Preferential Logic

Origin. Kraus, Lehmann, Magidor (1990). KLM framework. Ranked models. Non-monotonic consequence. Foundation of defeasible reasoning.

Models. Preferential structures. States with ordering. Most normal worlds. Conditional assertions. Ranked interpretations.

Formalism.

Preferential model: ⟨S, l, ≺⟩ where: S: states. l: S → 2^Atoms (labeling). ≺: strict partial order (preference). Minimal states satisfy conditionals.

Preferential consequence: α |~ β: "if α, normally β." True iff minimal α-states satisfy β. Non-monotonic. Defeasible.

KLM postulates: Reflexivity: α |~ α. Left Logical Equivalence: if ⊢ α ↔ β, then (α |~ γ iff β |~ γ). Right Weakening: if α |~ β and ⊢ β → γ, then α |~ γ. And: if α |~ β and α |~ γ, then α |~ β ∧ γ. Or: if α |~ γ and β |~ γ, then α ∨ β |~ γ. Cautious Monotonicity: if α |~ β and α |~ γ, then α ∧ β |~ γ.

Rational monotonicity: If α |~ γ and α |/~ ¬β, then α ∧ β |~ γ. Stronger than preferential. Ranked models.

System P: Preferential consequence relation. Characterized by preferential models. Weakest reasonable system.

System R: Adds rational monotonicity. Ranked models. Lexicographic closure.

Symbols.

Metatheory. Preferential models. KLM postulates. Rational closure. Representation theorems.

Applies to. Non-monotonic reasoning. Default logic. AI. Commonsense reasoning.

Limitations. Irrelevance problem. Drowning problem. Computational complexity. Multiple extensions.

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