Definition:Consistent (Logic)/Proof System

Definition

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

Then $\mathscr P$ is consistent if and only if:

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

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

Suppose that in $\mathscr P$, the Rule of Explosion (Variant 3) holds.

Then $\mathscr P$ is consistent if and only if:

For every logical formula $\phi$, not both of $\phi$ and $\neg \phi$ are theorems of $\mathscr P$

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).