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

The next two of these terms can easily lead to confusion about the precise meaning of "logical", and are therefore also discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

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