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. (shrink)

This paper suggests a way of formalizing the amount of information that can be conveyed to each player along every possible play of an extensive game. The information given to each player i when the play of the game reaches node x is expressed as a subset of the set of terminal nodes. Two definitions are put forward, one expressing the minimum amount of information and the other the maximum amount of information that can be conveyed without violating the constraint (...) represented by the information sets. Our definitions provide intuitive characterizations of such notions as perfect recall, perfect information and simultanetty. (shrink)

We investigate an axiomatization of the notion of common belief that makes use of no rules of inference and highlight the property of the set of accessibility relations that characterizes each axiom.

The problem of Common Knowledge will be considered in two classes of models: a class K.* of Kripke models and a class S of Scott models. Two modal logic systems will be defined. Those systems, KC and MC, include an axiomatisation of Common Knowledge. We prove determination of each system by the corresponding class of models. MSC: 03B45, 68T25.