# Curry's Paradox

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## Paradox

Let $P$ be an arbitrary proposition.

Let a proposition $C$ be defined:

- $C \implies P$

Aiming for a contradiction, suppose that $\neg C$.

Then $C \implies P$ is vacuously true.

By definition, $C$ is true, a contradiction.

This contradiction shows $C$ to be true.

By definition, $C \implies P$.

By Modus Ponendo Ponens, we conclude $P$, where $P$ is arbitrary.

But this means that any proof system expressing the above is inconsistent.

This needs considerable tedious hard slog to complete it.resolutionTo 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 `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also known as

Curry's Paradox can also be seen referred to as Löb's Paradox after Martin Hugo Löb.

## Also see

## Source of Name

This entry was named for Haskell Brooks Curry.

## Sources

- Jun 1942:
*The Combinatory Foundations of Mathematical Logic*(*J. Symb. Logic***Vol. 7**,*no. 2*) www.jstor.org/stable/2266302

Work In ProgressCan we identify who Laurence Goldstein is -- either the Michigan professor of English or the philosophy prof based in University of Kent, or neither?You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing 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 `{{WIP}}` from the code. |

- 1986: Laurence Goldstein:
*Epimenides and Curry*(*Analysis***Vol. 46**,*no. 3*: pp. 117 – 121) www.jstor.org/stable/3328637