Da Costa's Paraconsistent Logics (Cₙ)

Origin. Newton da Costa (1963). Brazilian school. Hierarchy of paraconsistent logics. Controlled contradiction. Foundation of South American paraconsistency.

Models. Contradiction doesn't explode. Hierarchy C₁ ⊋ C₂ ⊋ ... ⊋ C_ω of strictly decreasing strength. Consistency operator. Gradual weakening away from classical logic. Controlled inconsistency.

Formalism.

Paraconsistency: A, ¬A ⊬ B (explosion fails). Contradictions contained. Non-trivial inconsistent theories. Controlled reasoning.

Hierarchy Cₙ: C₁ ⊋ C₂ ⊋ ... ⊋ C_ω, strictly decreasing in theorems. C₁: the strongest, closest to classical. Cₙ₊₁: weaker — its consistency requirement is harder to meet, so fewer formulas behave classically. C_ω: the intersection of all Cₙ, the weakest of the family.

Consistency operator (°): A° ("A is consistent"). A° ∧ A ∧ ¬A ⊢ B (if A consistent, explosion for A). Marks "safe" formulas. Classical behavior when consistent.

C₁ axioms: Positive intuitionistic core. A → (¬A → B) fails in general. But A° → (A → (¬A → B)). Conditional explosion.

Propagation of consistency: (A₁° ∧ ... ∧ Aₙ°) → (A₁ ∧ ... ∧ Aₙ)°, and likewise for ∨ and →. Consistency of the parts propagates to the compound. Higher Cₙ iterate the consistency requirement (A⁽ⁿ⁾), which is harder to satisfy, so each level has fewer theorems than the last.

Semantics: Valuation semantics. v(A) ∈ {0, 1}. v(¬A) = 1 doesn't require v(A) = 0. Non-truth-functional negation.

Applications: Inconsistent databases. Naive set theory. Dialectical reasoning. Information fusion.

Symbols.

Metatheory. Strictly decreasing hierarchy C₁ ⊋ C₂ ⊋ ... ⊋ C_ω. Consistency operator. Controlled explosion.

Applies to. Inconsistent theories. Database reasoning. Philosophy. Information systems.

Limitations. Complexity. Non-standard negation. Multiple systems. Philosophical debates.

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