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:

1. Start with premises A
2. Correctly assess what's in E (Lφ for believed, ¬Lφ for not believed)
3. 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.

Metatheory. A stable expansion of A is a fixed point of E = Cn(A ∪ {Lφ : φ ∈ E} ∪ {¬Lφ : φ ∉ E}), hence a stable set in Stalnaker's sense — deductively closed, positively and negatively introspective — and so determined completely by its objective, modal-free part; the modal logic of such sets is KD45. Expansions need neither exist nor be unique: {Lp → p} has two, one containing p and one not, while {¬Lp → p, ¬p} has none, so the system supplies a semantics rather than a consequence relation with a deduction theorem. Gottlob (1992) settled the complexity: deciding membership in some expansion is Σ₂ᵖ-complete and membership in every expansion Π₂ᵖ-complete, one level above classical propositional reasoning. Konolige (1988) related the system to Reiter's default logic, and Gelfond and Lifschitz (1988) carried the same fixed-point construction over to logic programs, where it became the stable model semantics.

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

Limitations. The definition is a fixed-point condition on a completed belief set, not a derivation relation: no proof system generates an expansion step by step, and because expansions may fail to exist or fail to be unique, "follows from A" splits into credulous and sceptical readings that come apart. Expansions are deductively closed, so the agent is logically omniscient, and negative introspection commits it to ¬Lφ for every one of infinitely many unbelieved formulas. Nothing enforces groundedness: {Lp → p} has an expansion containing p supported by nothing beyond the belief in p itself, and the theory gives no principle for rejecting it. The language has one modality and one agent, so nested and multi-agent belief are inexpressible, and the logic is non-monotonic, so adding a premise can destroy an expansion outright.

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