Definition:Consistent (Logic)/Set of Formulas/Propositional Logic/Definition 2
Jump to navigation
Jump to search
Definition
Let $\LL$ be the language of propositional logic.
Let $\mathscr P$ be a proof system for $\LL_0$.
Let $\FF$ be a collection of logical formulas.
Suppose that in $\mathscr P$, the Rule of Explosion (Variant 3) holds.
Then $\FF$ is consistent for $\mathscr P$ if and only if:
- For every logical formula $\phi$, not both of $\phi$ and $\neg \phi$ are $\mathscr P$-provable consequences of $\FF$
Sources
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.11$: Compactness