Definition:Consistent/Set of Formulas/Propositional Logic/Definition 2

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

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

Let $\mathcal F$ be a collection of logical formulas.

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

Then $\mathcal F$ is consistent for $\mathscr P$ if and only if:

For every logical formula $\phi$, not both of $\phi$ and $\neg \phi$ are $\mathscr P$-provable consequences of $\mathcal F$