Definition:Theory

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, for every $\phi \in \LL$:


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

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

Also defined as
Some sources do not impose on a theory the restriction of being closed under $\mathscr M$-semantic consequence.

Then, theory is just a term describing an arbitrary set of $\LL$-formulas.

On, it was decided that the version with semantic consequence will be called theory.

In other sources, it always needs to be carefully inspected if a theory is considered to be closed under semantic consequence.

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