Curry's paradox
Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F". The paradox requires only a few apparently-innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic.
For Paul Curry's optical illusion and dissection puzzle, see Missing square puzzle.The paradox is named after the logician Haskell Curry, who wrote about it in 1942.[1] It has also been called Löb's paradox after Martin Hugo Löb,[2] due to its relationship to Löb's theorem.
In formal logics[edit]
Sentential logic[edit]
The example in the previous section used unformalized, natural-language reasoning. Curry's paradox also occurs in some varieties of formal logic. In this context, it shows that if we assume there is a formal sentence (X → Y), where X itself is equivalent to (X → Y), then we can prove Y with a formal proof. One example of such a formal proof is as follows. For an explanation of the logic notation used in this section, refer to the list of logic symbols.