# Common Knowledge Logic

**Origin.** Lewis (1969), Aumann (1976). Knowledge shared by all and known to be shared. Fixed point: everyone knows everyone knows... Foundation for coordination and convention.

**Models.** Kripke structures with multiple agents. Common knowledge C: infinite conjunction of nested knowledge. Fixed point characterization.

**Formalism.**

*Operators:*
Kᵢφ: agent i knows φ
Eφ: everyone knows φ (E = ∧ᵢKᵢ)
Cφ: common knowledge of φ

*Everyone knows:*
Eφ ≡ K₁φ ∧ K₂φ ∧ ... ∧ Kₙφ

*Common knowledge:*
Cφ ≡ Eφ ∧ EEφ ∧ EEEφ ∧ ...
Infinite conjunction of iterated everyone-knows.

*Fixed point:*
Cφ ↔ E(φ ∧ Cφ)
Common knowledge is greatest fixed point.

*Semantics:*
Define ~ᵢ as indistinguishability for agent i.
~_E = ∪ᵢ~ᵢ (union)
~_C = (~_E)* (transitive closure)
M, s ⊨ Cφ iff ∀t. s ~_C t implies M, t ⊨ φ

*Distributed knowledge:*
Dφ: distributed knowledge (intersection)
What agents would know if they pooled information.
~_D = ∩ᵢ~ᵢ

*Example:*
"Coordinated attack problem"
C(attack at dawn) needed for coordination.
Cannot achieve via finite message exchange.

**Symbols.**

| Symbol | Unicode | Name | Meaning |
|--------|---------|------|---------|
| Kᵢ | — | Knows | Agent i knows |
| E | — | Everyone | All know |
| C | — | Common | Common knowledge |
| D | — | Distributed | Pooled knowledge |

**Metatheory.** Axiomatizable with fixed-point rule. Complete for common knowledge. EXPTIME model checking. Not finitely axiomatizable without rule.

**Applies to.** Game theory. Conventions. Coordination problems. Distributed systems. Social epistemology.

**Limitations.** Idealized rationality. Finite communication insufficient. Common knowledge hard to achieve. Computational complexity.

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