Definition:Consistent (Logic)/Proof System/Propositional Logic/Definition 2

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Let $\LL$ be the language of propositional logic.

Let $\mathscr P$ be a proof system for $\LL_0$.

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 $\mathscr P$