Definition:Complete Theory

Definition
Let $\LL$ be a language.

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

Let $T$ be an $\LL$-theory.

$T$ is complete (with respect to $\LL$ and $\mathscr M$) :
 * $T$ is satisfiable for $\mathscr M$
 * for every $\LL$-sentence $\phi$, either $T \models_{\mathscr M} \phi$ or $T \models_{\mathscr M} \neg \phi$

where $T \models_{\mathscr M} \phi$ denotes semantic entailment.