Definition:Consistent (Logic)/Set of Formulas/Propositional Logic/Definition 1

Definition
Let $\mathcal L_0$ 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.

Then $\mathcal F$ is consistent for $\mathscr P$ :


 * There exists a logical formula $\phi$ such that $\mathcal F \not \vdash_{\mathscr P} \phi$

That is, some logical formula $\phi$ is not a $\mathscr P$-provable consequence of $\mathcal F$.