Friday, April 12, 2013

The Sack of Samarkand: A Lesson in the Abuse of Logic.

Note: The following legend is as imaginary as the paradox it illustrates.

Genghis Khan was a good man at heart, and he hated to massacre a city without warning it first. So when he reached the gates of Samarkand one Sunday evening, he summoned a messenger from the city. "One day this week," he said, "I will sack the city. I'm not going to tell you which day it will be, because it will be a surprise attack. But tell your king to be prepared." The terrified messenger raced back to the city and went into the royal court, where he told the Shah the bad news. To his great surprise, the Shah laughed.

"Open the gates," he said. "The city is saved."

"How is that?" asked the vizier.

"Genghis Khan," said the Shah, "is an honest man, and I expect him to keep his word that he will attack us by surprise. But he can't surprise us! Imagine that that it were Thursday night, and that Genghis Khan had not attacked. That would mean that the attack would have to be on Friday. But that can't be, because he said it will be a surprise, and we would know that it was coming. Genghis, therefore, cannot logically attack on Friday. But if it's not on Friday, then we know that it can't be on Thursday either. For if Wednesday night arrived without an attack, then we would know (since it couldn't be on Friday) that it had to be on Thursday. Again, that wouldn't be a surprise. The same logic means, in fact, that it can't be on Wednesday either, or on Tuesday, or on Monday! It is therefore logically impossible for Genghis to launch a surprise attack next week. Glory to God!"

A banquet was immediately prepared, and the city spent the night celebrating. And it was to everyone's astonishment when Genghis Khan attacked at noon on Tuesday. Facing no resistance, his soldiers burned the mosque, murdered the city's citizens, and stacked their heads in towering pyramids. Genghis gave a hearty laugh: he had counted on the Shah's cleverness all along.

This seems like a paradox. On the one hand, the Shah's reasoning was flawless. It should have been logically impossible to catch the Samarkandians by surprise. But he did surprise them in the end—how can that be?

Perhaps there is a problem with the Shah's reasoning or with Genghis's original promise. A sprawling Wikipedia article documents the efforts of philosophical contortionists to point these problems out. "Genghis's definition of surprise changes," says one. "He unwittingly shifts his ground!" "Surprise is a meaningless concept," charges another. "This paradox shows its fundamental instability." "This paradox," says the most grotesque of them all, "shows us that we don't really know what a surprise is. If we work harder and think more clearly, we'll find a way out of our confusion."

But the way to solve the problem involves no logical analysis. We must instead pay attention to what we already know: the real function of the concept surprise in speech. Genghis, in particular, applies the term to an event that he thinks is likely to catch the city off guard. Since he's speculating about the behavior of other people, it's not something that he can guarantee, but it is something that he can predict.

How should we test Genghis's proposition? We must not attempt to tease out what the city's preparedness logically must be. People, after all, don't get surprised in logical space. If I drew a picture of a toothless alligator, it wouldn't protect me in the Everglades.

Instead, we must determine the real likelihood that the attack will shock the Samarkandians. Genghis's own probabilistic analysis, which is reasonably good, goes like this:
Case I: If the Shah is an untutored ignoramus, then he will tremble in fear, prepare the city, and be surprised by the attack whenever it comes.
Case II: But if the Shah is a master of logic, then he will deduce that the attack just can't happen, and end up just as surprised. 
No matter what the Shah does, Genghis is justified in believing that his attack will be a surprise. And the Shah goes wrong when he concludes that Genghis Khan can't surprise him—not because his logic is flawed, but because it's completely disconnected from reality.

Does anyone still see a paradox?
Logic always loses to life.

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