Definition:Consistent (Logic)/Proof System

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

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

Then $\mathscr P$ is consistent :


 * There exists a logical formula $\phi$ such that $\not \vdash_{\mathscr P} \phi$

That is, some logical formula $\phi$ is not a theorem of $\mathscr P$.

Also defined as
Consistency is obviously necessary for soundness in the context of a given semantics.

Therefore it is not surprising that some authors obfuscate the boundaries between a consistent proof system (in itself) and a sound proof system (in reference to the semantics under discussion).

Also see

 * Definition:Sound Proof System