Belief Closure - a Semantics of Common Knowledge for Modal Propositional Logic

The paper investigates the relations between iterate and fixed-point accounts of common belief and common knowledge, using the formal tools of epistemic modal logic. Its main logical contribution is to introduce and axiomatize the following (fixed-point) notion of common belief. We first define a proposition to be belief-closed if everybody believes it in every world where it is true. We then define a proposition to be common belief in a world if it is implied by a belief-closed proposition that everybody believes in that world. Using the belief closure semantics of common belief, the paper proves soundness and completeness theorems for modal logics of varying strength. The weakest system involves a monotonicity assumption on individual belief; the strongest system is based on S5. Axiomatizations of common knowledge are secured by adding the truth axiom to any system. The paper also discusses anticipations of the belief closure semantics in the economic and game-theoretic literatures.