Definition:Theorem/Formal System

Definition
Let $\mathcal L$ be a formal language.

Let $\mathscr P$ be a proof system for $\mathcal L$.

A theorem of $\mathscr P$ is a well-formed formula of $\mathcal L$ which can be deduced from the axioms and axiom schemata of $\mathscr P$ by means of its rules of inference.

That a WFF $\phi$ is a theorem of $\mathscr P$ may be denoted as:


 * $\vdash_{\mathscr P} \phi$

Also known as
A theorem $\phi$ of $\mathscr P$ is also called provable from $\mathscr P$.

Also see

 * Definition:Theorem of Logic is a specific case of this concept, if one defines the apparatus of logic as a formal system.


 * Definition:Provable Consequence