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: w at least as plausible as v — the ≤-minimal worlds are the most plausible. 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 radical (lexicographic) upgrade with ψ, φ holds. [↑ψ]φ: after conservative upgrade with ψ.

Upgrade types: Hard (!ψ): eliminate non-ψ worlds. Radical (⇑ψ): all ψ-worlds become more plausible than all non-ψ worlds, order within each block preserved. Conservative (↑ψ): only the best ψ-worlds are promoted to the top; the rest keeps its order.

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]	—	Radical upgrade	Lexicographic revision
⇑	U+21D1	Radical upgrade	All ψ-worlds promoted above all non-ψ worlds
↑	U+2191	Conservative upgrade	Only the best ψ-worlds promoted
[!ψ]	—	Hard update	Eliminate the non-ψ worlds
□	U+25A1	Safe belief	True at all worlds at least as plausible
⦃	U+2983	Truth-set bracket, left	Opening of ⦃φ⦄
⦄	U+2984	Truth-set bracket, right	Closing of ⦃φ⦄
Min≤	—	Minimal worlds	The ≤-least worlds of a set
⊆	U+2286	Inclusion	Min≤(W) ⊆ ⦃φ⦄ defines belief
W	—	Domain	Set of worlds
⊤	U+22A4	Verum	Trivial condition: Bφ is B^⊤φ

Metatheory. Belief and conditional belief are definable from the ordering alone — Bφ is B^⊤φ — so the static language is a conditional doxastic logic, sound and complete for total preorders. Friedman and Halpern set plausibility measures in a framework subsuming probability, possibility and ranking functions, and showed that the qualitative plausibility measures are exactly those whose induced default consequence relation satisfies the KLM postulates of System P. Safe belief is the modality of the ordering itself, so on a reflexive, transitive and connected order it is S4.3, and it sits strictly between belief and knowledge. Baltag and Smets (2008) axiomatize the dynamic operators by reduction axioms, so completeness and decidability transfer from the static base. Radical upgrade induces a revision satisfying all eight AGM postulates and, iterated, the Darwiche–Pearl postulates that AGM leaves open; conservative upgrade agrees with it on plain belief and diverges on conditional belief.

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

Limitations. The ordering is a primitive of the model, constrained only to be a total preorder, so the logic records a belief state rather than explaining one and any pattern of plausibility whatever is a model. Totality is itself a commitment: it excludes incomparable worlds, and with them the state of an agent that ranks neither of two hypotheses above the other. Minima must exist for belief to be defined, so infinite descending chains have to be ruled out by fiat. Hard, radical and conservative update are all definable and all agree on plain belief, and nothing in the theory decides which policy an agent uses. Belief is closed under consequence, so the agent is logically omniscient, and success fails for doxastic input, since upgrading with a Moore sentence falsifies it. Adding common belief breaks the reduction axioms.