Definition:Consistent/Set of Formulas

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Definition

Let $\mathcal L$ be a logical language.

Let $\mathscr P$ be a proof system for $\mathcal L$.

Let $\mathcal F$ be a collection of logical formulas.


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

There exists a logical formula $\phi$ such that $\mathcal F \nvdash_{\mathscr P} \phi$.

That is, some logical formula $\phi$ is not a provable consequence of $\mathcal F$.


Propositional Logic

Suppose that in $\mathscr P$, the Rule of Explosion (Variant 3) holds.


Then $\mathcal F$ 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 $\mathcal F$


Also see