A family of paraconsistent deductive systems that include some syntactic method of representing the consistency of a formula $\phi$, e.g., the unary consistency connective $\circ$. While logics of formal inconsistency (or LFIs) reject the principle of explosion, they share a property of gentle explosion, that is, while one may not infer an arbitrary formula $\psi$ from a contradiction $\phi \wedge \neg \phi$, one may infer $\psi$ from a contradiction $\phi \wedge \neg \phi$ in conjunction with the assertion that $\phi$ is consistent.

More formally, a logic of formal inconsistency is a deductive system $\text{L}$ with a negation connective $\neg$ such that

• • $\text{L}$ is not explosive with respect to $\neg$, i.e., there are formulae $\phi$ and $\psi$ such that $\phi ,\neg \phi {⊬}_{\text{L}}\psi$

• • there is a set of formulae $◯\left(p\right)$ with atomic variable $p$ such that for all formulae $\phi$ and $\psi$, $◯\left(\phi \right),\phi ,\neg \phi {⊢}_{\text{L}}\psi$

where $◯\left(\phi \right)$ is the set of formulae in which $p$ is uniformly replaced by $\phi$.

The family of C-systems—in which $◯\left(\phi \right)=\left\{\circ \phi \right\}$—is the most well-known collection of logics of formal inconsistency.