# Definition:Consistent (Logic)

## Definition

Let $\LL$ be a logical language.

Let $\mathscr P$ be a proof system for $\LL$.

### 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 $\FF$ 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 $\FF$.

## 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.

## Also see

- Results about
**logical consistency**can be found**here**.

## Sources

- 1944: Eugene P. Northrop:
*Riddles in Mathematics*... (previous) ... (next): Chapter One: What is a Paradox?

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- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**consistent**(in logic)