Today, I'll try to tell you, "I am a liar." The
University of Houston's College of Engineering
presents this series about the machines that make our
civilization run, and the people whose ingenuity
created them.
Paradoxes reveal skeletons
in the closet of reason, says philosopher Justin
Leiber. They expose how logic lies at the mercy of
the circularity of language. Take the the paradox
of the "Surprise Execution": On Saturday, the judge
finds a prisoner guilty and condemns him to be
hanged. He says, "The hangman will arrive in your
cell on one of the next seven days to take you
away. I won't tell you when. His arrival will be a
complete surprise."
Then, the prisoner realizes he can't be hanged next
Saturday, because, if he hasn't yet been hanged on
one of the first six days, the hangman's arrival on
Saturday will be no surprise.
But wait: since he ruled out Saturday, a Friday
hanging couldn't be a surprise after Thursday. Thus
he moves back through the week until it's clear no
hanging is possible.
So has he really slipped the noose? To expose the
illogic of the situation, consider what happens
after the prisoner draws his conclusion. He smiles
and relaxes -- sits back to read the papers and
waits for the week to pass. Then, on Saturday the
hangman does arrive, and he is seriously surprised
after all.
Try a variation: A father says, "Son, I have a
great present for your birthday. It'll be a
complete surprise. You'll never guess it. It's that
bike you've always wanted." Days pass and the boy's
bafflement rises. "My father doesn't lie. How can I
make sense of this? What'll he give me?" The day
comes and his father gives him the bicycle. The
father hasn't lied: the present is the bike, and
the boy is surprised.
So, you might say, the very reasoning that gave the
prisoner hope condemned him in the end. A better
way of looking at it is that what one person knows
to be true, another does not.
Perhaps that's also at the root of the paradox in
which Bill says, "Joe always lies," and Joe shoots
back, "Bill always tells the truth." Then we
realize: if Joe's a liar, then Bill does not tell
the truth. That means Joe is in fact honest, and
... Well, you get the picture. Joe and Bill leave
us scrabbling to figure out who the liar really is.
And so mathematicians write books about paradoxes.
For a computer engineer, logical paradoxes promise
serious mischief in a machine's operation.
Paradoxes do more than make nice games. They're
profoundly important just because they expose
skeletons in the closets of reason. They warn us
what a minefield language is. The most minor
failing of words opens the door to logical
conflict.
I began by saying, "I am a liar." But if I am, then
you have to negate the sentence. I am in fact a
truth teller. But if I'm a truth teller, that means
I am in truth a liar. And if I keep playing this
game -- you won't get to hear the news.
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.
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