Definition:Complete Proof System

Definition
Let $\mathcal L$ be a logical language.

Let $\mathscr P$ be a proof system for $\mathcal L$, and let $\mathscr M$ be a formal semantics for $\mathcal L$.

Then $\mathscr P$ is said to be complete for $\mathscr M$ iff:


 * Every $\mathscr M$-tautology is a $\mathscr P$-theorem.

Symbolically, this can be expressed as the statement that, for every logical formula $\phi$ of $\mathcal L$:


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

Strongly Complete Proof System
$\mathscr P$ 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 $\mathcal F$ of logical formulas, and every logical formula $\phi$ of $\mathcal L$:


 * $\mathcal F \models_{\mathscr M} \phi$ implies $\mathcal F \vdash_{\mathscr P} \phi$

Also known as
Many sources obfuscate the distinction between complete and strongly complete.

Also see

 * Definition:Sound Proof System