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.
Leiber, J., Paradoxes. London: Gerald Duckworth & Co., Ltd., 1993.
Gardner, M., The Paradox of the Unexpected Hanging. The Unexpected Hanging and Other Mathematical Diversions. Chicago: University of Chicago Press, 1991, Chapter 1.