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 $\mathscr M$ iff:


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

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


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

Also known as
An unsatisfiable collection of logical formulas is often referred to as contradictory.

However, this term is also commonly used to describe the notion of inconsistency in the context of a proof system.

It is therefore discouraged as a synonym of unsatisfiable on

Also see

 * Definition:Inconsistent