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On the Logic of Common Belief and Common Knowledge

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  • LISMONT, Luc

    (G.R.E.Q.E., Ecoles 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 and Università di Palermo)

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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Bibliographic Info

Paper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 1994005.

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Date of creation: 01 Jan 1994
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Handle: RePEc:cor:louvco:1994005

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Related research

Keywords: common belief; common knowledge; Kripke structures; neighbourhood structures; partitional model; modal propositional logic; epistemic logic;

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Cited by:
  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. Mongin, P., 1998. "Does Optimization Imply Rationality?," Papers 9817, Paris X - Nanterre, U.F.R. de Sc. Ec. Gest. Maths Infor..
  3. Heifetz, Aviad, 1996. "Common belief in monotonic epistemic logic," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 109-123, October.
  4. LISMONT, Luc & MONGIN, Philippe, 1993. "Belief Closure : A Semantics of Common Knowledge for Modal Propositional Logic," CORE Discussion Papers 1993039, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  5. Kaneko, Mamoru, 1999. "Epistemic considerations of decision making in games," Mathematical Social Sciences, Elsevier, vol. 38(2), pages 105-137, September.
  6. Klaus Nehring & Giacomo Bonanno & Massimiliano Marcellino, 2003. "Varieties of Interpersonal Compatibility of Beliefs," Working Papers 992, University of California, Davis, Department of Economics.
  7. Olivier Gossner & Elias Tsakas, 2012. "Reasoning-based introspection," Theory and Decision, Springer, vol. 73(4), pages 513-523, October.
  8. Giacomo Bonanno & Klaus Nehring, . "Intersubjective Consistency Of Knowledge And Belief," Department of Economics 98-03, California Davis - Department of Economics.

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