semantics.

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Notes

  1. 1.

    By (K)-rules and (Nec)-rules for \(\Box \), \(\mathsf {F}\) and \(@_{n}\), the replacement of equivalence holds in \(\mathsf {H}\mathbf {EFL}\).

  2. 2.

    Given a set \(\varGamma \cup \{{\varphi }\}\) of formulas, we say that \(\varphi \) is deducible in \(\mathsf {H}\mathbf {EFL}\) from \(\varGamma \) if there exist finite formulas \(\psi _{1}\), \(\ldots \), \(\psi _{n} \in \varGamma \) such that \((\psi _{1}\wedge \ldots \wedge \psi _{n}) \rightarrow \varphi \) is provable in \(\mathsf {H}\mathbf {EFL}\). Then it is easy to see that the deduction theorem holds in \(\mathsf {H}\mathbf {EFL}\).

  3. 3.

    We do not need to assume that each of our models is named in the sense that each agent is named by an agent nominal.

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Acknowledgments

I would like to thank the anonymous reviewers for their careful reading of the manuscript and their many useful comments and suggestions. I presented the contents of this paper first at the 48th MLG meeting at Kaga, Ishikawa, Japan on 6th December 2013 and then at Kanazawa Workshop for Epistemic Logic and its Dynamic Extensions, Kanazawa, Japan on 22nd February 2014. I would like to thank Jeremy Seligman and Fenrong Liu for fruitful discussions of the topic. All errors, however, are mine. The work of the author was partially supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) Grant Number 15K21025 and JSPS Core-to-Core Program (A. Advanced Research Networks).

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Authors and Affiliations

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

    Katsuhiko Sano

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  1. Katsuhiko Sano
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Correspondence to Katsuhiko Sano .

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Editors and Affiliations

  1. Institute for Logic, Language and Computation, University of Amsterdam , Amsterdam, Noord-Holland, The Netherlands

    Alexandru Baltag

  2. Department of Philosophy, The University of Auckland, Auckland, Auckland, New Zealand

    Jeremy Seligman

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

    Tomoyuki Yamada

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Sano, K. (2017). Axiomatizing Epistemic Logic of Friendship via Tree Sequent Calculus. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_16

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