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The barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell himself as an illustration of the paradox, though he attributes it to an unnamed person who suggested it to him. It shows that an apparently plausible scenario is logically impossible. Specifically, it describes a barber who is defined such that he both shaves himself and does not shave himself.
Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things:
- shaving himself; or
- being shaved by the barber.
Also, "The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves."
From this, asking the question "Who shaves the barber?" results in a paradox because according to the statement above, he can either shave himself, or go to the barber (which happens to be himself). However, neither of these possibilities are valid: they both result in the barber shaving himself, but he cannot do this because he shaves only those men "who do not shave themselves".
Despite its popular name, however, Barber paradox is not really a paradox in the true sense of this word. A man who shaves exactly those men who do not shave themselves simply cannot and does not exist, and there are virtually no reasons to expect the opposite. This is in contrast with the set of all sets that do not contain themselves (from Russell's paradox), whose existence cannot be painlessly dismissed as it follows from the very intuitive and widely relied upon axioms of naive set theory.
This paradox is often attributed to Bertrand Russell (e.g., by Martin Gardner in Aha!). It was suggested to him as an alternative form of Russell's paradox, which he had devised to show that set theory as it was used by Georg Cantor and Gottlob Frege contained contradictions. However, Russell denied that the Barber's paradox was an instance of his own:
That contradiction [Russell's paradox] is extremely interesting. You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning.
— Bertrand Russell, The Philosophy of Logical Atomism
This point is elaborated further under Applied versions of Russell's paradox.
In first-order logic
This sentence is unsatisfiable (a contradiction) because of the universal quantifier . The universal quantifier y will include every single element in the domain, including our infamous barber x. So when the value x is assigned to y, the sentence can be rewritten to , which simplifies to , a contradiction.
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In Prolog, one aspect of the barber paradox can be expressed by a self-referencing clause:
shaves(barber, X) :- male(X), not shaves(X,X). male(barber).
In his book Alice in Puzzleland, the logician Raymond Smullyan had the character Humpty Dumpty explain the apparent paradox to Alice. Smullyan argues that the paradox is akin to the statement "I know a man who is both five feet tall and six feet tall," in effect claiming that the "paradox" is merely a contradiction, not a true paradox at all, as the two axioms above are mutually exclusive.
A paradox is supposed to arise from plausible and apparently consistent statements; Smullyan suggests that the "rule" the barber is supposed to be following is too absurd 