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Let $\mathcal L$ be a logical language.

Let $\mathscr P$ be a proof system for $\mathcal L$.

A collection $\mathcal F$ of logical formulas is inconsistent for $\mathscr P$ if and only if:

For every logical formula $\phi$, $\mathcal F \vdash_{\mathscr P} \phi$.

That is, every logical formula $\phi$ is a provable consequence of $\mathcal F$.

Also known as

Inconsistent collections of logical formulas are often called contradictory.

Likewise, a logical formula which is inconsistent by itself is often called a contradiction.

Since these terms are also often used to describe unsatisfiability in the context of a formal semantics, they are discouraged as synonyms of inconsistent on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see