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Thebarber paradox is apuzzle derived fromRussell's paradox. It was suggested toBertrand Russell as an illustration of theparadox, but he deemed it an invalid modification of his paradox.[1] The puzzle 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, which implies that no such barber exists.[2][3]
The barber is the "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself?[1]
Any answer to this question results in acontradiction:
In its original form, this paradox has no solution, as no such barber can exist. The question is aloaded question in that it assumes the existence of a barber who could not exist, which is a vacuous proposition, and hence false. There are other non-paradoxical variations, but those are different.[3]
This paradox is often incorrectly attributed toBertrand Russell (e.g., byMartin Gardner inAha!). It was suggested to Russell as an alternative form ofRussell's paradox,[1] which Russell had devised to show thatset theory as it was used byGeorg Cantor andGottlob 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[1]
This point is elaborated further underApplied versions of Russell's paradox.
This sentence says that a barberx exists. Itstruth value is false, as the existential clause is unsatisfiable (a contradiction) because of theuniversal quantifier. The universally quantifiedy will include every single element in the domain, including our infamous barberx. So when the valuex is assigned toy, the sentence in the universal quantifier can be rewritten to, which is an instance of the contradiction. Since the sentence is false for the biconditional, the entire universal clause is false. Since the existential clause is a conjunction with one operand that is false, the entire sentence is false. Another way to show this is to negate the entire sentence and arrive at atautology. Nobody is such a barber, so there is no solution to the paradox.[2][3]
