\begin{definition}[Definition:Complete Proof System/Strongly Complete]
Let $\LL$ be a logical language.
Let $\mathscr P$ be a proof system for $\LL$, and let $\mathscr M$ be a formal semantics for $\LL$.
$\mathscr P$ is '''strongly complete''' for $\mathscr M$ {{iff}}:
:Every $\mathscr M$-semantic consequence is a $\mathscr P$-provable consequence.
Symbolically, this can be expressed as the statement that, for every collection $\FF$ of logical formulas, and every logical formula $\phi$ of $\LL$:
:$\FF \models_{\mathscr M} \phi$ implies $\FF \vdash_{\mathscr P} \phi$
\end{definition}