In some deductive systems a unary connective $\circ$ corresponding to the consistency of a formula $\phi$. In the context of paraconsistent logic—in which a formula $\phi$ and its negation $\neg \phi$ may both be true—the truth of $\circ \phi$ indicates that such an inconsistent state of affairs does not obtain. The ability to express the consistency of a formula enables a deductive system $\text{L}$ to reject the principle of explosion while remaining gently explosive, that is, while a contradiction $\phi \wedge \neg \phi$ is not sufficient to infer everything, the contradiction in conjunction with the statement of the consistency of $\phi$ does license this inference. Such a connective features prominently in logics of formal inconsistency, including C-systems.