# 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 $\mathcal M$ be an $\mathcal L$-structure.

The **$\mathcal L$-theory of $\mathcal M$** is the $\mathcal L$-theory consisting of those $\mathcal L$-sentences $\phi$ such that:

- $\mathcal M \models \phi$

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

This theory can be denoted $\operatorname{Th} \left({\mathcal M}\right)$ when the language $\mathcal L$ 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 $\neg \phi \in T$.