Which Alice?

ALICE IN PUZZLE-LAND
was innocent. This means that Case 1 or Case 3 holds, and A must
be guilty (though Tweedledum couldn't know this, because as far as
he was concerned, Case 5 could also hold and B could be guilty).
We now see that if Tweedledum asked about B, then (since he
didn't solve the problem) either Case 1 or Case 2 holds. If he asked
about C, then either Case 1 or Case 3 holds. Now, Humpty Dumpty
asked whether Tweedledum inquired about B or about C. If he had
been told that Tweedledum inquired about B, then Humpty
Dumpty would know that either Case 1 or Case 2 holds, and hence
that either A or B was guilty, but he couldn't have told which. But
Humpty Dumpty was able to solve the problem; therefore, he must
have been told that Tweedledum inquired about C; so Humpty
Dumpty then knew that either Case 1 or Case 3 held, and that A is
the guilty one in each case. This proves that A was guilty.
Chapter 11
A QUESTION Yes, they do follow. Let us first consider
Proposition 1: Suppose a person believes he is awake. Either he
really is awake or he isn't. Suppose he is awake. Then his belief is
correct, but anyone who has a correct belief while awake must be of
Type A. Suppose, on the other hand, that he is asleep. Then his
belief is incorrect, but anyone who has an incorrect belief while
asleep must be of Type A. Therefore, whether he is awake or
asleep, he must be of Type A. This proves Proposition 1.
As for Proposition 2, suppose a person believes he is of Type A.
If he really is of Type A, then his belief is correct, but a person of
Type A can have a correct belief only while he is awake. On the
other hand, if he is of Type B, then his belief is incorrect, but a
person of Type B can have an incorrect belief only while he is
awake. So in either case he is awake, which proves Proposition 2.
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