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On the Logic of Common Belief and Common Knowledge

Author

Listed:
  • LISMONT, Luc

    (G.R.E.Q.E., Ecoles des Hautes Etudes en Sciences Sociales, Marseille)

  • MONGIN, Philippe

    (Centre National de la Recherche Scientifique (France) and CORE, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium and Università di Palermo)

Abstract

The paper surveys the currently available axioma.tizations of common belief (CB) and common knowledge (CK) by means of modal propositionallogic:s. (Throughout, knowledge - whether individual or common - is defined as true belief.) Section 1 introduces the formal method ofaxiomatization followed by epistemic logicians, especially the syntaxe-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while bridly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in sections 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r .t. the (ormer structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought - only mono tonicity has thusfar proved indispensable. Section 5 explains another consequence of general releo.-ance: despite the "infinitary" nature of CB, the axiom systems of this paper admit of effecth-e decision procedures, i.e., they are decidable in the logician's sense.

Suggested Citation

  • LISMONT, Luc & MONGIN, Philippe, 1994. "On the Logic of Common Belief and Common Knowledge," LIDAM Discussion Papers CORE 1994005, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:1994005
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    Cited by:

    1. Mongin, P., 1998. "Does Optimization Imply Rationality?," Papers 9817, Paris X - Nanterre, U.F.R. de Sc. Ec. Gest. Maths Infor..
    2. Battigalli, Pierpaolo & Bonanno, Giacomo, 1999. "Recent results on belief, knowledge and the epistemic foundations of game theory," Research in Economics, Elsevier, vol. 53(2), pages 149-225, June.
    3. Giacomo Bonanno & Klaus Nehring, "undated". "Varieties of Interpersonal Compatibility of Beliefs," Department of Economics 99-02, California Davis - Department of Economics.
    4. Giacomo Bonanno & Klaus Nehring, "undated". "Agreeing To Disagree: A Survey," Department of Economics 97-18, California Davis - Department of Economics.
    5. Olivier Gossner & Elias Tsakas, 2012. "Reasoning-based introspection," Theory and Decision, Springer, vol. 73(4), pages 513-523, October.
    6. Fukuda, Satoshi, 2020. "Formalizing common belief with no underlying assumption on individual beliefs," Games and Economic Behavior, Elsevier, vol. 121(C), pages 169-189.
    7. Giacomo Bonanno & Klaus Nehring, "undated". "Intersubjective Consistency Of Knowledge And Belief," Department of Economics 98-03, California Davis - Department of Economics.
    8. Giacomo Bonanno, 2007. "A Syntactic Approach to Rationality in Games," Working Papers 247, University of California, Davis, Department of Economics.
    9. Bonanno, Giacomo & Nehring, Klaus, 1998. "Assessing the truth axiom under incomplete information," Mathematical Social Sciences, Elsevier, vol. 36(1), pages 3-29, July.
    10. Lismont L. & Mongin, P., 1996. "Belief closure: A semantics of common knowledge for modal propositional logic," Mathematical Social Sciences, Elsevier, vol. 31(1), pages 60-60, February.
    11. Luciana C. Fiorini & José A. Rodrigues-Neto, 2014. "Self-Consistency and Common Prior in Non-Partitional Knowledge Models," ANU Working Papers in Economics and Econometrics 2014-621, Australian National University, College of Business and Economics, School of Economics.
    12. Kaneko, Mamoru, 1999. "Epistemic considerations of decision making in games," Mathematical Social Sciences, Elsevier, vol. 38(2), pages 105-137, September.
    13. Giacomo Bonanno & Klaus Nehring, "undated". "Intersubjective Consistency Of Knowledge And Belief," Department of Economics 98-03, California Davis - Department of Economics.
    14. Heifetz, Aviad, 1996. "Common belief in monotonic epistemic logic," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 109-123, October.
    15. Giacomo Bonanno & Klaus Nehring, "undated". "Introduction To The Semantics Of Belief And Common Belief," Department of Economics 97-19, California Davis - Department of Economics.
    16. Bonanno, G. & Nehring, K., 1995. "Intersubjective Consistency of Beliefs and the Logic of Common Belief," Department of Economics 95-08, California Davis - Department of Economics.
    17. Giacomo Bonanno & Klaus Nehring, "undated". "Epistemic Foundations Of Solution Concepts In Game Theory: An Introduction," Department of Economics 97-21, California Davis - Department of Economics.
    18. Bonanno, Giacomo & Nehring, Klaus, 1998. "On the logic and role of Negative Introspection of Common Belief," Mathematical Social Sciences, Elsevier, vol. 35(1), pages 17-36, January.
    19. Jean Baccelli & Marcus Pivato, 2021. "Philippe Mongin (1950–2020)," Theory and Decision, Springer, vol. 90(1), pages 1-9, February.
    20. Fiorini, Luciana C. & Rodrigues-Neto, José A., 2017. "Self-consistency, consistency and cycles in non-partitional knowledge models," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 11-21.
    21. Giacomo Bonanno & Klaus Nehring, "undated". "Varieties of Interpersonal Compatibility of Beliefs," Department of Economics 99-02, California Davis - Department of Economics.

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