A Theorem and a Paradox

Löb’s theorem and Curry’s paradox are two very closely related results in logic. Both are surprising, but one—Löb’s—is considered acceptable while the other—Curry’s—is not. In fact, both should fail.

Löb’s Theorem and Curry’s Paradox

If a sentence says of itself that it is not true, there is little choice but to take it for its word. But what if a sentence says of itself that it is true, or, in any case, provable? Logically, there is an inconsistency in proving this statement. Martin Löb and Haskell Curry were two mathematical logicians who sought to examine this question.

When Curry met Abel

Based on his Inclosure Schema and the Principle of Uniform Solution (PUS), Priest has argued that Curry’s paradox belongs to a different family of paradoxes than the Liar. Pleitz (2015, The Logica Yearbook 2014, pp. 233–248) argued that Curry’s paradox shares the same structure as the other paradoxes and proposed a scheme of which the Inclosure Schema is a particular case and he criticizes Priest’s position by pointing out that applying the PUS implies the use of a paraconsistent logic that does not validate Contraction, but that this can hardly seen as uniform. In this paper, we will develop some further reasons to defend Pleitz’ thesis that Curry’s paradox belongs to the same family as the rest of the self-referential paradoxes & using the idea that conditionals are generalized negations. However, we will not follow Pleitz in considering doubtful that there is a uniform solution for the paradoxes in a paraconsistent spirit. We will argue that the paraconsistent strategies can be seen as special cases of the strategy of restricting Detachment and that the latter uniformly blocks all the connective-involving self-referential paradoxes, including Curry’s.