# Definition:Inconsistent (Logic)

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

## Also see

• Results about inconsistent in the context of logic can be found here.