Common Belief Revisited

Thomas Ågotnes

cs.AIFeb 17, 2026

Common Belief Revisited

Authors: Thomas Ågotnes

Organizations: University of Bergen and Shanxi University

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 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.

Figures & tables

Panel of Figure 1 — Figure 1: Simple KD45 model.

all instances of propositional tautologies	$Prop$
$C(\phi\rightarrow\psi)\rightarrow(C\phi\rightarrow C\psi)$	$K$
$C\phi\rightarrow\neg C\neg\phi$	$D$
$C\phi\rightarrow CC\phi$	$4$
$C(C\phi\rightarrow\phi$ )	$Cc$
$(C\bigwedge_{1\leq i<j\leq n+1}(C\phi_{i}\vee C\phi_{j}))\rightarrow\bigvee_{1\leq i\leq n+1}C\phi_{i}$	$Cn$
From $\phi\rightarrow\psi$ and $\phi$ , derive $\psi$	$MP$

Table 1: Axiomatisation

\mathrm{CB}_{n}

over the language

\mathcal{L}_{C}

Panel of Figure 2 — Figure 2: Part of the model construction. Each cluster (denoted $\cdots$ ) has incoming transitions of only one type, has internal universal access for the same type, and has one outgoing transition of the opposite type for each node in the cluster.

Epistemic Logic

Epistemic logic studies how agents know, believe, and revise information, and how those states change through communication and action.

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Feb 26, 2026J/K move · Enter open · S save

The logic of KM belief update is contained in the logic of AGM belief revision

Giacomo Bonanno

University of California, Davis, USA

Belief Revision Logical Reasoning Epistemic Logic

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For each axiom of KM belief update we provide a corresponding axiom in a modal logic containing three modal operators: a unimodal belief operator

B

, a bimodal conditional operator

>

and the unimodal necessity operator

\square

. We then compare the resulting logic to the similar logic obtained from converting the AGM axioms of belief revision into modal axioms and show that the latter contains the former. Denoting the latter by

\mathcal L_{AGM}

and the former by

\mathcal L_{KM}

we show that every axiom of

\mathcal L_{KM}

is a theorem of

\mathcal L_{AGM}

. Thus AGM belief revision can be seen as a special case of KM belief update. For the strong version of KM belief update we show that the difference between

\mathcal L_{KM}

and

\mathcal L_{AGM}

can be narrowed down to a single axiom, which deals exclusively with unsurprising information, that is, with formulas that were not initially disbelieved.

Abstract

Epistemic Logic

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