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Belief Closure : A Semantics of Common Knowledge for Modal Propositional Logic

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  • LISMONT, Luc

    (G.R.E.Q.U.E; Ecole 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)

Abstract

The paper axiomatizes individual and common belief by means of modal propositional logic systems of varying strength. The weakest system of all just requires the monotonicity of individual belief on top of the axiom and rules of common belief. It is proved to be sound and complete with respect to a specially devised variant of neighbourhood semantiC's. The remaining systems include a K-system for each individual. They are shown to be sound and complete with respect to suitable variants of Kripke semantics. The specific features of either neighbourhood or Kripke semantics in this paper relate to the validation clause for common belief. Informally, we define a proposition to be belief closed if everybody believes it at every world where it is true, and we define a proposition to be common belief at a world if it is implied by a belief closed proposition that everybody believes at that particular world. This "fixed-point" or "circular" account of common belief is seen to imply the more standard "iterate" account in terms of countably infinite sequences of share beliefs. Axiomatizations of common knowledge can be secured by adding the truth axiom of individual belief to any system. The paper also briefly discusses game-theoretic papers which anticipated the belief closure semantics.

Suggested Citation

  • LISMONT, Luc & MONGIN, Philippe, 1993. "Belief Closure : A Semantics of Common Knowledge for Modal Propositional Logic," LIDAM Discussion Papers CORE 1993039, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:1993039
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. Stephen Morris & Hyun Song Shin, "undated". "Approximate Common Knowledge and Co-ordination: Recent Lessons from Game Theory," CARESS Working Papres 97-8, University of Pennsylvania Center for Analytic Research and Economics in the Social Sciences.
    3. Giacomo Bonanno & Klaus Nehring, "undated". "Introduction To The Semantics Of Belief And Common Belief," Department of Economics 97-19, California Davis - Department of Economics.
    4. Giacomo Bonanno & Klaus Nehring, "undated". "Intersubjective Consistency Of Knowledge And Belief," Department of Economics 98-03, California Davis - Department of Economics.
    5. Giacomo Bonanno & Klaus Nehring, "undated". "Agreeing To Disagree: A Survey," Department of Economics 97-18, California Davis - Department of Economics.
    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. 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.
    8. Jean Baccelli & Marcus Pivato, 2021. "Philippe Mongin (1950–2020)," Theory and Decision, Springer, vol. 90(1), pages 1-9, February.
    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. Giacomo Bonanno & Klaus Nehring, "undated". "Intersubjective Consistency Of Knowledge And Belief," Department of Economics 98-03, California Davis - Department of Economics.
    11. Colombetti, Marco, 1999. "A modal logic of intentional communication," Mathematical Social Sciences, Elsevier, vol. 38(2), pages 171-196, September.
    12. Heifetz, Aviad, 1996. "Common belief in monotonic epistemic logic," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 109-123, October.

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