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On the logic of common belief and common knowledge

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  • LISMONT, L.
  • MONGIN, P.

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.
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Suggested Citation

  • Lismont, L. & Mongin, P., 1994. "On the logic of common belief and common knowledge," LIDAM Reprints CORE 1104, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:1104
    DOI: 10.1007/978-1-4613-1139-3_1
    Note: In : (Special Issue on Logic and the Epistemic Foundations of game Theory, edited by M. Bacharach and P. Mongin), Theory and Decision, 37, 75-106, 1994
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    Cited by:

    1. Giacomo Bonanno & Klaus Nehring, "undated". "Varieties of Interpersonal Compatibility of Beliefs," Department of Economics 99-02, California Davis - Department of Economics.
    2. Olivier Gossner & Elias Tsakas, 2012. "Reasoning-based introspection," Theory and Decision, Springer, vol. 73(4), pages 513-523, October.
    3. Giacomo Bonanno & Klaus Nehring, "undated". "Intersubjective Consistency Of Knowledge And Belief," Department of Economics 98-03, California Davis - Department of Economics.
    4. Bonanno, Giacomo & Nehring, Klaus, 1998. "Assessing the truth axiom under incomplete information," Mathematical Social Sciences, Elsevier, vol. 36(1), pages 3-29, July.
    5. 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.
    6. 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.
    7. 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.
    8. Mongin, P., 1998. "Does Optimization Imply Rationality?," Papers 9817, Paris X - Nanterre, U.F.R. de Sc. Ec. Gest. Maths Infor..
    9. Bonanno, G. & Nehring, K., 1995. "Intersubjective Consistency of Beliefs and the Logic of Common Belief," Papers 95-08, California Davis - Institute of Governmental Affairs.
    10. 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.
    11. Jean Baccelli & Marcus Pivato, 2021. "Philippe Mongin (1950–2020)," Theory and Decision, Springer, vol. 90(1), pages 1-9, February.
    12. Giacomo Bonanno & Klaus Nehring, "undated". "Agreeing To Disagree: A Survey," Department of Economics 97-18, California Davis - Department of Economics.
    13. 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.
    14. Fukuda, Satoshi, 2020. "Formalizing common belief with no underlying assumption on individual beliefs," Games and Economic Behavior, Elsevier, vol. 121(C), pages 169-189.
    15. Giacomo Bonanno, 2007. "A Syntactic Approach to Rationality in Games," Working Papers 247, University of California, Davis, Department of Economics.
    16. Kaneko, Mamoru, 1999. "Epistemic considerations of decision making in games," Mathematical Social Sciences, Elsevier, vol. 38(2), pages 105-137, September.
    17. Giacomo Bonanno & Klaus Nehring, "undated". "Intersubjective Consistency Of Knowledge And Belief," Department of Economics 98-03, California Davis - Department of Economics.
    18. Heifetz, Aviad, 1996. "Common belief in monotonic epistemic logic," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 109-123, October.
    19. Giacomo Bonanno & Klaus Nehring, "undated". "Introduction To The Semantics Of Belief And Common Belief," Department of Economics 97-19, California Davis - Department of Economics.
    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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