Which Alice?
Which Alice?
Which Alice?
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Solutions to the Puzzles<br />
proposition that all three are sane? Therefore it is contradictory to<br />
assume the Hatter sane; the Hatter must really be mad.<br />
Since the Hatter is mad, his belief is wrong, and therefore the<br />
March Hare does believe that all three are sane. Of course the March<br />
Hare is wrong (since the Hatter is not sane), and so the March Hare is<br />
also mad. Then the Dormouse, believing the March Hare is sane, is<br />
also mad, and so all three are mad (which is not too surprising!).<br />
19<br />
AND THESE THREE? To begin with, the Gryphon and the<br />
Mock Turtle must be alike, because the Mock Turtle believes the<br />
Gryphon is sane. If the Mock Turtle is sane, his belief is correct,<br />
which means that the Gryphon is also sane. If the Mock Turtle is<br />
mad, his belief is wrong, which means that the Gryphon is not really<br />
sane, but also mad. Therefore the two are alike.<br />
Now I shall prove that the Lobster is mad. Well, suppose he were<br />
sane. Then his belief is correct, hence the Gryphon does believe that<br />
exactly one of the three is sane. But this is impossible, because if the<br />
Gryphon is sane, so is the Mock Turtle (and also the Lobster), so it is<br />
false that exactly one is sane (all three are), so the sane Gryphon<br />
couldn't believe it. On the other hand, if the Gryphon is mad, then<br />
it is true that exactly one is sane (namely the Lobster, since the Mock<br />
Turtle is also mad), but a mad creature can't believe a true<br />
statement! Therefore the assumption that the Lobster is sane leads<br />
to a contradiction, so the Lobster can't be sane; he must be mad.<br />
Now we know the Lobster is mad. Therefore it is not really true<br />
that the Gryphon believes that exactly one of the three is sane. If<br />
the Gryphon is mad, then the Mock Turtle is also mad, which<br />
means that all three are mad, and hence it is false that exactly one<br />
of them is sane. This means that the Gryphon, being mad, must<br />
believe all false propositions—in particular that exactly one of<br />
the three is sane, but we have already proven that he doesn't.<br />
This is a contradiction, hence the Gryphon can't be mad.<br />
Therefore the Gryphon is sane, and the Mock Turtle (being of the<br />
same type as<br />
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