Definition:Inconsistent (Logic)

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 $\mathscr P$ iff:


 * 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.

Also see

 * Definition:Provable Consequence
 * Definition:Consistent


 * Definition:Unsatisfiable