# Carroll Paradox

## Contents

## Paradox

Modus Ponendo Ponens leads to infinite regress.

## Proof

To be proven: $q$.

- $(1).\quad$ Assume $p \implies q$.

- $(2).\quad$ Assume $p$.

- $(3).\quad$ $p \land \left({p \implies q}\right) \vdash q$.

- $(4).\quad$ From $(2)$ and $(1)$, $p \land \left({p \implies q}\right)$.

- $(5).\quad \left({p \land \left({p \implies q}\right) \land \left({p \land \left({p \implies q}\right)}\right) \vdash q}\right) \vdash q$.

- $(6).\quad$ From $(4)$ and $(3)$, $(p \land (p \implies q))\land ((p \land (p \implies q)) \vdash q)$.

$\ldots$

## Source of Name

This entry was named for Lewis Carroll.

## Sources

- 1979: Douglas R. Hofstadter:
*Gödel, Escher, Bach: an Eternal Golden Braid*: Two-Part Invention, quoting*What the Tortoise Said to Achilles*by Lewis Carroll