Definition:Theory (Logic)/Structure

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Let $\mathcal L$ be a logical language.

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.