Common Belief Revisited
This work addresses a foundational issue in formal logic and multi-agent systems, offering a precise characterization that resolves a long-standing open problem, though it is incremental in refining existing theoretical frameworks.
The paper tackles the problem of characterizing the logic of common belief, showing that contrary to common belief, it is not KD4 and requires an additional axiom dependent on the number of agents, providing a complete characterization that settles an open problem.
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: $C(Cφ\rightarrow φ)$ -- 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.