Definition:Consistent/Proof System/Propositional Logic/Definition 1

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Let $\mathcal L_0$ be the language of propositional logic.

Let $\mathscr P$ be a proof system for $\mathcal L_0$.

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 $\mathscr P$.