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Found this on Yahoo!Answers.
Why is this a paradox? Can you solve it?
There is a village where the barber shaves all those and only those who do not shave themselves. Who shaves the barber?
Answer& link
https://answers.yahoo.com/question/index;_ylc=X3oDMTB2Y2t1c212BF9TAzIxMTUzMDA5ODgEc2VjA2ZwBHNsawN0b2RheXE-?qid=20090625164520AAgNZ4A
OK. take a deep breath and read this slowly, because it is very tricky. First two definitions:
def 1) All and only those who shave themselves = S
def 2) All people in the village, including the barber, are either members of S or members of not-S. there is no one in the village that is a member of both.
If the barber shaved himself then he would be a member of S.
If the barber does not shave himself then he would be a member of not-S.
A barber who must shave all and only not-S can not shave any S
The barber can not shave himself, because he would be shaving a member of S
if the barber is not a self shaver then he is a member of not-S
The barber must shave himself because he must shave all members of not-S
Thus he must both shave and not shave himself
This is the paradox.
No one else in the village can shave the barber because only the barber can shave members of not-S.
This is a variation of Russel's paradox, and there are several ways to solve, or attempts, to solve this depending on how it is interpreted. I'm not even going to attempt a solution here. Someone could, and probably has, written a doctoral thesis on the subject.