Definition:Provable Consequence

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

Let $\mathcal F$ be a collection of WFFs of $\mathcal L$.

Denote with $\mathscr P \left({\mathcal F}\right)$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\mathcal F$ as axioms.

Let $\phi$ be a theorem of $\mathscr P \left({\mathcal F}\right)$.

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

$\mathcal F \vdash_{\mathscr P} \phi$

Note in particular that for $\mathcal F = \varnothing$, 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:

  • $\mathcal F$ proves $\phi$
  • $\phi$ is provable from $\mathcal F$

to describe the concept of provable consequence.

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

Also see