Definition:Consistent (Logic)/Set of Formulas/Propositional Logic/Definition 1
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Definition
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 $\FF$.