Which Alice?

And That's the Beautiful Part of It!
"Now look," said Humpty Dumpty, a bit irritatedly, "suppose I
told you that there was a man, and he was six feet tall, and he wasn't
six feet tall—what would you say?"
"Obviously that there was no such man," replied Alice.
"Good! And suppose I told you that there was a barber who
neither shaved himself nor didn't shave himself—what would you
say?"
"That there was no such barber," replied Alice.
"Well, that's exactly the barber of this story! Such a barber could
neither shave himself nor not shave himself. Ergo, there is no such
barber. That's logic for you!"
This completely convinced Alice.
"There is a related problem which will put this problem in a still
better light," continued Humpty Dumpty. "In a certain town there
are two barbers—call them Barber A and Barber B. It is given that
Barber A shaves all the inhabitants who don't shave themselves, but
it is not given that he doesn't shave some other inhabitants as well.
As for Barber B, he never shaves any inhabitant who shaves
himself, but he doesn't necessarily shave all inhabitants who don't
shave themselves. Now, it is perfectly possible that Barbers A and B
exist; there is no contradiction in assuming they do."
"Then what's the problem?" asked Alice.
"The problem has two parts: Does Barber A shave himself or
doesn't he? And does Barber B shave himself or doesn't he?"
Alice thought about this for a bit. "Barber A shaves himself and
Barber B doesn't," answered Alice, quite proudly.
"Good! Very good!" Humpty Dumpty exclaimed. "Now, can you
tell me why?"
"Because," replied Alice, quite confidently, "if Barber A didn't
shave himself, then he would be one of those who don't shave
themselves, but since he shaves all such people, he would have to
shave himself. This is a contradiction. Therefore, he doesn't shave
himself. As for Barber B, if he shaves himself, then he would be
shaving someone who shaves himself, which he never does. Therefore,
Barber B can't shave himself."
"Now you're learning!" said Humpty Dumpty. "It's really lucky
for you that you have such a good teacher!"
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