Definition:Provable Consequence

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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


Sources