Definition:Theory (Logic)/Structure

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Definition

Let $\LL$ be a logical language.

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.