Common Belief Revisited

Abstract

Contrary to common belief, common belief is not KD4. If individual belief is KD45, common belief does indeed lose the 5 property and keep the D and 4 properties -- and it has none of the other commonly considered properties of knowledge and belief. But it has another property: $C(C\phi \rightarrow \phi)$ -- corresponding to so-called shift-reflexivity (reflexivity one step ahead). This observation begs the question: is KD4 extended with this axiom a complete characterisation of common belief in the KD45 case? If not, what \emph{is} the logic of common belief? In this paper we show that the answer to the first question is ``no'': there is one additional axiom, and, furthermore, it relies on the number of agents. We show that the result is a complete characterisation of common belief, settling the open problem.