# Definition:Consistent (Logic)/Set of Formulas

## 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 $\FF$