Which Alice?

Solutions to the Puzzles
Now we use Alice's statement (which is given to be true) which is
that the White Rabbit and the Duchess are either both right or both
wrong. They can't be both right (since the Rabbit was not right),
hence they must be both wrong. Since the Duchess was wrong, then
the Gryphon must have stolen the tarts.
Chapter 3
THE CATERPILLAR AND THE LIZARD The Caterpillar believes
that he and the Lizard are both mad. If the Caterpillar were
sane, it would be false that he and the Lizard were both mad, hence
the Caterpillar (being sane) couldn't possibly believe this false fact.
Therefore the Caterpillar must be mad. Since he is mad, his belief is
wrong, so it is not true that both are mad. So the other one (the
Lizard) must be sane. Therefore the Caterpillar is mad and the
Lizard is sane.
THE COOK AND THE CAT If the Cook were mad, then it
would be true that at least one of the two is mad, and we would have
a mad person holding a true belief, which is not possible. Therefore
the Cook must be sane. Since she is sane, her belief is correct, hence
one of the two really is mad. Since it is not the Cook, it must be
the Cheshire Cat. Therefore the Cook is sane and the Cheshire Cat
is mad.
THE FISH-FOOTMAN AND THE FROG-FOOTMAN It is impossible
to determine from the given facts whether the Fish-Footman
is sane or mad, but we will prove that the Frog-Footman must be
sane. We prove this as follows:
There are two possibilities: Either the Fish-Footman is sane or he
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