The barber paradox

Paradoxes seen in 'Portal' as a means of confusing logic-based systems.

In a particular imaginary town, it is a policy that all men must be shaven. There are only two options available to each man - either the man must shave himself, or he must visit the barber to be shaven. There is only one barber in town, and his business is predictably doing well.

However, consider the barber. Who shaves him?

Perhaps the barber shaves himself. If a man shaves himself, then he must not be shaven by the barber - but the barber cannot help be shaven by the barber if he shaves himself, since he is the barber. Conversely, perhaps the barber is shaven by the barber instead. If a man is shaven by the barber, he must not shave himself - but in this case the barber does shave himself, since he is the barber. We are left with a paradox, since neither alternative is possible.

This is a classic exposition of the well-known ‘Russell’s Paradox.’ Trivial answers can be given in the context of the story, such as ‘the barber is a woman, and doesn’t need to shave’ or ‘the rules don’t apply to him, as he doesn’t live in the town.’ However, this is to miss the point of the story. Assume what is implied in the story - that the barber needs to shave at some point, and that he is a man who lives in the town, and abides by its rules. The paradox takes a plausible scenario, and shows it to be logically impossible.

Paradoxes of this sort arise due to the inclusion of what is called ‘self-reference’. Self-reference is present in the story because the barber is part of the rules governing all the men in the town, yet he himself is one of the men, and thus must abide by those rules too. Other paradoxes demonstrate the self-reference problem too - the elegant and simple claim, ‘this sentence is false.’ If it’s true, then it’s false, and if it’s false, then it’s true. (The Liar paradox).