Dynamic Doxastic Logic

Origin. Segerberg (1995), Baltag, Smets. Belief revision as action. Dynamic modalities. Plausibility models. Foundation of dynamic epistemics.

Models. Belief change as action in logic. [*φ]ψ: after revising with φ, ψ holds. Plausibility orderings. Conditional belief. Dynamic epistemic actions.

Formalism.

Plausibility models: M = (W, ≤, V). W: worlds. ≤: plausibility ordering (more plausible ≤ less). V: valuation.

Belief operator: Bφ: agent believes φ. True at w iff φ true at all most plausible worlds. Min_≤(W) ⊆ [[φ]]. Best worlds satisfy φ.

Conditional belief: B^ψ φ: believes φ given ψ. Most plausible ψ-worlds satisfy φ. Min_≤([[ψ]]) ⊆ [[φ]]. Hypothetical belief.

Revision action: [*φ]: revise with φ. Reorders plausibility. φ-worlds become most plausible. Dynamic modality.

Semantics of revision: After [*φ], new ordering ≤' where: φ-worlds more plausible than ¬φ-worlds. Within φ-worlds: old order. Within ¬φ-worlds: old order.

Reduction axioms: [*φ]p ↔ p (atoms unchanged). [*φ]¬ψ ↔ ¬[*φ]ψ. [*φ]Bψ ↔ B^φ[*φ]ψ. Recursive reduction.

Public announcement: [!φ]: truthful announcement. Different from revision. Eliminates ¬φ-worlds. Hard information.

Symbols.

Symbol	Unicode	Meaning
B	—	belief
B^φ	—	conditional belief
[*φ]	—	revision with φ
≤	U+2264	plausibility

Metatheory. Plausibility semantics. Dynamic revision. Conditional belief. Reduction axioms.

Applies to. Belief dynamics. Multi-agent epistemics. AI. Game theory.

Limitations. Plausibility source. Idealization. Complexity. Multiple revision policies.