Which Alice?

ALICE IN PUZZLE-LAND
is mad. We will show that in either case the Frog-Footman must be
sane.
Suppose the Fish-Footman is sane. Then his belief is correct,
which means that the Frog-Footman really is like the Fish-Footman,
which means the Frog-Footman is sane.
On the other hand, suppose the Fish-Footman is mad. Then his
belief is wrong, so the Frog-Footman is the opposite of the Fish-
Footman. Since the Fish-Footman is mad and the Frog-Footman is
the opposite, the Frog-Footman must be sane.
We therefore see that in either case (whether the Fish-Footman is
sane or whether he is mad) the Frog-Footman must be sane.
Incidentally, if the Fish-Footman had believed that the Frog-
Footman was opposite to him, rather than like him, what would this
have made the Frog-Footman?
Answer: The Frog-Footman would have had to be mad. I'll leave
this as an exercise for you to prove.
17
THE KING AND QUEEN OF DIAMONDS It is impossible that
anyone in this setup could believe he or she is mad, because a
sane person would know the truth that he is sane, and a mad person
would mistakenly believe that he is sane. Therefore the Queen
did not really believe she was mad, so the King was mad to believe
she did.
Nothing can be deduced about the sanity of the Queen.
18
WHAT ABOUT THESE THREE? Suppose the Hatter is sane.
Then his belief is correct, which means that the March Hare does
not believe that all three are sane. Then the March Hare must be
sane, because if he were mad, he would believe the false proposition
that all three are sane. Then the Dormouse, believing that the
March Hare is sane, must be sane, which makes all three sane. But
then how could the sane March Hare fail to believe the true
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