Which Alice?

ALICE IN PUZZLE-LAND
means that the Dormouse stole the eggs. This means also that the
March Hare stole the milk.
So the March Hare stole the milk; the Hatter stole the butter (and
told the truth); and the Dormouse stole the eggs (and lied).
THE FINAL TALE If the White Rabbit had known a little
more logic, he could never have said that Bill was right and the
Knave was wrong, because it is logically impossible that Bill could be
right and the Knave could be wrong! In other words, I am saying
that if Bill is right, then the Knave has to be right. Let me prove this.
Suppose Bill the Lizard is right. Then what he said is true, which
means that either the March Hare or the Dormouse is right (or
possibly both). Suppose the March Hare is right. Then the Cook
must be right (because the March Hare said that the Cook and the
Cheshire Cat are both right). On the other hand, if the Dormouse is
right, then again the Cook is right (because the Dormouse said so).
So in either case (whether the March Hare or the Dormouse is
right) the Cook must be right. But either the March Hare or the
Dormouse is right. Therefore the Cook must be right in either case.
This proves that the Cook is right (assuming, of course, that the
Lizard was right, which we are doing). Also, the March Hare said
that the Cheshire Cat (as well as the Cook) is right, and the
Dormouse said that the Caterpillar (as well as the Cook) is right.
Therefore either the Cheshire Cat or the Caterpillar is right
(because either the March Hare or the Dormouse is right: if the
former, the Cheshire Cat is right; if the latter, the Caterpillar is
right). Well, the Hatter said that either the Cheshire Cat or the
Caterpillar is right, so the Hatter is right. This means that the Cook
and Hatter are both right—which is exactly what the Knave of
Hearts said! So the Knave of Hearts is right (all providing, of
course, that Bill is right).
So we have proved that if Bill the Lizard is right, then the Knave
of Hearts must be right. Therefore the White Rabbit was quite
wrong when he said that Bill was right and the Knave was wrong. So
the Rabbit was wrong.
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