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

From ProofWiki
Jump to navigation Jump to search


Let $\LL_0$ be the language of propositional logic.

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

Let $\FF$ be a collection of logical formulas.

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

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

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