Which Alice?

ALICE IN PUZZLE-LAND
satisfy the five basic conditions which characterize Looking-Glass
logicians."
"Why is that?" asked Alice.
"Oh, that's very easy to prove!" replied Humpty Dumpty.
"Suppose a person is completely honest and also believes all and
only those statements which are false. Since he is honest, then of
course he satisfies Condition One. As for Condition Two, suppose
he claims a statement to be true. Then he really believes the
statement (because he is honest). Therefore it is false that he doesn't
believe the statement. But he believes everything that is false—even
false things about his own beliefs! So, since it is false that he doesn't
believe the statement, and since he believes everything which is
false, then he must believe the false fact that he doesn't believe the
statement—in other words, he believes that he doesn't believe the
statement. And since he believes that he doesn't believe the statement,
then he claims he doesn't believe it (because, we recall, he is
honest). Therefore he satisfies Condition Two.
"As for Condition Three, take any true statement. Since it is true,
he cannot believe it. Since he doesn't believe it, then he must believe
that he does believe it (because all his beliefs are wrong!). Then, since
he believes that he believes it, then he must claim that he believes it
(again, since he is honest). This proves that he satisfies Condition
Three.
"Conditions Four and Five are obvious," continued Humpty
Dumpty. "Take any statement and its opposite. One of them must
be true and the other must be false. Therefore he believes the false
one and doesn't believe the true one. So he doesn't believe both of
them, hence he satisfies Condition Four, but he believes at least one
of them, so he satisfies Condition Five.
"And that," concluded Humpty Dumpty, "is the entire story: A
Looking-Glass logician is honest but totally deluded. Conversely,
anyone who is both honest and totally deluded satisfies the five
conditions of being a Looking-Glass logician. Now you have the
key."
"One thing still puzzles me," said Alice. "Why is it that a Looking-
Glass logician never claims any statement and also claims its
opposite, and yet he will claim that the statement and its opposite
are both true?"
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