Category:Definitions/Complete Proof Systems

This category contains definitions related to Complete Proof Systems.
Related results can be found in Category:Complete Proof Systems.

Let $\LL$ be a logical language.

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

Then $\mathscr P$ is said to be complete for $\mathscr M$ if and only if:

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

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

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