Although the language is not always used this way, I find it helpful (when considering paraconsistent logics) to strictly distinguish a *contradiction* from an *inconsistency* as follows:

- $\phi$ is
*contradictory*if, for some $\psi$, $\phi \vdash \psi$ and $\phi \vdash \neg{\psi}$ (equivalently, $\phi \vdash \psi \wedge \neg{\psi}$ if $\wedge$ obeys the usual rules); - $\phi$ is
*inconsistent*if, for every $\chi$, $\phi \vdash \chi$ (equivalently, $\phi \vdash \bot$ if $\bot$ obeys the usual rules); - $\phi$ is
*paraconsistent*if it is contradictory but not inconsistent.

(Since the definition of contradiction depends on the operator $\neg$, one might also say a *$\neg$-contradiction* in case of multiple candidates for negation, as in linear logic.)

One can generalise this to speak of more general contexts or even entire logics as being contradictory, inconsistent, or paraconsistent; then we recover the usual meaning of when a logic is paraconsistent.

]]>Thanks; I added some remarks about paraconsistent logic.

]]>I have created stubs for *inconsistency* and *contradiction*