Definition:Theory

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Definition

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 $\mathcal F$ is an $\mathcal L$-theory iff, for every $\phi \in \mathcal L$:

$\mathcal F \models_{\mathscr M} \phi \implies \phi \in \mathcal F$

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


Theory of Set of Formulas

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 known as

In cases where the language $\mathcal L$ is obvious, one usually speaks of a theory.


Sources