Definition:Unsatisfiable/Set of Formulas

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

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

A collection $\mathcal F$ of logical formulas of $\mathcal L$ is unsatisfiable for $\mathscr M$, or contradictory for $\mathscr M$''', iff:


 * There is no $\mathscr M$-model $\mathcal M$ for $\mathcal F$

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


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