Which Alice?

Looking-Glass Logic
Lewis Carroll has told us very little about the other White
Knight; all he told us was that he once tried to put on the first White
Knight's helmet, which was very careless, considering that the first
White Knight was in it at the time.
Well, when Alice met him, she was totally baffled! So many
statements he made seemed to be wrong! Could he be one of those
people who always lie? thought Alice. No, she rejected this outright,
since her intuition told her that he was a completely sincere person.
But the things he said! First of all he told Alice that she was a
unicorn. When Alice asked him, "Do you really believe I am a
unicorn?" he answered, "No." Next he claimed that the White King
was asleep and dreaming of Alice, but then he said the White King
was not dreaming of anything. Then two contradictory statements
came up (I forget just what they were) and he first claimed that one
of them was true, then he claimed the other was false, and then he
claimed that they were both true.
At first, Alice thought that he was simply being inconsistent, but
she could never catch him in a direct inconsistency—that is, she
could find no statement which he claimed to be true and also
claimed to be false, though he would claim that the statement was
both true and false! Still, she could not get him to make separate
claims—one that the statement was true, and the other that the
statement was false.
After several hours of questioning, Alice gathered an enormous
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