Definition:Unsatisfiable/Formula

Definition
Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

A logical formula $\phi$ of $\LL$ is unsatisfiable for $\mathscr M$ :


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

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


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

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 (Logic)