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

  • Lismont, Luc
  • Mongin, Philippe

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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Article provided by Elsevier in its journal Mathematical Social Sciences.

Volume (Year): 30 (1995)
Issue (Month): 2 (October)
Pages: 127-153

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Handle: RePEc:eee:matsoc:v:30:y:1995:i:2:p:127-153
Contact details of provider: Web page: http://www.elsevier.com/locate/inca/505565

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  1. D. Samet, 1987. "Ignoring Ignorance and Agreeing to Disagree," Discussion Papers 749, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  2. 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, June.
  3. 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.
  4. Milgrom, Paul, 1981. "An Axiomatic Characterization of Common Knowledge," Econometrica, Econometric Society, vol. 49(1), pages 219-22, January.
  5. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
  6. Modica, Salvatore & Rustichini, Aldo, 1999. "Unawareness and Partitional Information Structures," Games and Economic Behavior, Elsevier, vol. 27(2), pages 265-298, May.
  7. 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.
  8. LISMONT, Luc & MONGIN, Philippe, 1994. "On the Logic of Common Belief and Common Knowledge," CORE Discussion Papers 1994005, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  9. John Geanakoplos, 1992. "Common Knowledge," Journal of Economic Perspectives, American Economic Association, vol. 6(4), pages 53-82, Fall.
  10. Robert J Aumann, 1999. "Agreeing to Disagree," Levine's Working Paper Archive 512, David K. Levine.
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