Curry's Paradox

Paradox
Let $P$ be an arbitrary proposition.

Consider the following argument: "This argument is valid. Therefore, $P$." Assume for contradiction that the argument is invalid.

Then by the definition of argument validity, its premise must be true and its conclusion false.

But if its premise is true then the argument is valid, contradicting our assumption.

Thus the argument must be valid.

This validates the argument's premise, allowing us to conclude $P$ for an arbitrary proposition $P$.

One can also state this paradox in terms of the material conditional.

Let $P$ be an arbitrary proposition.

Let our argument $C$ be the statement:
 * $\paren {C \to P}$

so that:
 * $C \iff \paren {C \to P}$

Assume $\neg C$ for contradiction.

Then the definition of $C$ gives:
 * $\neg \paren {C \to P}$

and by negated implication:
 * $C$ and $\neg P$

But then we have $C \text{ and } \neg C$, a contradiction.

Thus, our assumption of $\neg C$ was false, and we have $C$.

The definition of $C$ gives:
 * $\paren {C \to P}$

and modus ponens gives:
 * $P$

We have thus proven an arbitrary proposition $P$.