Common Belief Revisited
Thomas Ågotnes
TL;DR
This work shows that common belief in KD45 is not simply KD4; it admits a shift-reflexivity property $C(C\phi\rightarrow\phi)$ and, crucially, its complete axiomatization depends on the number of agents. By introducing two new axioms, $Cc$ and $Cn$, the authors define a family ${\rm CB}_n$ that is sound and complete for KD45$_n$ (for all $n\ge2$) in the language with a single grand-coalition common-belief operator. The completeness proof uses a canonical-model construction with a novel label-and-cluster method to simulate multi-agent transitions and a proxy-state mechanism enabled by $C2$ (the $n=2$ case of $Cn$), then extends to general $n$. A reflexive-transitive closure variant yields a corresponding S4-based result, and the approach clarifies the core properties of common belief independent of individual belief operators, with clear avenues for future work on related logics.
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: $C(Cφ\rightarrow φ)$ -- 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.