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Belief closure: A semantics of common knowledge for modal propositional logic

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  • Lismont L.
  • Mongin, P.

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.
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Suggested Citation

  • 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.
    • Handle: RePEc:eee:matsoc:v:31:y:1996:i:1:p:60a-60a
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    Cited by:

    1. Bonanno, Giacomo & Nehring, Klaus, 1998. "Assessing the truth axiom under incomplete information," Mathematical Social Sciences, Elsevier, vol. 36(1), pages 3-29, July.
    2. Giacomo Bonanno & Klaus Nehring, "undated". "Intersubjective Consistency Of Knowledge And Belief," Department of Economics 98-03, California Davis - Department of Economics.
    3. Stephen Morris & Hyun Song Shin, "undated". "Approximate Common Knowledge and Co-ordination: Recent Lessons from Game Theory," Penn CARESS Working Papers 72042421d029130510780dde2, Penn Economics Department.
    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. 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. Colombetti, Marco, 1999. "A modal logic of intentional communication," Mathematical Social Sciences, Elsevier, vol. 38(2), pages 171-196, September.
    8. Fukuda, Satoshi, 2024. "On the axiomatization of an unawareness structure from knowing-whether operators," Journal of Mathematical Economics, Elsevier, vol. 115(C).
    9. 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.
    10. Fukuda, Satoshi, 2020. "Formalizing common belief with no underlying assumption on individual beliefs," Games and Economic Behavior, Elsevier, vol. 121(C), pages 169-189.
    11. Giacomo Bonanno & Klaus Nehring, "undated". "Introduction To The Semantics Of Belief And Common Belief," Department of Economics 97-19, California Davis - Department of Economics.
    12. Heifetz, Aviad, 1996. "Common belief in monotonic epistemic logic," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 109-123, October.
    13. Jean Baccelli & Marcus Pivato, 2021. "Philippe Mongin (1950–2020)," Theory and Decision, Springer, vol. 90(1), pages 1-9, February.

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