Definition:Satisfiable/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 satisfiable for $\mathscr M$ iff:

$\phi$ is valid in some structure $\mathcal M$ of $\mathscr M$

That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that:

$\mathcal M \models_{\mathscr M} \phi$

Also known as

It is sometimes convenient to refer to satisfiability for $\mathscr M$ in a single adjective.

In such cases, $\mathscr M$-satisfiable is often seen.