Definition:Complete Proof System/Strongly Complete

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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 $\mathscr M$ if and only if:

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 $\FF \vdash_{\mathscr P} \phi$

Also see