Common Belief Defies KD4: New Axioms

🔑 Enhanced 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.
• •Historical context traces to David Lewis's 1969 work on common knowledge, extended to common belief by Aumann in 1976, with axiomatization debates persisting through works by Mocnik (1987) and van Ditmarsch et al.
• •The new axioms align with neighborhood semantics for common belief, distinguishing it from fixed-point semantics used for common knowledge (KD45).

🛠️ 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

⏳ Timeline