Barber paradox


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. 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.

Paradox

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?
Answering this question results in a contradiction. The barber cannot shave himself as he only shaves those who do not shave themselves. Thus, if he shaves himself he ceases to be the barber. Conversely, if the barber does not shave himself, then he fits into the group of people who would be shaved by the barber, and thus, as the barber, he must shave himself.
A barber is one who shaves those who do not shave themselves. So, can a barber shave himself?
This paradox is often incorrectly attributed to Bertrand Russell. It was suggested to Gardner as an alternative form of Russell's paradox, which Russell 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:
This point is elaborated further under Applied versions of Russell's paradox.

In first-order logic

This sentence is unsatisfiable 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 is an instance of the contradiction.