Belief Logic

Origin. Hintikka's epistemic logic extends to belief (1962). KD45 as standard doxastic logic. Weaker than knowledge: belief can be false. Introspection: beliefs about beliefs. Foundation for belief revision and multi-agent reasoning.

Models. Doxastic logic. Knowledge Kφ: φ true in all epistemically possible worlds. Belief Bφ: φ true in all doxastically possible worlds. Belief can be mistaken: Bφ ∧ ¬φ possible. Introspection: Bφ → BBφ, ¬Bφ → B¬Bφ.

Formalism.

KD45 system:

K: B(φ → ψ) → (Bφ → Bψ) (distribution)
D: Bφ → ¬B¬φ (consistency: no contradictory beliefs)
4: Bφ → BBφ (positive introspection)
5: ¬Bφ → B¬Bφ (negative introspection)

Compared to knowledge (S5): S5 adds: Kφ → φ (veridicality/truth). KD45 omits veridicality: belief can be false.

Semantics: Kripke model (W, R, V) where R is serial, transitive, Euclidean.

Serial (D): ∀w.∃v. wRv (always has doxastic alternative)
Transitive (4): R ∘ R ⊆ R
Euclidean (5): wRv ∧ wRu → vRu

Multi-agent belief: Bᵢφ: agent i believes φ. BᵢBⱼφ: i believes j believes φ. Common belief: CBφ = EBφ ∧ EBEBφ ∧ ...

Graded belief: Bⁿφ: believe φ to degree n. Ranking functions: plausibility ordering.

Belief vs knowledge: Bφ ∧ φ: true belief. Bφ ∧ JBφ ∧ φ: justified true belief (challenged by Gettier).

Symbols.

Symbol	Unicode	Name	Meaning
B	—	Believes	Doxastic operator
K	—	Knows	Epistemic (contrast)
D	—	Consistency	¬B⊥
4	—	Positive introspection	Bφ → BBφ
5	—	Negative introspection	¬Bφ → B¬Bφ
CB	—	Common belief	Iterated mutual belief

Metatheory. KD45 complete for serial, transitive, Euclidean frames. Decidable (PSPACE). Weaker than S5. Compatible with false beliefs. Introspection controversial philosophically.

Applies to. Epistemic logic foundations. Multi-agent systems. Belief revision. Game theory. Philosophy of mind. AI agents. Dialogue systems.

Limitations. Logical omniscience: agents believe all consequences. Introspection may be too strong. Single belief operator idealized. Belief change needs dynamics. Graded belief not standard.