\begin{definition}[Definition:Consistent (Logic)/Proof System/Propositional Logic/Definition 2]
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''' {{iff}}:
:For every logical formula $\phi$, not ''both'' of $\phi$ and $\neg \phi$ are theorems of $\mathscr P$
\end{definition}