Definition:Theory

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Definition

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

Let $\FF$ be a set of $\LL$-formulas.


Then $\FF$ is an $\LL$-theory if and only if, for every $\phi \in \LL$:

$\FF \models_{\mathscr M} \phi \implies \phi \in \FF$

where $\models_{\mathscr M}$ denotes $\mathscr M$-semantic consequence.


Theory of Set of Formulas

Let $\FF$ be a set of $\LL$-formulas.


Then the $\LL$-theory of $\FF$, denoted $\map T {\FF}$ is the set:

$\set {\phi \in \LL: \FF \models_{\mathscr M} \phi}$

where $\models_{\mathscr M}$ denotes $\mathscr M$-semantic consequence.


Also known as

In cases where the language $\LL$ is obvious, one usually speaks of a theory.


Sources