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, ... Cω. Consistency operator. Gradual recovery of classical logic. Controlled inconsistency.

Formalism.

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

Hierarchy Cₙ: C₁: basic paraconsistent. C₂: more classical. ... Cω: intersection of all Cₙ. Increasing classicality.

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.

Higher Cₙ: Cₙ₊₁ requires (A₁ ∧ ... ∧ Aₙ)° → (A₁° ∧ ... ∧ Aₙ°). Consistency propagates. More classical theorems.

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.

Symbol	Unicode	Meaning
Cₙ	—	paraconsistent logic level n
A°	—	A is consistent
¬	U+00AC	paraconsistent negation
⊢	U+22A2	derivability

Metatheory. Hierarchy. Consistency operator. Gradual classicality. Controlled explosion.

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

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