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

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

Let $\mathscr P$ be a proof system for $\mathcal L_0$. Suppose that in $\mathscr P$, the Rule of Explosion holds.

Then $\mathscr P$ is consistent :


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