Common Belief Defies KD4: New Axioms

🔑 Key Takeaways

• •The paper 'Common Belief Defies KD4: New Axioms' by Andreas Herzig and others, published on arXiv on 2026-02-18, demonstrates that common belief under KD45 individual beliefs lacks the 5 axiom but satisfies D, 4, and a new shift-reflexivity axiom C(Cφ → φ).
• •KD4 extended with shift-reflexivity is proven incomplete for common belief, necessitating an additional axiom that depends on the number of agents n, fully axiomatizing the logic.
• •This result settles a long-standing open problem in epistemic logic dating back to the 1990s on the precise logical characterization of common belief operators.

🛠️ Technical Deep Dive

• •Individual beliefs modeled as KD45 logic: serial (D: ¬B_i φ → B_i ¬B_i φ), transitive (4: B_i φ → B_i B_i φ), Euclidean (5: B_i φ → B_i B_j φ for i≠j).
• •Common belief operator Cφ defined semantically as true at worlds where φ holds throughout all accessible worlds under the common accessibility relation R_C = ∩_{i=1}^n R_i.
• •Shift-reflexivity axiom C(Cφ → φ): Derived from R_C ⊆ ⋃_{i=1}^n R_i, ensuring reflexivity after one shift.
• •Completeness theorem: For n agents, add axiom C^n φ → Cφ (where C^k is k-fold common belief), sound and complete w.r.t. canonical model.
• •Proof techniques involve bisimulation and canonical model construction, showing no finite axiomatization independent of n.

🔮 Future ImplicationsAI analysis grounded in cited sources

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