Common Belief Revisited
arXiv:2602.15403v1 Announce Type: new 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.
arXiv:2602.15403v1 Announce Type: new 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.
Executive Summary
The article 'Common Belief Revisited' challenges the conventional understanding of common belief in the context of knowledge and belief systems. The authors question the assumption that common belief is equivalent to KD4, a logical system that combines knowledge and belief properties. Through a rigorous analysis, they demonstrate that common belief, when individual belief is KD45, loses the 5 property but retains the D and 4 properties. Moreover, it possesses a new property, shift-reflexivity. The authors then investigate whether the extended KD4 logic with shift-reflexivity provides a complete characterization of common belief. They find that it does not and identify an additional axiom that relies on the number of agents, leading to a complete characterization of common belief.
Key Points
- ▸ The conventional understanding of common belief as equivalent to KD4 is disputed
- ▸ Common belief retains the D and 4 properties when individual belief is KD45, but loses the 5 property
- ▸ Shift-reflexivity is identified as a new property of common belief
Merits
Strength
The article provides a comprehensive and rigorous analysis of the common belief problem, challenging established assumptions and shedding new light on the topic. The authors' use of logical systems and axiomatic approaches enhances the theoretical foundations of the discussion.
Demerits
Limitation
The article's focus on formal logical systems may limit its accessibility to readers without a strong background in mathematical logic. Additionally, the reliance on specific axioms and properties may not generalize to other contexts or applications.
Expert Commentary
The article 'Common Belief Revisited' makes a significant contribution to the field of epistemic logic by challenging the conventional understanding of common belief. The authors' rigorous analysis and use of logical systems provide a deeper understanding of the properties of common belief and shed new light on its characterization. While the article's focus on formal logical systems may limit its accessibility, the findings have important implications for the design of artificial intelligence systems and policy decisions related to cooperation and collective action. Recommendations for future research include exploring the generalizability of the article's findings to other contexts and developing more practical applications of the extended KD4 logic with shift-reflexivity.
Recommendations
- ✓ Further research is needed to explore the generalizability of the article's findings to other contexts and applications.
- ✓ Developing more practical applications of the extended KD4 logic with shift-reflexivity, such as in artificial intelligence or decision-making systems, can help to realize the full potential of the article's contributions.
