Autoepistemic Logic

Origin. Robert Moore introduced autoepistemic logic (1985) to formalize reasoning about one's own beliefs. An ideally rational agent reasoning about what they know and don't know. Non-monotonic: "I don't believe P, so P is probably false." Related to default logic but based on belief rather than defaults. Foundation for introspective reasoning in AI.

Models. An ideally rational agent's beliefs about their own beliefs. The agent believes P if P follows from their base beliefs plus introspection about what they believe. "If I believed P, I would believe Q" — but I don't believe P, so I don't believe Q. Self-referential but grounded in stable states of belief.

Formalism.

Language: Propositional logic extended with modal operator L (or B):

Lφ: "I believe φ" / "φ is believed"

Intended interpretation: An ideally rational agent with:

Perfect introspection: knows what they believe and don't believe
Logical closure: believes all consequences of their beliefs
No beliefs beyond what's warranted

Stable expansions: Given a set of premises A, a stable expansion E is a set of beliefs such that: E = Cn(A ∪ {Lφ : φ ∈ E} ∪ {¬Lφ : φ ∉ E})

E is what you'd believe if you:

Start with premises A
Correctly assess what's in E (Lφ for believed, ¬Lφ for not believed)
Close under logical consequence

This is a fixed point: E assumes knowledge of E.

Example: A = {¬Lp → q} "If I don't believe p, then q."

Stable expansion: If p ∉ E, then ¬Lp ∈ E, so q ∈ E. But there's no reason for p ∈ E. So E contains q and ¬Lp.

Multiple or no expansions: A = {Lp → p} Two stable expansions: one with p, one without.

A = {¬Lp → p, ¬p} No stable expansion (contradiction either way).

Symbols.

Symbol	Unicode	Name	Meaning
L	—	Belief operator	"I believe that"
B	—	Belief (alt)	Alternative notation
¬L	—	Disbelief	"I don't believe that"
Cn	—	Consequence	Logical closure
E	—	Expansion	Stable belief set
⊢	U+22A2	Derives	Within expansion

Metatheory. Determining if a formula is in some/all stable expansions is Σ₂ᵖ/Π₂ᵖ-complete. Autoepistemic logic is related to default logic; translations exist. Stable expansions correspond to models of a modal logic (KD45 + introspection). Every default theory can be translated to autoepistemic logic. Not all autoepistemic theories have stable expansions.

Applies to. Introspective AI agents. Formalizing self-knowledge. Knowledge representation with closed-world reasoning. Agent belief dynamics. Meta-reasoning systems.

Limitations. Ideal rationality is unrealistic — real agents aren't logically omniscient. Multiple expansions represent genuine ambiguity but complicate reasoning. Non-existence of expansions is problematic. The fixed-point definition is circular; finding expansions is hard. Less directly usable than default logic for many applications.