# Carroll Paradox

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## Contents

## Paradox

Modus Ponendo Ponens leads to infinite regress.

## Proof

To be proven: $q$.

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

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

- $(3).\quad$ $p \land \paren {p \implies q} \vdash q$ by Modus Ponendo Ponens.

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

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

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

$\ldots$

and so *ad infinitum* (or, as Lewis Carroll put it, *ad nauseaum*).

## Resolution

This is an antinomy.

It arises because of confusion between an axiom and a rule of inference.

In this context, Modus Ponendo Ponens is a rule of inference.

## Also known as

This is also found in the literature as the **Achilles paradox**, from the nature of its exposition by Lewis Carroll.

## Source of Name

This entry was named for Lewis Carroll.

## Historical Note

The **Carroll Paradox** was originally presented by Charles Lutwidge Dodgson, writing under the name Lewis Carroll.

He presented it as a whimsical conversation between Achilles and the tortoise he had been racing in the Achilles Paradox.

## 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

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Achilles paradox**:**2.**