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On the Logic of Common Belief and Common Knowledge

  • 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)

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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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
Date of revision:
Handle: RePEc:cor:louvco:1994005
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