Definition:Provable Consequence

Definition
Let $\mathscr P$ be a proof system for a formal language $\LL$.

Let $\FF$ be a collection of WFFs of $\LL$.

Denote with $\map {\mathscr P} \FF$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\FF$ as axioms.

Let $\phi$ be a theorem of $\map {\mathscr P} \FF$.

Then $\phi$ is called a provable consequence of $\FF$, and this is denoted as:


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

Note in particular that for $\FF = \O$, this notation agrees with the notation for a $\mathscr P$-theorem:


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

Also defined as
While this definition is adequate for most proof systems, it is more natural for some of them to define provable consequence in a different way.

For example, the tableau proof system based on propositional tableaus.

Also known as
One also encounters phrases like:


 * $\FF$ proves $\phi$
 * $\phi$ is provable from $\FF$

to describe the concept of provable consequence.

A provable consequence is also known as a derivable formula or a provable formula.

Also see

 * Definition:Theorem (Formal Systems)


 * Definition:Semantic Consequence