CURRY'S PARADOX

The paradox is generated by a conditional: (C) If (C) is true, then p, where p is an arbitrarily chosen proposition—say, one which is just plain false. Classically we can now argue: suppose (C) is true. Then, if (C) is true then p . So p, by modus ponens . So, by the rule of conditional proof, we can infer (Q): if (C) is true, then p . That is, we have obtained (C), and hence (C) is true. Now, since we have (C) is true, and we have (Q), we can infer p by modus ponens. That is, we have proved any arbitrary proposition by logic alone. The paradox is sometimes attributed to the mathematicians Kleene and Rosser, and sometimes called Löb's paradox. It is noteworthy as clearly arising from a vicious self-reference, but not involving negation. See also semantic paradoxes.