Definition:Sound Proof System/Strongly Sound

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Let $\LL$ be a logical language.

Let $\mathscr P$ be a proof system for $\LL$.

let $\mathscr M$ be a formal semantics for $\LL$.

$\mathscr P$ is strongly sound for $\mathscr M$ if and only if:

Every $\mathscr P$-provable consequence is an $\mathscr M$-semantic consequence.

Symbolically, this can be expressed as the statement that, for every collection of logical formulas $\FF$, and logical formula $\phi$ of $\LL$:

$\FF \vdash_{\mathscr P} \phi$ implies $\FF \models_{\mathscr M} \phi$

Also see