# Definition:Unsatisfiable

## Definition

Let $\LL$ be a logical language.

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

### Unsatisfiable Formula

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

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

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

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

### Unsatisfiable Set of Formulas

A collection $\FF$ of logical formulas of $\LL$ is unsatisfiable for $\mathscr M$ if and only if:

There is no $\mathscr M$-model $\MM$ for $\FF$

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

$\MM \not \models_{\mathscr M} \FF$

### Unsatisfiable for Boolean Interpretations

Let $\mathbf A$ be a WFF of propositional logic.

$\mathbf A$ is called unsatisfiable (for boolean interpretations) if and only if:

$v \left({\mathbf A}\right) = F$

for every boolean interpretation $v$ for $\mathbf A$.

In terms of validity, this can be rendered:

$v \not\models_{\mathrm{BI}} \mathbf A$

that is, $\mathbf A$ is invalid in every boolean interpretation of $\mathbf A$.