On the logic of common belief and common knowledge

Theory and Decision 37 (1):75-106 (1994)
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Abstract

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge- whether individual or common- is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly 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 former 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 monotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the "infinitary" nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they are decidable in the logician's sense.

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Author's Profile

Philippe Mongin
Last affiliation: Centre National de la Recherche Scientifique

Citations of this work

Neighborhood Semantics for Modal Logic.Eric Pacuit - 2017 - Cham, Switzerland: Springer.
Doxastic Logic.Michael Caie - 2019 - In Jonathan Weisberg & Richard Pettigrew (eds.), The Open Handbook of Formal Epistemology. PhilPapers Foundation. pp. 499-541.
Does optimization imply rationality?Philippe Mongin - 2000 - Synthese 124 (1-2):73 - 111.

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References found in this work

Convention: A Philosophical Study.David Kellogg Lewis - 1969 - Cambridge, MA, USA: Wiley-Blackwell.
Knowledge and belief.Jaakko Hintikka - 1962 - Ithaca, N.Y.,: Cornell University Press.

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