Definition:Theory (Logic)

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 $\neg \phi \in T$.