Adaptive Logic

Origin. Diderik Batens and the Ghent Centre for Logic and Philosophy of Science developed adaptive logics from the early 1980s, starting with inconsistency-adaptive systems that reason as classically as a possibly inconsistent premise set allows while localizing the damage. The programme was consolidated into the "standard format" — a triple of lower limit logic, abnormality set and strategy — in Batens, "A general characterization of adaptive logics" (2001), and "A universal logic approach to adaptive logics" (2007), which prove the metatheory once for every logic presented in the format rather than case by case.

Models. Reasoning that presumes normality and withdraws the presumption when abnormality is forced. A premise set is read through those lower-limit models that are as normal as the premises permit: either models whose abnormalities all lie among the unreliable ones, or models whose abnormality set is minimal under inclusion. The proof theory is dynamic. Lines carry the conditions on which they were derived, a line is marked when its condition turns out unsafe, and the set of currently derived formulas can shrink as the proof lengthens.

Formalism.

Standard format: An adaptive logic is a triple ⟨LLL, Ω, strategy⟩. LLL: lower limit logic, monotonic, never given up. Ω: abnormalities, given by a logical form. Strategy: Reliability or Minimal Abnormality.

Upper limit logic: ULL = LLL + ¬A for every A ∈ Ω. The adaptive logic sits between: Cn_LLL(Γ) ⊆ Cn_AL(Γ) ⊆ Cn_ULL(Γ).

Conditional derivation: Proof lines are quadruples: number, formula, justification, condition Δ ⊆ Ω. A on condition Δ means: A follows in LLL from Γ together with ¬Dab(Δ). Unconditional lines carry Δ = ∅.

Dab-formulas: Dab(Δ) = the disjunction of the members of Δ. Dab(Δ) is a minimal Dab-consequence of Γ if Γ ⊢_LLL Dab(Δ) and no proper subset of Δ has that property.

Reliability strategy: U(Γ) = union of the Δ of all minimal Dab-consequences of Γ. Mark a line with condition Δ if Δ ∩ U(Γ) ≠ ∅. Semantics: LLL-models M with Ab(M) ⊆ U(Γ).

Minimal Abnormality strategy: Semantics: LLL-models M with Ab(M) minimal under ⊆. Marking is defined through the choice sets of the minimal Dab-consequences. At least as strong as Reliability, strictly stronger on some premise sets.

Dynamic proof: Marked: currently retracted. Unmarked: currently derived. Final derivability: A is finally derived at a line if that line is unmarked in some extension of the proof and stays unmarked in every further extension.

Inconsistency-adaptive example (CLuN): LLL = CLuN, classical logic without ex falso quodlibet. Ω = {A ∧ ¬A : A a formula}. ULL = CL. Contradictions are tolerated only where the premises force them; everywhere else the logic reasons classically.

Symbols.

Symbol	Unicode	Name	Meaning
Ω	U+03A9	Omega / Abnormalities	Formulas presumed false unless forced
Δ	U+0394	Delta / Condition	Finite set of abnormalities a line rests on
Γ	U+0393	Gamma / Premise set	The premises being reasoned from
⊆	U+2286	Subset	Condition drawn from Ω; minimality of abnormality
∩	U+2229	Set intersection	Marking test against the unreliable formulas
∅	U+2205	Empty set	Unconditional line
∈	U+2208	Membership	Abnormality belonging to Ω
≠	U+2260	Inequality	Non-empty overlap triggers marking
∧	U+2227	Conjunction	Form of a contradictory abnormality
∨	U+2228	Disjunction	Dab-formulas are disjunctions of abnormalities
¬	U+00AC	Negation	Denial of an abnormality
⊢	U+22A2	Derivability	Consequence in the lower limit logic
⊢^a	—	Adaptive consequence	Final derivability in the adaptive logic
LLL	—	Lower limit logic	Monotonic base, never retracted
ULL	—	Upper limit logic	LLL plus the denial of every abnormality
Dab(Δ)	—	Disjunction of abnormalities	Disjunction of the members of Δ
U(Γ)	—	Unreliable formulas	Disjuncts of the minimal Dab-consequences of Γ
Ab(M)	—	Abnormal part	Abnormalities true in the model M
Cn	—	Consequence operator	Closure under a given consequence relation
CLuN	—	Paraconsistent base	Classical logic without ex falso quodlibet
CL	—	Classical logic	Upper limit of the inconsistency-adaptive logics

Metatheory. The point of the standard format is that a logic presented in it inherits its metatheory rather than proving it afresh. The consequence set is bracketed, Cn_LLL(Γ) ⊆ Cn_AL(Γ) ⊆ Cn_ULL(Γ), and the two ends coincide when Γ has a lower-limit model verifying no abnormality, so the adaptive logic behaves exactly like the upper limit on normal premise sets. Batens proved the dynamic proof theory sound and complete with respect to the corresponding selection semantics for both strategies: final derivability of A from Γ coincides with truth of A in all reliable models of Γ, respectively in all minimally abnormal models. Reassurance holds — if Γ has lower-limit models it has selected models, so the adaptive consequence set is non-trivial whenever the lower-limit one is — and Strong Reassurance strengthens this to the claim that every unselected model is beaten by a selected one. Minimal Abnormality is at least as strong as Reliability and strictly stronger on some premise sets. Horsten and Welch (2007) showed that final derivability is undecidable already at the propositional level, even when the lower limit logic and the abnormality set are both decidable.

Applies to. Paraconsistency. Belief revision. Abduction. Scientific reasoning. Inductive generalization. Reasoning from inconsistent databases.

Limitations. Final derivability is a property of the whole space of proof extensions, not of any proof: a line unmarked now may be marked later, so what a proof exhibits at a given stage is provisional, and Horsten and Welch showed the final relation undecidable even propositionally with a decidable base. There is thus no effective procedure certifying a conclusion, only one refuting a particular marking. The marking conditions depend on the minimal Dab-consequences of the entire premise set, so one added premise can re-mark arbitrarily many earlier lines. The two strategies determine different consequence relations, and Ω must be stipulated in advance by a logical form; nothing in the framework selects it, and different choices yield different logics on the same premises. The upper limit caps what can ever be concluded: no adaptive logic in the format reaches beyond LLL plus the denial of all abnormalities, however well supported a stronger conclusion may be.