Which Alice?

Which Alice?
"Now, just a minute," said Michael, "you don't expect me to
believe the Red King's theory, do you?"
"Why not?" I replied, scarcely able to restrain a smile. "It's the
most ridiculous theory I ever heard in my life!" "Why?" I asked.
"Isn't the theory logically possible?" "Of course not!" cried
Michael. "The whole theory is crazy from beginning to end!"
"But the King proved that his theory was correct, didn't he?" I
asked.
There was a long, thoughtful pause. Alice was the first to break
the silence.
"Not really," she said, "the Red King's proof was fallacious." "Just
where was the fallacy?" I innocently inquired. "The entire
argument was circular," replied Alice. "The proofs that a person
who believes he is of Type A must be awake, and that a person who
believes he is awake must be of Type A—these proofs depend on
the theory being correct in the first place!" "Very good!" I replied.
"That's exactly the fallacy!" "So I was right!" exclaimed Michael.
"The theory is false!" "No, no!" I sharply corrected. "Alice didn't
prove that the theory is false; she merely proved that the Red
King failed to prove that the theory is true. But just because the
King's proof was wrong doesn't mean that the theory itself is
wrong."
"It's the silliest theory I ever heard!" said Michael quite emphatically.
"Silly is one thing; logically impossible is another," I
answered. "I
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