The Barbershop paradox was proposed by Lewis Carroll in a three-page essay titled “A Logical Paradox,” which appeared in the July 1894 issue of Mind. The name comes from the ‘ornamental’ short story that Carroll uses to illustrate the paradox (although it had appeared several times in more abstract terms in his writing and correspondence before the story was published). Carroll claimed that it illustrated “a very real difficulty in the Theory of Hypotheticals” in use at the time. Modern logicians would not regard it as a paradox but simply as a logical error on the part of Carroll.
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 two “hypotheticals” seem “incompatible” with each other. So, by contradiction, Carr must logically be in.
However, the correct conclusion to draw from the incompatibility of the two “hypotheticals” is that what is hypothesised in them (– that Allen is out) must be false under our assumption that Carr is out. Then our logic simply allows us to arrive at the conclusion “If Carr is out, then Allen must necessarily be in”.
In modern logic theory this scenario is not a paradox. The law of implication reconciles what Uncle Joe claims are incompatible hypotheticals. This law states that “if X then Y” is logically identical to “X is false or Y is true” (¬X ∨ Y). For example, given the statement “if you press the button then the light comes on”, it must be true at any given moment that either you have not pressed the button, or the light is on.
In short, what obtains is not that ¬C yields a contradiction, only that it necessitates A, because ¬A is what actually yields the contradiction.
In this scenario, that means Carr doesn’t have to be in, but that if he isn’t in, Allen has to be in.
A more detailed discussion of this apparent paradox may be found on Wikipedia.