In a certain small town, there is a barber named Bill. Since Bill is the only barber in the town, he decides that he will shave all the town's residents who do not shave themselves, but (obviously) not the ones who do.

If Bill follows this rule, will he shave himself or not?

There is another possible solution (based on a time-honoured approach to similar paradoxes such as the Liar and Russell's). We can give up what is known as Bivalence- i.e. the claim that every sentence is either determinately true or determinately false. So sentences, and in particular, the crucial sentence "Bill shaves himself" can come out either true, false, or neither (think of neither, intutively, as "is a borderline case" or "sorta"- there is a very well worked out area of mathematical logic dealing with such middle possibilities.) Anyway, imagine now that Bill shaves the left side of his face every morning, but does nothing to the right side and lets the beard there grow long. Then we might say that the sentence "Bill shaves himself" is neither true nor false, it gets the middle value, and as a result, the sentence "Bill shaves himself if and only if it is not the case that Bill shaves himself" can turn out true, since it is an "if and only if" sentence where both sides have the same truth value (i.e. "Neither" or "sorta").

Interesting fact: I once attended a party at a university philosophy department where costumes were encouraged. One of the philosophers shaved half his beard and claimed he was dressed as the Barber who shaves only those who fail to shave themselves.

One interesting aspect of this sort of solution is that it is open to what are called 'revenge' problems. So, even assuming that we accept that some people shave themselves, other people do not shave themselves, and still other people neither shave themselves nor do not shave themselves, we can get a new paradox by assuming that Bill has a cousin in a neighbouring town who decides to shave exactly those people who either do not shave themselves, or who neither shave themselves nor do not shave themselves.

Posted by RoyCook on 2003-10-06 13:39:13