1. Any one of the family of paraconsistent logics introduced by philosopher Newton da Costa (1929– ), in which the positive fragment of intuitionistic logic is enriched with a paraconsistent negation. The most well-known C-system ${\text{C}}_{\omega }$ adds to positive intuitionistic logic the axiom schema of the principle of excluded middle and the principle of double negation, i.e.:

• • $\phi \vee \neg \phi$

• • $\neg \neg \phi \to \phi$

From a syntactic perspective, ${\text{C}}_{\omega }$ can be identified as a type of dualization of intuitionistic logic. For example, the axiom schema that da Costa endorses are plausibly dual to the schema:

• • $\neg \left(\phi \wedge \neg \phi \right)$

• • $\phi \to \neg \neg \phi$

both of which are valid with respect to intuitionistic negation. An important feature of C-systems is the appearance of a consistency connective—a hallmark of Brazilian-style paraconsistent logics, including the logics of formal inconsistency.

2. With respect to a deductive system $\text{L}$, a deductive system ${\text{L}}^{\prime }$ with a consistency connective $\circ$ such that

1. 1 ${\text{L}}^{\prime }$ conservatively extends $\text{L}$

2. 2 $\left\{\phi ,\neg \phi ,\circ \phi \right\}⊢\psi$ for all $\phi$ and $\psi$.

For example, da Costa’s logics ${\text{C}}_n$ are C-systems with respect to the positive fragment of intuitionistic logic. A related notion is that of a dC-system (i.e., a definable C-system) in which the consistency connective $\circ$ is definable from the other connectives.