# 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.In particular: 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: Haskell B. Curry:
*The Combinatory Foundations of Mathematical Logic*(*J. Symb. Logic***Vol. 7**,*no. 2*: pp. 49 – 64) www.jstor.org/stable/2266302

Work In ProgressIn particular: Can 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