Definition:Sound 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 sound for $\mathscr M$ iff:

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

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

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

Strongly Sound Proof System

$\mathscr P$ is strongly sound for $\mathscr M$ iff:

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 $\mathcal F$, and logical formula $\phi$ of $\mathcal L$:

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

Also known as

Many sources obfuscate the distinction between sound and strongly sound.

Some sources speak of consistent proof systems. However, on $\mathsf{Pr} \infty \mathsf{fWiki}$, consistency is a term only applied to (sets of) formulas.