## Curry's Paradox

Setup: (S) If it is true that S, then Paris is the capital of Italy.

Self-referential statements of this form seem to be able to prove that any statement placed in the consequent is true. If we assume S is true, then the antecedent is true by the assumption, and so the consequent follows since the implication is true by the assumption. If we assume S is false, then by the negation of material implication, we obtain "It is true that S and Paris is not the capital of Italy." But this contradicts that S is false, so this case can't happen.

The contradiction step can be avoided, while still showing the consequent is true by noting that assuming S is true, Paris is the capital of Italy, as was shown in the last paragraph. But this is exactly the statement made by S, so S is true, and therefore Paris is indeed the capital of Italy whether or not S is true.

The problem is that not all statements can be assigned a truth value. Statement S is self-referential, and this property causes problems for whichever truth value is assigned to it.