# Definition:Inconsistent (Logic)

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## Definition

Let $\LL$ be a logical language.

Let $\mathscr P$ be a proof system for $\LL$.

### Definition 1

A set $\FF$ of logical formulas is **inconsistent for $\mathscr P$** if and only if:

- For every logical formula $\phi$, $\FF \vdash_{\mathscr P} \phi$.

That is, *every* logical formula $\phi$ is a provable consequence of $\FF$.

### Definition 2

A set $\FF$ of logical formulas is **inconsistent for $\mathscr P$** if and only if:

- There exists a logical formula $\phi$ such that both
- $\FF \vdash_{\mathscr P} \paren {\phi \land \neg \phi}$

## Also known as

**Inconsistent** sets 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.