Common Belief Revisited

TL;DR

This paper revisits the logic of common belief, revealing it is not KD4 but requires an additional agent-dependent axiom for a complete characterization, settling an open problem.

Contribution

It identifies the precise axiomatic extension needed to fully characterize common belief in multi-agent systems.

Findings

Common belief is not KD4 but has shift-reflexivity.

An additional agent-dependent axiom completes the logic.

The result settles an open problem in the field.

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.