Definition:Theory/Set of Formulas

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Let $\mathcal L$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\mathcal L$.

Let $\mathcal F$ be a set of $\mathcal L$-formulas.

Then the $\mathcal L$-theory of $\mathcal F$, denoted $T \left({\mathcal F}\right)$ is the set:

$\left\{{\phi \in \mathcal L: \mathcal F \models_{\mathscr M} \phi}\right\}$

where $\models_{\mathscr M}$ denotes $\mathscr M$-semantic consequence.

Also see