Löb'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.

The standard method for proving conditional sentences (sentences of the form "if A, then B") is called a "conditional proof". For the sake of argument, A is assumed and then with that assumption B is shown to be true.

Because A refers to the overall sentence, this means that assuming A is the same as assuming "If A, then B". Therefore, in assuming A, we have assumed both A and "If A, then B". Therefore B is true, and we have proven "If this sentence is true, then 'Germany borders China' is true." Therefore Germany borders China.

To put it another way, let us consider both cases. If we assume the sentence is true, then Germany must border China. But Germany does not border China. However, in formal logic, when the second part of a conditional claim is proven false, the sentence as a whole is given the benefit of the doubt and declared true. So, the very fact that Germany does not border China proves our original sentence: "if this sentence is true, then Germany borders China."

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.

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