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 $\neg \phi \in T$.