Adaptive Logic

Origin. Batens (1980s+). Dynamic proof theory. Abnormalities. Paraconsistent dynamics. Foundation of defeasible formal reasoning.

Models. Proofs that adapt. Abnormality handling. Conditional derivations. Marking and unmarking. Dynamic consequence.

Formalism.

Components: Lower limit logic (LLL): monotonic base. Abnormalities Ω: formulas treated specially. Strategy: how to handle abnormalities. Adaptive consequence.

Abnormality: Formulas in Ω (e.g., contradictions). "Abnormal" if true. Assumed false unless forced. Dynamic retraction.

Conditional derivation: Derive A on condition Δ. Δ ⊆ Ω: abnormalities assumed false. If Δ shown true, retract A. Defeasible conclusions.

Marking: Line marked if condition Δ problematic. Unmarked: currently derived. Marked: currently retracted. Dynamic status.

Reliability strategy: Condition Δ reliable if no Dab(Δ) derived. Dab(Δ): disjunction of abnormalities. Minimize abnormalities.

Minimal abnormality: Choose models with minimal abnormality sets. Skeptical or credulous. Multiple extensions possible.

CLuN example: LLL = CL (classical). Ω = {A ∧ ¬A : A any formula}. Paraconsistent adaptive. Tolerate contradictions when needed.

Symbols.

Symbol	Unicode	Meaning
LLL	—	lower limit logic
Ω	U+03A9	abnormalities
Dab	—	disjunction of abnormalities
⊢^a	—	adaptive consequence

Metatheory. Dynamic proofs. Marking. Strategies. Final derivability.

Applies to. Paraconsistency. Belief revision. Abduction. Scientific reasoning.

Limitations. Computational complexity. Strategy choice. Final derivability undecidable. Non-monotonic complications.