Curry's paradox

Curry's paradox is a paradox that occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-referring sentence and some apparently innocuous logical deduction rules. The paradox is named after the logician Haskell Curry. The paradox may be expressed in natural language and in various mathematical settings, including certain forms of set theory, lambda calculus, and combinatory logic.

It has also been called Löb's paradox after Martin Hugo Löb, due to its relationship to Löb's theorem.

Claims of the form "if A, then B" are called conditional claims. Curry's paradox uses a particular kind of self-referential conditional sentence, as demonstrated in this example:

Even though Germany does not border China, the example sentence certainly is a natural-language sentence, and so the truth of that sentence can be analyzed. The paradox follows from this analysis. The analysis consists of two steps.

The claim "Germany borders China" could be replaced by any other claim, and the sentence would still be provable; thus every sentence appears to be provable, similar to the principle of explosion. Because the proof uses only well-accepted methods of deduction, and because none of these methods appears to be incorrect, this situation is paradoxical.

The following analysis is used to show that the sentence "If this sentence is true, then Germany borders China" is itself true. The quoted sentence is of the form "If A, then B" where A refers to the sentence itself and B refers to "Germany borders China". Such sentences are called conditional sentences, and the standard method for proving them is called conditional proof. To apply this method, the first step is to assume for the sake of argument that the hypothesis (A) is true. The goal is then to show that the conclusion (B) can be proven from that assumption. Therefore, for the purpose of the proof, assume A.

...
Wikipedia