Definition:Unsatisfiable

Definition

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

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 $\mathscr M$:

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

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 $\mathscr M$:

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

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

$\map v {\mathbf A} = \F$

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 $\mathbf A$.