Which Alice?
Which Alice?
Which Alice?
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Looking-Glass Logic<br />
So <strong>Alice</strong> wrote the following:<br />
Proposition 2—Given any true statement, a Looking-Glass logician<br />
believes that he believes the statement.<br />
"Now," said Humpty Dumpty, "do you see why it is impossible for<br />
a Looking-Glass logician ever to believe a true statement?"<br />
"Not quite," admitted <strong>Alice</strong>.<br />
"This follows easily from Proposition One, Proposition Two, and<br />
Condition Four," he replied. "Take any statement that a Looking-<br />
Glass logician believes. By Proposition One, he believes that he<br />
doesn't believe the statement. Then he can't also believe that he<br />
does believe the statement (because by Condition Four he can<br />
never believe anything and also believe its opposite). Since he<br />
doesn't believe that he believes it, then the statement cannot be true,<br />
for if it were true, then by Proposition Two he would believe that he<br />
believes it. But he doesn't believe that he believes it—therefore it<br />
can't be true. So, you see, a Looking-Glass logician never believes<br />
any true statement; all things believed by a Looking-Glass logician<br />
are false."<br />
<strong>Alice</strong> pondered over this for some time. "That's a rather difficult<br />
proof!" she said.<br />
"Oh, you'll get used to it in time!"<br />
<strong>Alice</strong> thought some more about this. "Tell me something else,"<br />
said <strong>Alice</strong>. "Does a Looking-Glass logician necessarily believe all<br />
false statements, or is it just that he believes only false statements?"<br />
"That's a good question, child," replied Humpty Dumpty, "and<br />
the answer is yes. Take any false statement. By Condition Five, he<br />
either believes the statement or he believes its opposite. He can't<br />
believe its opposite, because its opposite is true! Therefore he<br />
believes the false statement."<br />
"How extraordinary!" exclaimed <strong>Alice</strong>. "So a Looking-Glass<br />
logician believes all false statements and no true ones!"<br />
"Exactly," said Humpty Dumpty, "and that's the beautiful part<br />
of it!"<br />
"Another interesting thing," added Humpty Dumpty, "is that<br />
anyone who believes all false statements and no true ones, and<br />
who is also honest in expressing his beliefs—any such person must<br />
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