Barbershop Paradox - The Paradox

The Paradox

Briefly, the story runs as follows: Uncle Joe and Uncle Jim are walking to the barber shop. There are three barbers who live and work in the shop—Allen, Brown, and Carr—but not all of them are always in the shop. Carr is a good barber, and Uncle Jim is keen to be shaved by him. He knows that the shop is open, so at least one of them must be in. He also knows that Allen is a very nervous man, so that he never leaves the shop without Brown going with him.

Uncle Joe insists that Carr is certain to be in, and then claims that he can prove it logically. Uncle Jim demands the proof. Uncle Joe reasons as follows.

Suppose that Carr is out. If Carr is out, then if Allen is also out Brown would have to be in—since someone must be in the shop for it to be open. However, we know that whenever Allen goes out he takes Brown with him, and thus we know as a general rule that if Allen is out, Brown is out. So if Carr is out then the statements "if Allen is out then Brown is in" and "if Allen is out then Brown is out" would both be true at the same time.

Uncle Joe notes that this seems paradoxical; the hypotheticals seem "incompatible" with each other. So, by contradiction, Carr must logically be in.