Definition:Unsatisfiable/Formula

Definition
Let $\mathcal L$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\mathcal L$.

A logical formula $\phi$ of $\mathcal L$ is unsatisfiable for $\mathscr M$ iff:


 * $\phi$ is valid in none of the structures of $\mathscr M$

That is, for all structures $\mathcal M$ of $\mathscr M$:


 * $\mathcal M \not\models_{\mathscr M} \phi$

Also known as
Unsatisfiable formulas are also referred to as:


 * contradictions;
 * logical falsehoods;
 * logical falsities;
 * inconsistent formulas.

Because the term contradiction also commonly refers to the concept of inconsistency in the context of a proof system, it is discouraged as a synonym of unsatisfiable formula on.

The next two of these terms can easily lead to confusion about the precise meaning of "logical", and are therefore also discouraged on.

Also see

 * Definition:Bottom (Logic), a symbol often used to represent contradictions in logical languages.
 * Definition:Tautology
 * Definition:Satisfiable Formula
 * Definition:Falsifiable Formula


 * Definition:Inconsistent