IDEAS home Printed from https://ideas.repec.org/p/cor/louvco/1993039.html
   My bibliography  Save this paper

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp1993.html
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Bacharach, Michael, 1985. "Some extensions of a claim of Aumann in an axiomatic model of knowledge," Journal of Economic Theory, Elsevier, vol. 37(1), pages 167-190, October.
    2. Kaneko, Mamoru & Nagashima, Takashi, 1991. "Final decisions, the Nash equilibrium and solvability in games with common knowledge of logical abilities," Mathematical Social Sciences, Elsevier, vol. 22(3), pages 229-255, December.
    3. Lismont, L. & Mongin, P., 1991. "La connaissance commune: une sémantique pour la logique modale," LIDAM Reprints CORE 1093, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Samet, Dov, 1990. "Ignoring ignorance and agreeing to disagree," Journal of Economic Theory, Elsevier, vol. 52(1), pages 190-207, October.
    5. Ronald Fagin & Joseph Y. Halpern & Yoram Moses & Moshe Y. Vardi, 2003. "Reasoning About Knowledge," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262562006, April.
    6. 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).
    7. MONGIN, Philippe, 1992. "Some connections between epistemic logic and the theory of nonadditive probability," LIDAM Discussion Papers CORE 1992048, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. Milgrom, Paul, 1981. "An Axiomatic Characterization of Common Knowledge," Econometrica, Econometric Society, vol. 49(1), pages 219-222, January.
    9. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
    10. MERTENS, Jean-François & ZAMIR, Shmuel, 1985. "Formulation of Bayesian analysis for games with incomplete information," LIDAM Reprints CORE 608, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    11. Robert J Aumann, 1999. "Agreeing to Disagree," Levine's Working Paper Archive 512, David K. Levine.
    12. John Geanakoplos, 1992. "Common Knowledge," Journal of Economic Perspectives, American Economic Association, vol. 6(4), pages 53-82, Fall.
    13. Modica, Salvatore & Rustichini, Aldo, 1999. "Unawareness and Partitional Information Structures," Games and Economic Behavior, Elsevier, vol. 27(2), pages 265-298, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. John Geanakoplos, 1993. "Common Knowledge," Cowles Foundation Discussion Papers 1062, Cowles Foundation for Research in Economics, Yale University.
    3. Giacomo Bonanno & Klaus Nehring, "undated". "Agreeing To Disagree: A Survey," Department of Economics 97-18, California Davis - Department of Economics.
    4. Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
    5. Áron Tóbiás, 2023. "Cognitive limits and preferences for information," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 221-253, June.
    6. Guilhem Lecouteux, 2018. "Bayesian game theorists and non-Bayesian players," The European Journal of the History of Economic Thought, Taylor & Francis Journals, vol. 25(6), pages 1420-1454, November.
    7. 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.
    8. Fukuda, Satoshi, 2019. "Epistemic foundations for set-algebraic representations of knowledge," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 73-82.
    9. Bernard Walliser, 1991. "Logique épistémique et théorie des jeux," Revue Économique, Programme National Persée, vol. 42(5), pages 801-832.
    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. Fukuda, Satoshi, 2024. "The existence of universal qualitative belief spaces," Journal of Economic Theory, Elsevier, vol. 216(C).
    12. Zimper, Alexander, 2009. "Half empty, half full and why we can agree to disagree forever," Journal of Economic Behavior & Organization, Elsevier, vol. 71(2), pages 283-299, August.
    13. Shyam Sunder, 2001. "Knowing What Others Know: Common Knowledge, Accounting, and Capital Markets," Yale School of Management Working Papers ysm326, Yale School of Management, revised 01 Feb 2002.
    14. Heifetz, Aviad & Meier, Martin & Schipper, Burkhard C., 2013. "Unawareness, beliefs, and speculative trade," Games and Economic Behavior, Elsevier, vol. 77(1), pages 100-121.
    15. Áron Tóbiás, 2021. "Meet meets join: the interaction between pooled and common knowledge," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(4), pages 989-1019, December.
    16. Satoshi Fukuda, 2024. "On the consistency among prior, posteriors, and information sets," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 78(2), pages 521-565, September.
    17. Billot, Antoine & Walliser, Bernard, 1999. "Epistemic properties of knowledge hierarchies," Journal of Mathematical Economics, Elsevier, vol. 32(2), pages 185-205, October.
    18. Heifetz, Aviad & Samet, Dov, 1999. "Hierarchies of knowledge: an unbounded stairway," Mathematical Social Sciences, Elsevier, vol. 38(2), pages 157-170, September.
    19. Dominiak, Adam & Lefort, Jean-Philippe, 2015. "“Agreeing to disagree” type results under ambiguity," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 119-129.
    20. Halpern, Joseph Y. & Rego, Leandro Chaves, 2008. "Interactive unawareness revisited," Games and Economic Behavior, Elsevier, vol. 62(1), pages 232-262, January.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cor:louvco:1993039. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Alain GILLIS (email available below). General contact details of provider: https://edirc.repec.org/data/coreebe.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.