Curry's Paradox

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: resolution 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 {{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

• Löb's Theorem

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 Progress In 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