Carroll Paradox

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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 logical axiom and a rule of inference.

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


This article, or a section of it, needs explaining.
In particular: exactly where Achilles's thinking goes wrong, in refusing to accept that $P, P \implies Q \vdash Q$ without first needing to accept as a logical axiom $\paren {P \land P \implies Q} \implies Q \vdash Q$
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Also known as

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

However, the term Achilles paradox is also well established as the name of the paradox of Zeno which questions how Achilles can ever overtake a tortoise in a running race.


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

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