Carroll Paradox

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 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.