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
Bᵢ	—	Agent i believes	Indexed doxastic operator
Bⁿ	—	Graded belief	Belief to degree n
R	—	Accessibility relation	Serial, transitive, Euclidean
⊥	U+22A5	Falsum	D as ¬B⊥
→	U+2192	Material conditional	Antecedent of the K axiom

Metatheory. KD45 is sound and complete with respect to serial, transitive, Euclidean frames. D, 4 and 5 are Sahlqvist formulas, so the logic is canonical and the frame class is elementary, and completeness follows from the canonical model. Every point-generated KD45 frame is a root together with one non-empty cluster that the root sees and that sees only itself, which yields the finite model property and an exponential model bound. Single-agent KD45 satisfiability is NP-complete (Halpern and Moses 1992), as Ladner (1977) showed for S5, because the cluster structure makes a small model suffice; Halpern and Moses also showed that for two or more agents KD45ₙ is PSPACE-complete. Since T implies D, KD45 ⊊ KT45 = S5, so belief with veridicality collapses into knowledge. Stalnaker (2006) showed that in a combined system with strong belief — Bφ → BKφ — the knowledge fragment is forced to be S4.2 rather than S5.

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

Limitations. Every normal doxastic logic is logically omniscient: necessitation gives Bφ for each validity and K closes belief under implication. Worse, K agglomerates, so Bφ ∧ Bψ → B(φ ∧ ψ), and D then makes the agent's entire belief set consistent — the logic cannot represent an agent whose beliefs are jointly inconsistent, which is exactly the case belief revision exists to handle. Negative introspection, ¬Bφ → B¬Bφ, requires the agent to survey the whole of its own doxastic state, and it is what forces the root-plus-cluster frame shape. The system is static: it has no operator for belief change, the AGM postulates are not derivable in it, and iterated revision needs a plausibility ordering the language does not contain. Graded belief is not expressible — Bⁿ is no part of KD45.