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.