Carroll Paradox

Paradox
Modus Ponendo Ponens leads to infinite regress.

Proof
To be proven: $q$.


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


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


 * $(3).\quad$ $p \land \left({p \implies q}\right) \vdash q$.


 * $(4).\quad$ From $(2)$ and $(1)$, $p \land \left({p \implies q}\right)$.


 * $(5).\quad \left({p \land \left({p \implies q}\right) \land \left({p \land \left({p \implies q}\right)}\right) \vdash q}\right) \vdash q$.


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

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