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