Carroll Paradox

Paradox
Modus Ponens leads to infinite regress.

Proof
To be proven: $q$.

1. Assume $p \implies q$.

2. Assume $p$.

3. $p \land (p \implies q) \vdash q$.

4. From 2 and 1, $p \land (p \implies q)$.

5. $(p \land (p \implies q) \land (p \land (p \implies q) \vdash q)) \vdash q$.

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

$\ldots$