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The Paradox of the Unexpected Hanging Cambridge University Press 978-0-521-75613-6 - Knots and Borromean Rings, Rep-Tiles, and Eight Queens: Martin Gardner's Unexpected Hanging Martin Gardner Excerpt More information CHAPTER ONE The Paradox of the Unexpected Hanging “A new and powerful paradox has come to light.” This is the opening sentence of a mind-twisting article by Michael Scriven that appeared in the July 1951 issue of the British philosophical jour- nal Mind. Scriven, who bears the title of “professor of the logic of science” at the University of Indiana, is a man whose opinions on such matters are not to be taken lightly. That the paradox is indeed powerful has been amply confirmed by the fact that more than 20 articles about it have appeared in learned journals. The authors, many of whom are distinguished philosophers, disagree sharply in their attempts to resolve the paradox. Since no consensus has been reached, the paradox is still very much a controversial topic. No one knows who first thought of it. According to the Harvard University logician W. V. Quine, who wrote one of the articles (and who discussed paradoxes in Scientific American for April 1962), the paradox was first circulated by word of mouth in the early 1940s. It often took the form of a puzzle about a man condemned to be hanged. The man was sentenced on Saturday. “The hanging will take place at noon,” said the judge to the prisoner, “on one of the seven days of next week. But you will not know which day it is until you are so informed on the morning of the day of the hanging.” The judge was known to be a man who always kept his word. The prisoner, accompanied by his lawyer, went back to his cell. As soon as the two men were alone the lawyer broke into a grin. “Don’t you see?” he exclaimed. “The judge’s sentence cannot possibly be car- ried out.” “I don’t see,” said the prisoner. “Let me explain. They obviously can’t hang you next Saturday. Saturday is the last day of the week. On Friday afternoon you would 1 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-75613-6 - Knots and Borromean Rings, Rep-Tiles, and Eight Queens: Martin Gardner's Unexpected Hanging Martin Gardner Excerpt More information 2 Knots and Borromean Rings, Rep-Tiles, and Eight Queens Figure 1. The prisoner eliminates all possible days. still be alive and you would know with absolute certainty that the hanging would be on Saturday. You would know this before you were told so on Saturday morning. That would violate the judge’s decree.” “True,” said the prisoner. “Saturday, then is positively ruled out,” continued the lawyer. “This leaves Friday as the last day they can hang you. But they can’t hang you on Friday because by Thursday afternoon only two days would remain: Friday and Saturday. Since Saturday is not a possi- ble day, the hanging would have to be on Friday. Your knowledge of that fact would violate the judge’s decree again. So Friday is out. This leaves Thursday as the last possible day. But Thursday is out because if you’re alive Wednesday afternoon, you’ll know that Thursday is to be the day.” “I get it,” said the prisoner, who was beginning to feel much bet- ter. “In exactly the same way I can rule out Wednesday, Tuesday and Monday. That leaves only tomorrow. But they can’t hang me tomor- row because I know it today!” In brief, the judge’s decree seems to be self-refuting. There is nothing logically contradictory in the two statements that make up his decree; nevertheless, it cannot be carried out in practice. That is how the paradox appeared to Donald John O’Connor, a philosopher at the University of Exeter, who was the first to discuss the para- dox in print (Mind, July 1948). O’Connor’s version of the paradox concerned a military commander who announced that there would be a Class A blackout during the following week. He then defined a © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-75613-6 - Knots and Borromean Rings, Rep-Tiles, and Eight Queens: Martin Gardner's Unexpected Hanging Martin Gardner Excerpt More information The Paradox of the Unexpected Hanging 3 Class A blackout as one that the participants could not know would take place until after 6 p.m. on the day it was to occur. “It is easy to see,” wrote O’Connor, “that it follows from the announcement of this definition that the exercise cannot take place at all.” That is to say, it cannot take place without violating the def- inition. Similar views were expressed by the authors of the next two articles (L. Jonathan Cohen in Mind for January 1950, and Peter Alexander in Mind for October 1950), and even by George Gamow and Marvin Stern when they later included the paradox (in a man- to-be-hanged form) in their book Puzzle Math (Viking, 1958). Now, if this were all there was to the paradox, one could agree with O’Connor that it is “rather frivolous.” But, as Scriven was the first to point out, it is by no means frivolous, and for a reason that completely escaped the first three authors. To make this clear, let us return to the man in the cell. He is convinced, by what appears to be unimpeachable logic, that he cannot be hanged without contra- dicting the conditions specified in his sentence. Then on Thursday morning, to his great surprise, the hangman arrives. Clearly he did not expect him. What is more surprising, the judge’s decree is now seen to be perfectly correct. The sentence can be carried out exactly as stated. “I think this flavour of logic refuted by the world makes the paradox rather fascinating,” writes Scriven. “The logician goes pathetically through the motions that have always worked the spell before, but somehow the monster, Reality, has missed the point and advances still.” In order to grasp more clearly the very real and profound linguis- tic difficulties involved here, it would be wise to restate the paradox in two other equivalent forms. By doing this we can eliminate vari- ous irrelevant factors that are often raised and that cloud the issue, such as the possibility of the judge’s changing his mind, of the pris- oner’s dying before the hanging can take place, and so on. The first variation of the paradox, taken from Scriven’sarticle, can be called the paradox of the unexpected egg. Imagine that you have before you ten boxes labeled from 1 to 10. While your back is turned, a friend conceals an egg in one of the boxes. You turn around. “I want you to open these boxes one at a time,” your friend tells you, “in serial order. Inside one of them I guarantee that you will find an unexpected egg. By ‘unexpected’ I mean that you will not be able to deduce which box it is in before you open the box and see it.” © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-75613-6 - Knots and Borromean Rings, Rep-Tiles, and Eight Queens: Martin Gardner's Unexpected Hanging Martin Gardner Excerpt More information 4 Knots and Borromean Rings, Rep-Tiles, and Eight Queens Figure 2. The paradox of the unexpected egg. Assuming that your friend is absolutely trustworthy in all state- ments, can this prediction be fulfilled? Apparently not. Your friend obviously will not put the egg in box 10, because after you have found the first nine boxes empty you will be able to deduce with certainty that the egg is in the only remaining box. This would con- tradict your friend’s statement. Box 10 is out. Now consider the situ- ation that would arise if your friend were so foolish as to put the egg in box 9. You find the first eight boxes empty. Only 9 and 10 remain. The egg cannot be in box 10. Ergo it must be in 9. You open 9. Sure enough, there it is. Clearly it is an expected egg, and so your friend is again proved wrong. Box 9 is out. But now you have started on your inexorable slide into unreality. Box 8 can be ruled out by precisely the same logical argument, and similarly boxes 7, 6, 5, 4, 3, 2 and 1. Confident that all ten boxes are empty, you start to open them. What have we here in box 5? A totally unexpected egg! Your friend’s predic- tion is fulfilled after all. Where did your reasoning go wrong? To sharpen the paradox still more, we can consider it in a third form, one that can be called the paradox of the unexpected spade. Imagine that you are sitting at a card table opposite a friend who is holding all the 13 spades. After shuffling them, fanning them so you can’t see the faces and dealing a single card face down on the table, your friend asks you to name slowly the 13 spades, starting with the ace and ending with the king. Each time you fail to name the card on the table your friend will say “No,” and when you name the card correctly, “Yes.” “I’ll wager a thousand dollars against a dime,” your friend says, “that you will not be able to deduce the name of this card before I respond with ‘Yes.’” © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-75613-6 - Knots and Borromean Rings, Rep-Tiles, and Eight Queens: Martin Gardner's Unexpected Hanging Martin Gardner Excerpt More information The Paradox of the Unexpected Hanging 5 2 3 4 5 6 A 7 8 9 J10 Q K Figure 3.
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