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?
Maybe it's just me, but the way I'm interpreting this problem, there is a definite solution. I see three subgroups of people in the town:
1) people who don't need to shave ever (most ladies and some guys)
2) people who have chosen never to shave (bearded guys and maybe bearded ladies if the town has a permanent freak show)
3) people who choose to shave.
If you are in group 3, your choices for shavers are either yourself or Bill. For Bill, the choice is only himself.
The fact the problem is told in the present tense means Bill is applying his rule starting now. This is key to the problem having a solution. If the problem is told in the past tense, then the contradiction holds.
If Bill is in group 3, he shaves himself. Therefore according to his rule, going forward he cannot shave himself anymore. Because there are no more options, he must stop shaving all together, and he moves into group 2 permanently and doesn't shave himself.
If Bill is in group 2 he will either stay in group 2 forever and not shave, or he will at some point decide he wants to get rid of his beard. However he can't, since the instant he starts shaving himself he'll have to stop or he'll contradict the rule. So he stays in group 2.
If Bill is in group 1, he'll either stay in 1 forever, or he'll start having the need to shave later on, at which point he moves either moves into 2 and then has to never shave, or tries to go into 3. But again, the instant he tries he has to stop, so he goes into 2 whether he wants to or not.
So the answer to the question "Will he shave himself" is no. Again, if the problem was told in the past tense, and the question was "DOES he shave himself", well then we have issues.
not a contradiction?
