Common Belief Revisited
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918342877511680 |
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| author | Ågotnes, Thomas |
| author_facet | Ågotnes, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15403 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Common Belief Revisited Ågotnes, Thomas Artificial Intelligence 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. |
| title | Common Belief Revisited |
| topic | Artificial Intelligence |
| url | https://arxiv.org/abs/2602.15403 |