Definition:Satisfiable/Set of Formulas

From ProofWiki
Jump to navigation Jump to search


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 satisfiable for $\mathscr M$ iff:

There is some $\mathscr M$-model $\mathcal M$ of $\mathcal F$

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

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

Also known as

Some sources refer to satisfiable as semantically consistent.

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

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

Also see