# Definition:Consistent

## Definition

Let $\mathcal L$ be a logical language.

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

### Proof System

Then $\mathscr P$ is **consistent** if and only if:

- There exists a logical formula $\phi$ such that $\not \vdash_{\mathscr P} \phi$

That is, some logical formula $\phi$ is **not** a theorem of $\mathscr P$.

### Set of Formulas

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 $\mathcal F$.

## Also defined as

Common confusion arises in the precise interpretation of **consistency** in logic.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, **consistency** is placed in the realm of proof systems.

It is also common to place it in the realm of formal semantics.

That is, to define **consistent** as what is satisfiable on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Because of the variation, any use of these terms in a source work should be treated with the utmost care and precision to determine the exact meaning.