Definition:Consistent (Logic)/Set of Formulas
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Definition
Let $\LL$ be a logical language.
Let $\mathscr P$ be a proof system for $\LL$.
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 $\FF \nvdash_{\mathscr P} \phi$.
That is, some logical formula $\phi$ is not a provable consequence of $\FF$.
Propositional Logic
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 $\FF$
Also see
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.11$ Some Strategies for Constructing Proofs: Definition $\text{II}.11.2$