# Definition:Inconsistent

(Redirected from Definition:Contradiction/Inconsistent)

## Definition

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 $\mathsf{Pr} \infty \mathsf{fWiki}$.