Plausibility Logic

Origin. Friedman and Halpern (1995), Baltag and Smets (2008). Graded belief and plausibility. More/less plausible worlds. Dynamic belief revision. Foundation for epistemic plausibility.

Models. Plausibility ordering on worlds. Most plausible worlds determine belief. Conditional belief: belief given evidence. Dynamic: evidence changes ordering.

Formalism.

Plausibility ordering: ≤: W × W → Bool w ≤ v: v at least as plausible as w. Typically: total preorder per agent.

Belief operator: Bφ: agent believes φ. True iff φ true in all most plausible worlds. Min≤(W) ⊆ ⦃φ⦄

Conditional belief: B^ψ φ: agent believes φ given ψ. Most plausible ψ-worlds satisfy φ. Min≤(⦃ψ⦄) ⊆ ⦃φ⦄

Safe belief: □φ: φ true in all worlds ≤-related to current. Knowledge-like but plausibility-based.

Dynamic operators: [!ψ]φ: after hard evidence ψ, φ holds. [⇑ψ]φ: after soft upgrade with ψ, φ holds. [↑ψ]φ: after radical upgrade with ψ.

Upgrade types: Hard: eliminate non-ψ worlds. Radical: ψ-worlds become most plausible. Conservative: ψ-worlds raised, others kept.

AGM connection: Revision: [⇑ψ] models belief revision. AGM postulates satisfied. Iterated revision.

Symbols.

Symbol	Unicode	Name	Meaning
≤	U+2264	Plausibility	World ordering
B	—	Belief	Plausibility belief
B^ψ	—	Conditional	Belief given ψ
[⇑ψ]	—	Upgrade	Soft revision

Metatheory. Sound and complete. Dynamics axiomatizable. AGM connection. Decidable.

Applies to. Belief revision. Multi-agent belief. Learning. Evidence dynamics. AI reasoning.

Limitations. Ordering assumption. Hard vs soft evidence. Multiple agent complexity. Iteration subtle.