# Definition:Consistent (Logic)/Set of Formulas/Propositional Logic

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## Definition

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: It is not necessary that the language is PropLog, it may also be an extension such as PredLogYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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.

### Definition 1

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 $\FF$.

### Definition 2

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 $\FF$