# Definition:Theory (Logic)

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## Definition

Let $\mathcal L$ be a logical language.

An **$\mathcal L$-theory** $T$ is a set of $\mathcal L$-sentences.

### Theory of Structure

Let $\MM$ be an $\LL$-structure.

The **$\LL$-theory of $\MM$** is the $\LL$-theory consisting of those $\LL$-sentences $\phi$ such that:

- $\MM \models \phi$

where $\models$ denotes that $\MM$ is a model for $\phi$.

This theory can be denoted $\map {\operatorname{Th} } \MM$ when the language $\LL$ is understood.

### Complete Theory

$T$ is **complete** if and only if:

- for every $\mathcal L$-sentence $\phi$, either $T \models \phi$ or $T \models \neg \phi$

where $T \models \phi$ denotes semantic entailment.

### Maximal Theory

$T$ is **maximal** if and only if:

- for every $\mathcal L$-sentence $\phi$, either $\phi \in T$ or $\neg \phi \in T$.