Labyrinths of Reason (19 page)

If A’s train ride is long enough, he can rule out
every
number.

When in doubt, simplify. The seven days or ten boxes (or alephnull integers!) are excess baggage. The paradox would “work” with six days/boxes, or five, or four. How far can it be trimmed? To two days? One day?

Let’s try it with one day. The judge sentences the prisoner to be executed on Saturday (the prisoner, of course, hears this). The prisoner is also not supposed to know the day beforehand. Of course, he does know it. The only way the executioner could surprise him would be not to hang him at all. That is ruled out from the start. Consequently there can be no surprise and no paradox. The judge has asked for something that cannot be. Saying “You will die on Saturday, and it will be a surprise” is like saying “You will die on Saturday, and 2 + 2 = 5.” The second part of the sentence is wrong, that’s all.

Go a little lighter on the pruning shears. Take the two-day case. The judge sentences the prisoner to die the following weekend, but the prisoner must not be able to deduce which day, Saturday or Sunday, beforehand. Does the paradox still exist?

It is again agreed that the prisoner will be executed on one of the two days no matter what. Sunrise comes Saturday with no hangman.
Then at breakfast Saturday the prisoner knows with certainty that he will be executed Sunday.

That, however, means that the sentence will
not
be carried out as specified: There will be no surprise. Conclusion: It is not possible to carry out the sentence by hanging the prisoner on Sunday.

Can the surprise requirement be met on Saturday? Well, it depends on whether the prisoner expects a Saturday execution. There are two possibilities. Either the prisoner expects the hangman on Saturday, or he doesn’t.

The prisoner might figure, “Well, I’m a goner all right,” and leave it at that. He may not have any opinion about which day it will be. In that case, the executioner need only hang him on Saturday to satisfy the judge. (Sunday is still out. Even the most stoic prisoner will realize that he will die Sunday if not hanged Saturday.)

The paradox’s hook lies in the alternative, that the prisoner
does
analyze his plight and expects the hangman on Saturday. Then the executioner will have failed to meet the surprise provision.

Forget that this a logic paradox for a moment. What would you do if you were the executioner? You have to execute the prisoner on Saturday or Sunday, and you have to follow the judge’s orders if at all possible.

Apparently, a reasonable executioner trying his level best to comply with the orders would almost have to choose Saturday for the hanging. There’s no chance of a Sunday hanging being unanticipated. At least with Saturday the executioner can hope that the prisoner didn’t give the matter much thought.

So the executioner takes the prisoner to the gallows at sunrise Saturday. As is customary, the prisoner is allowed to say his last words. He turns to the judge and says, “Your executioner failed! I expected to be hanged today. Only by hanging me today would there be any chance of my not anticipating it. But I did!”

Prisoner and executioner are locked in a battle of wits: Each can anticipate any reason the other might think up for choosing any specific day. The paradox can be short-circuited with a “stupid” prisoner, who doesn’t contemplate his fate or try to second-guess. But when prisoner and executioner are the perfect logicians of a logic puzzle, the paradox is profound indeed.

Scottish mathematician Thomas H. O’Beirne pointed out that it is possible for one person to make a prediction about future events
which is true but which others do not know to be true until afterward. The judge is correct in saying that the prisoner will be surprised, even though the prisoner doesn’t know it (yet).

This can be made clearer by reformulating the paradox thus: The judge sentences the prisoner to die sometime the following week (leaving the date to the executioner). Then the judge hops in a time machine and sets the dial for a week or more hence. Arriving in the near future, the judge steps out, gets a newspaper, and reads that the prisoner was hanged the Tuesday after sentencing. In the prisoner’s final interview, he said he was surprised at the date; he thought they’d let him live until the end of the week. A cruel idea pops into the judge’s head. “Suppose I go back to the day of sentencing and tell the prisoner that he won’t be able to guess the day of his execution,” he thinks. “That will be a correct statement because, here in the future, I know he was surprised. And just telling him that will drive him crazy!”

The judge gets in the time machine again and returns to the day of sentencing. He gets out and says to the prisoner (as in the original paradox), “You will be hanged next week but you will not be able to guess the day of execution beforehand.” The prisoner concludes he can’t be hanged, and he is wrong; the judge is right.

Anything wrong here? Well, the judge really saw the consequences of his
original
sentence (which did not say anything about the date being unexpected). Telling the prisoner that he will be surprised changes things—possibly insignificantly, possibly a lot. The prisoner’s surprise is no longer certain.

While in the future, the judge might also learn that a surprise party he threw for his sister on her birthday was unexpected. If he went back to the previous week and told his sister that, then obviously she would not be surprised. Imparting
some
valid information about the future can invalidate the information.

If the judge is free to use the time machine as he likes, this may not be much of a limitation. After telling the prisoner that he will be surprised, the judge can zip back into the future to make sure that the prediction still holds. If it does, fine. If not, he can come back and modify the sentence until the prediction and the reality coincide. The result would be a prediction which is true but which the prisoner cannot know to be true until after the fact.

Though it seems quite different, “Berry’s paradox” (after librarian G. G. Berry, who described it to Bertrand Russell) has something of the same flavor. Think of the least integer not namable in fewer than nineteen syllables. Certainly some integer qualifies as
just that. But the phrase “the least integer not namable in fewer than nineteen syllables” is a description of a certain number, and that description has eighteen syllables. Ergo, the least integer not namable in fewer than nineteen syllables can in fact be named in eighteen syllables!

Berry’s paradox defies facile resolution. Once you’re ready to write it off to ambiguous wording, bring in that stock character of paradoxes, the omniscient being. The being seemingly could be aware of every possible description of every number or sentence. To this being, one number
is
the least integer not namable in fewer than nineteen syllables! The being, like the judge, seems to know something forbidden to us.

All this perhaps shows that the judge can know what he is supposed to know in the paradox. More interesting, though, are the deductions of the prisoner and the executioner. Who, if anyone, is right?

The paradox of the unexpected hanging poses the question: What is knowledge? The prisoner is caught up in a web of second-guessing, third-guessing, and nth-guessing. He
thinks
he knows he can’t be hanged on Saturday. The executioner thinks he knows that the prisoner
can’t
know the day of his execution. The paradox raises the twin specters of being right for the wrong reason or being wrong for the right reason. These same concerns frequently pop up in the philosophy of science. There (more than in criminal justice!) we often “know” things by chains of reasoning as convoluted as the prisoner’s.

Like most common words, “know” has a lot of flexibility built into it. We all say things like “I know the Celtics are going to win the championship” when there is actually considerable doubt. In science, you usually want to know things with greater certainty than that.

For years philosophers defined knowledge by a set of three criteria called the “tripartite account.” The idea was that these criteria would be met if, and only if, you knew something.

Let’s take an example from the supposedly certain realm of mathematics. Suppose you know that 4,294,967,297 is a prime number (a number that cannot be divided evenly by any whole number other than 1 and itself). Then these three conditions must hold:

First
, you
believe
that 4,294,967,297 is prime. If you don’t even
believe it, you can’t know it. We wouldn’t say that a flat-earther knows the earth is round.

Second
, your belief that 4,294,967,297 is prime is
justified
. You must have good reason for believing it. You can’t believe it just because of a mistake in arithmetic. Nor can you believe it on the basis of a hunch, or reading tea leaves, or a fit of temporary insanity.

Third
, 4,294,967,297 really
is
prime. Obviously, if the statement is wrong, you don’t know it as fact.

The first time you hear of it, the tripartite account sounds almost too trite to be of any interest. But knowledge is a more complex business than it appears. The second criterion is most troublesome of the three. Why require “justified” belief at all? It might seem that it is enough to believe something and have it be true.

A two-criteria definition would include anyone who “lucked out” and was right about something for bad reasons. After Kennedy’s assassination (1963) and the Reagan assassination attempt (1981), several psychics came forward with claims that they had predicted the events. At least some had made predictions about the Presidents being in peril on the approximate dates, and these predictions had been published or appeared in press releases before the event. The same psychics also made scores of other predictions that did not come true. Washington psychic Jeane Dixon makes so many predictions each year that she cannot help being right many times. If this is “knowledge,” it is not a very useful kind.

It is not always easy to say what constitutes “good reason” for believing something. In 1640 French mathematician Pierre de Fermat felt he had reason to believe that 4,294,967,297 is prime. He noticed that you can produce prime numbers with the formula

Fermat’s formula has a multi-level exponent. A regular exponential expression, like 2
³
, means the lower number (2) multiplied by itself the number of times indicated by the little superscript number (3). 2
³
is 2×2×2, or 8. In Fermat’s formula, you choose any number for
n
, evaluate the topmost expression (2
ⁿ
) and multiply the bottom 2 by itself
that
many times. Then add 1.

For instance, 2
^{2 1}
+ 1 is 5, and 5 is prime. 2
^{2 2}
+ 1 is 17, 2
^{2 3}
+ 1 is 257, 2
^{2 4}
+ 1 is 65,537, and all are primes. Fermat suspected that 4,294,967,297 (2
^{2 5}
+ 1) and all the higher members of this series must be prime.

Lots of other people believed that too. There was empirical evidence
and the backing of authority. But as you’ve probably guessed, 4,294,967,297 isn’t prime at all. Swiss mathematician Leonhard Euler discovered that it is 641 times 6,700,417.
¹

Belief, justification, truth—the history of science includes examples of all permutations of these three conditions. Let’s use T to indicate that a condition holds and F to indicate that it doesn’t, and list the criteria in the order above.

TTT stands for a justified true belief or what is held to be true knowledge. In this category goes most scientific belief; that portion of it that is right, anyway.

FTT represents a justified truth that is disbelieved. There are many instances of this; one is creationism, the quasi-scientific cult that rejects evolution in the face of overwhelming evidence. The fuddy-duddyism of those who reject new discoveries (the French Academy’s refusal to accept meteorites; physicist Herbert Dingle’s crankish refutation of relativity) is FTT. So is the inertia that made physicist Max Planck complain (1949): “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.”
²

TFT is an unjustified true belief; being right for the wrong reasons, as in a psychic’s lucky guess. Again, there are many examples. Democritus of the fifth century
B.C
. held the true belief that all matter is made of particles so tiny as to be invisible: atoms. Although Democritus’ works are lost, it is unlikely he had anything we would consider valid evidence. His was a philosophical insight that turned out to be right. (The serendipity is less striking for the fact that the atoms of twentieth-century physics are not indivisible as Democritus thought.)

TTF is a justified belief that is
wrong
. It is interesting to reflect that many successive cosmological views have been in this category. The ancients had the justification of their own senses for believing that the sun moved around the earth. Although generations of schoolteachers have cited this as the epitome of wrongness, it took a
ertain intellectual daring to postulate that the sun is a physical body that circles the world at a great distance and creates night and day. When Copernicus placed the sun at the center of the universe, he had justification, but was just as wrong. Since TTF beliefs are false, I cannot cite a generally accepted current belief as an example. It should come as no surprise if large parts of our present cosmology are wrong too.