Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T19:55:38.940Z Has data issue: false hasContentIssue false
  • English
  • Français

SMALL INFINITARY EPISTEMIC LOGICS

Published online by Cambridge University Press:  01 February 2019

TAI-WEI HU ,
MAMORU KANEKO  and
NOBU-YUKI SUZUKI
TAI-WEI HU*
Affiliation:
Department of Economics, University of Bristol
MAMORU KANEKO*
Affiliation:
Faculty of Political Science and Economics, Waseda University
NOBU-YUKI SUZUKI*
Affiliation:
Department of Mathematics, Faculty of Science, Shizuoka University
*
*DEPARTMENT OF ECONOMICS UNIVERSITY OF BRISTOL BRISTOL, UK E-mail: [email protected]URL: https://taiweihu.weebly.com/
†FACULTY OF POLITICAL SCIENCE AND ECONOMICS WASEDA UNIVERSITY TOKYO, JAPAN E-mail: [email protected]URL: https://infoshako.sk.tsukuba.ac.jp/kaneko/
‡DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE SHIZUOKA UNIVERSITY SHIZUOKA, JAPAN E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a series of small infinitary epistemic logics to study deductive inference involving intra-/interpersonal beliefs/knowledge such as common knowledge, common beliefs, and infinite regress of beliefs. Specifically, propositional epistemic logics GL (Lα) are presented for ordinal α up to a given αo (αo ≥ ω) so that GL(L0) is finitary KDn with n agents and GL(Lα) (α ≥ 1) allows conjunctions of certain countably infinite formulae. GL(Lα) is small in that the language is countable and can be constructive. The set of formulae Lα is increasing up to α = ω but stops at ω We present Kripke-completeness for GL(Lα) for each α ≤ ω, which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL(Lα) has a sufficient expressive power to discuss intra-/interpersonal beliefs with infinite lengths. As applications, we discuss the explicit definability of Axioms T (truthfulness), 4 (positive introspection), 5 (negative introspection), and of common knowledge in GL(Lα) Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL(Lα), α ≤ ω.

Keywords

03B4203B4591A1091A26infinitary epistemic logiccompletenessRasiowa–Sikorski lemmaTanaka–Ono lemmacommon knowledgeexplicit definabilitygame theoryrationalizability
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 4 , December 2019 , pp. 702 - 735
Copyright
Copyright © Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Alberucci, L. (2002). The Modal μ-Calculus and the Logic of Common Knowledge. Ph.D. Thesis, Universität Bern, Institut für Informatik und angewandte Mathematik.Google Scholar
Alberucci, L., Krahenbuhl, J., & Studer, T. (2014). Justifying induction on modal-formulae. Logic Journal of IGPL, 22, 805–817.CrossRefGoogle Scholar
Antonakos, E. (2016). Pairing traditional and generic common knowledge. In Artemov, S. and Nerode, A., editors. Logical Foundations of Computer Science 2016. Basel: Springer International Publishing, pp. 14–26.CrossRefGoogle Scholar
Artemov, S. (2006). Justified common knowledge. Theoretical Computer Science, 357, 4–22.CrossRefGoogle Scholar
Bernheim, D. (1984). Rationalizable strategic behavior. Econometrica, 52, 1007–1028.CrossRefGoogle Scholar
Brünnler, K. & Studer, T. (2012). Syntactic cut-elimination for a fragment of the modal mu-calculus. Annals of Pure and Applied Logic, 163, 1838–1853.CrossRefGoogle Scholar
Enqvist, S., Seifan, F., & Venema, Y. (2016). Completeness for the μ-modal calculus: Separating the combinatorics from the dynamics. ILLC Preprint series 2016–33, U. Amsterdam. Available at https://www.illc.uva.nl/Research/Publications/Reports/PP-2016-33.text-Aug-2016.pdf.Google Scholar
Fagin, R., Halpern, J. Y., Moses, Y., & Verdi, M. Y. (1995). Reasoning about Knowledge. Cambridge: The MIT Press.Google Scholar
Fontaine, G. (2008). Continuous fragment of the μ-calculus. In Kaminski, M. and Martini, S., editors. Computer Science Logic, Vol. 5213. Heidelberg: Springer-Verlag, pp. 139–153.CrossRefGoogle Scholar
Halmos, P. R. (1963). Lectures on Boolean Algebras. Amsterdam: Van Nostrand.Google Scholar
Halpern, J. Y., Samet, D., & Segev, E. (2009). Defining knowledge in terms of belief: The modal logic perspective. Review of Symbolic Logic, 2, 469–487.CrossRefGoogle Scholar
Heifetz, A. (1999). Iterative and fixed point common belief. Journal of Philosophical Logic, 28, 61–79.CrossRefGoogle Scholar
Hu, T.-W. (2007). On p-rationalizability and approximate common certainty of rationality. Journal of Economic Theory, 136, 379–391.CrossRefGoogle Scholar
Hu, T.-W. & Kaneko, M. (2014). Game Theoretic Decidability and Undecidability, WINPEC Working Paper Series No. E1410. Available at http://www.waseda.jp/fpse/winpec/assets/uploads/2015/09/No.E1410Tai-Wei-Hu-and-Mamoru-Kaneko.pdf.Google Scholar
Hughes, G. E. & Cresswell, M. J. (1984). Companion to Modal Logic. London: Routledge.Google Scholar
Jäger, G., Kretz, M., & Studer, T. (2008). Canonical completeness of infinitary μ. Journal of Logic and Algebraic Programming , 76, 270–292.CrossRefGoogle Scholar
Jäger, G. & Studer, T. (2011). A Buchholz rule for modal fixed point logics. Log. Univers. 5, 1–19.CrossRefGoogle Scholar
Kaneko, M. (2002). Epistemic logics and their game theoretical applications: Introduction. Economic Theory, 19, 7–62.CrossRefGoogle Scholar
Karp, C. (1964). Languages with Expressions of Infinite Lengths. Amsterdam: North-Holland.Google Scholar
Kaneko, M. & Nagashima, T. (1996). Game logic and its applications I. Studia Logica, 57, 325–354.CrossRefGoogle Scholar
Kaneko, M. & Nagashima, T. (1997). Game logic and its applications II. Studia Logica, 58, 273–303.CrossRefGoogle Scholar
Kaneko, M. & Suzuki, N.-Y. (2003). Epistemic models of shallow depths and decision making in games: Horticulture. Journal of Symbolic Logic, 68, 163–186.CrossRefGoogle Scholar
Mendelson, E. (1970). Boolean Algebra and Switching Circuits. New York: McGraw-Hill Company.Google Scholar
Mendelson, E. (1988). Introduction to Mathematical Logic. Monterey: Wadsworh.Google Scholar
Meyer, J.-J. C. & van der Hoek, W. (1995). Epistemic Logic for AI and Computer Science. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Nash, J. F. (1951). Non-cooperative games. Annals of Mathematics, 54, 286–295.CrossRefGoogle Scholar
Osborne, M. & Rubinstein, A. (1994). A Course in Game Theory. Cambridge: MIT Press.Google Scholar
Pearce, D. (1984). Rationalizable strategic behavior and the problem of perfection. Econometrica, 52, 1029–1050.CrossRefGoogle Scholar
Rasiowa, H. & Sikorski, R. (1963). The Mathematics of Metamathematics. Warszawa: Państwowe Wydawnictwo Naukowe.Google Scholar
Sato, M. (1977). A study of kripke-type models for some modal logics by gentzen’s sequential method. Publication RIMS, 13, 381–468.CrossRefGoogle Scholar
Tan, T. C. & Werlang, S.-R. (1988). The bayesian foundation of solution concepts of games. Journal of Economic Theory, 45, 370–391.CrossRefGoogle Scholar
Tanaka, Y. (1999). Kripke completeness of infinitary predicate multi-modal logics. Notre Dame Journal of Formal Logic, 40, 326–340.CrossRefGoogle Scholar
Tanaka, Y. & Ono, H. (2001). Rasiowa-Sikorski lemma and Kripke completeness of predicate and infinitary modal logics. In Zakharyaschev, M. I., Segerberg, K., de Rijke, M. and Wansing, K., editors. Advances in Modal Logic, Vol. 2. Stanford: CSLI Publications, pp. 419–437.Google Scholar