Martin Gardner: execution by surprise and associated logical paradox

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"There was a great new paradox" - so began a little comprehensible to the uninitiated Michael Scriven article in the July issue of the British philosophical journal Mind for 1951. Scriven held the chair of philosophy of science at Indiana University, and in such matters with his opinion could not be ignored. The paradox was indeed magnificent. Sufficient proof - more than twenty articles about it in various journals. The authors, among whom were well-known philosophers, strongly disagreed about what should be considered as the decision of the paradox. Over the years, none failed to come to an agreement, so that the paradox and still is the subject of heated debate.

It is not known who first had the idea of ​​paradox. According to WV Quine, the logic of Harvard University, the author of one of the above-mentioned articles, the first of this paradox of talking in the early forties of this century, often formulating it as a puzzle of a man sentenced to death by hanging.

The condemned was thrown into prison on Saturday.
- You will be hanged at noon - the judge told him, - one of the seven days next week. But what kind of day it should happen, you will know only in the morning on the day of execution.
Judge was famous for the fact that always kept his word. Convicted he returned to the chamber, accompanied by a lawyer. As soon as they were left alone, the defender smiled contentedly.
- Do not you understand? - Exclaimed on.- After a judge's sentence can not be to enforce!
- How? I do not understand, - the prisoner muttered.
- I'll explain now. Obviously, the next Saturday you can not hang up: Saturday - the last day of the week, and on Friday afternoon you would have known for sure that you will be hanged on Saturday. Thus, on the day of execution you have learned to the formal notice on Saturday morning, therefore, ordered the judges would be violated.
- It's true - agreed to a prisoner.
- So, Saturday, of course, is no longer, - continued the lawyer - so Friday is the last day when you can hang. However, on Friday, you can not hang up, because after Thursday would have remained only two days - Friday and Saturday. Since Saturday afternoon can not be executed, you only have to hang on Friday. But once you become aware of this as early as Thursday, the judge will order again broken. Thus, Friday is also eliminated. So, the last day when you still could be executed, this Thursday. However, Thursday is also out, because staying in the living environment, you will understand at once that the execution should take place on Thursday.
- All clear! - Said the prisoner, vospryanuv duhom.- Similarly, I can exclude Wednesday, Tuesday and Monday. It only remains to tomorrow. But tomorrow I probably will not be hanged, because I know about it today!

In short, the judgment is self-contradictory. On the one hand, the two statements, from which it is composed, there is nothing logically contradictory, but on the other - to carry it out, turns out to be impossible. That is the paradox imagined DJ. O'Connor, a philosopher from the University of Exeter, published the first article on this paradox (Mind, July 1948). In the statement of the officer O'Connor appeared announcing his subordinates that the next week is to be held anxiety, which no one can know in advance until 18.00 the day on which it is assigned.

"It is easy to see - O'Connor wrote - from the definition, it follows that there is no alarm at all can not be." O'Connor, apparently meant that declare an alarm without violating the above conditions is impossible. A similar opinion is shared by the authors of the later articles.

If this paradox to be exhaustive, it would be possible to join the opinion of O'Connor, who the problem seemed "a mere trifle". However Scriven first noticed something that had escaped the attention of other authors and makes the problem is not so simple. To understand the essence of Scriven notes, back to the story of a man thrown into prison. Impeccably logical reasoning it would seem convinced that without breaking the sentence, the execution of the impossible. And suddenly, to the surprise of the convict, on Thursday morning in the camera is the executioner. Convicted Of course, this did not expect, but the most surprising, that the sentence was completely accurate - it is possible to enforce in full accordance with the wording. "It seems to me, - says Scriven - that is a gross intrusion of the outside world, destroying the subtle logical constructions, makes the paradox of a special piquancy Logic with touching regularity utters incantations which in the past led to the desired result, but the monster reality at this time refuses to obey. and continues to follow its own path. "

To understand the linguistic difficulties that we encounter in this paradox, should result in two new formulations of its equivalent first. This will help us to eliminate various kinds of factors that are not relevant to the case and a black-out the end result: the ability to change the judge sentence a prisoner to death penalty, etc.

_ Consider the first version of the paradox proposed Skryavenom, -paradoks with egg-surprise.

Imagine that you are standing in front of ten boxes, numbered with numbers from 1 to 10. You turn, and your friend puts in one of the boxes egg and asks you to turn back. "Open all the boxes one by one, - he says - first the first, then second, and so the order of a tenth guarantee that one of them is egg-surprise Calling egg surprise, I have a mind that you can not learn.. box room with the egg as long as you do not open the box and did not see the eggs. "

Suppose that your friend always tells the truth. Whether then it is feasible prediction? Obviously, no. He certainly does not put an egg in a box of 10, because, by opening the first nine frames and found nothing in them, you can say with certainty that the egg is the sole remaining in the box. It would be contrary to the prediction of your friend, so the tenth box excluded. Consider now what would happen if your buddy slowness hid an egg in the ninth box. The first eight boxes will be empty then, and remain in front of you two closed boxes: ninth and tenth. In the tenth box of eggs can not be, therefore, it is in the box 9. You open the ninth box and the egg, of course, there is. However, it is clear that the egg can not be considered a surprise. Thus, we have again proven that your buddy is wrong. Box 9 is also excluded. But at this very moment, you and "detached from reality": using similar reasoning can be eliminated first eighth of the box, then seventh, and so on, until the first! Finally, being absolutely sure that all the ten empty boxes, you start to open them one by one, and it is white ... What's in the box 5? Egg-surprise! So, in spite of all your reasoning prediction proved correct your friend. So you made a mistake, but what?

To make the paradox even more "paradoxical" form, consider the third version of its formulation, which may be called a paradox with unpredictable card. Imagine that at the table in front of you sits your friend and holds thirteen cards suit of spades. Shuffle the cards and spread them in the hand fan, the pictures to her, he puts on the table, one hole card. You must slow in order to list all thirteen cards, starting with the Ace (Ace corresponding to 1, Jack - 11, Queen - 12 and the King - 13 points.) And ending with King. When you call the card lying on the table, your friend has to say "yes", in all other cases, he says "no."

- I'll bet you a thousand dollars against ten cents - he says - that you can not identify the card as long as I will not say "yes."

Suppose that your friend will do everything in his power not to lose money. Can it in this condition to put on King Buffet peak? Obviously, no. After you list the first twelve cards will only have a king, and you confidently of his call. Maybe inverted card - lady? No, because after will be called Jack, only two cards remain: the king and queen. As king, you have already ruled out, unknown card can only be a lady. It seems that everything is correct, you win $ 1,000 again. Similarly excluded all other possibilities. It turns out that regardless of the map you know it beforehand. The above chain of reasoning seems invulnerable. On the other hand, it is clear that, looking at the back side of the inverted cards, you do not have the slightest idea about what kind of card!

Even in a simplified form etoyu paradox (with the two days, with two boxes, or with just two cards), it is difficult to escape the feeling of some very peculiar ambiguity. Let your friend have only an ace and a deuce. If he puts on the table the two, then you really win. Calling ace, you have it thus excluded and can confidently say: "I came to the conclusion that on the table two". In making this conclusion, you start from the assumption that the following statement is true: "Lying in front of me the card must be either a peak ace or deuce peak". (In the three respective paradox option assumes that the convict be hanged, the cards will be only for what called your friend, and that in one of the boxes will certainly lay an egg.) You are in no way transgressed against the logic and has the right to hope that you can win your buddy $ 1,000.

Suppose, however, that your friend put pa table ace of spades. Can you just figure out what they laid out a map - it is an ace? Of course, your friend would not have to risk $ 1,000, putting the two. Therefore, unknown card must be Ace. You speak these words aloud and hear a "yes" answer. Do you have any reason to believe that you have won the bet?

Oddly enough, but such grounds, you do not. Trying to understand the reasons for this strange assertion, we come to the heart of our paradox. Your previous conclusion was based on the fact that the card can be either an ace or a deuce, so if an unknown card is not an ace, then it must necessarily be two. However, here you have used one additional assumption:
Do you think that your friend is telling the truth, or, to put it simply, is doing everything in its power not to lose $ 1,000. But if you are by logical argument set that is on the table is an ace, then save your $ 1000 your friend will not be able, even if he did not lay out a deuce and ace. Since your friend in any way deprived of their money, it has no reason to prefer one card with another. It is necessary to understand how your confidence, that is on the table ace, once it becomes very shaky. However, you are doing quite reasonable, betting that an unknown card - the ace, because it can actually be an ace. But to win it requires much more: you have to prove that they came to their conclusion by using the "iron" logic, but this is impossible. Thus, a vicious circle is contained in your reasoning. First, you assume that your friend predicted the event correctly, and, based on their assumption, the conclusion, according to which an unknown card must be Ace. But if the table is an ace, your friend was wrong in his prediction, and therefore you do not have to rely on guessing when inverted cards. But that's not all. Since you can not detect the card, then your friend's prediction is true. Therefore, you are back to the starting point, and the whole circle begins again. In this sense the situation resembles a vicious circle in the argument relating to the well-known paradox of the proposed first English mathematician P. EB Jourdain in 1913. The reasoning is similar to those described above, you walk in a circle, always returning to the starting position: define a logical way, what card is on the table, it is impossible. It is possible, of course, you guess it. Knowing his friend, you can come to the conclusion that on the table, most likely, is the ace. However, no self-respecting logic circuit will call your reasoning perfectly rigorous.

All of invalidity of your reasoning becomes particularly evident in the example of the ten boxes. First you "infer" that the egg is in the box 1, but this box is empty. From this you conclude that the egg boxed 2, but it does not find anything. It pushes you to think that the egg is in the box 3, and so on. D. (Everything is happening as if for a moment before you look into the box, where, in your opinion, should lay an egg, some completely incomprehensible shifts his way into the box with a higher number.) Finally, you will find a welcome in the egg carton 8. is it now possible to call this event foreseen in advance, and all your arguments considered perfect from the point of view of logic? Certainly not, because you are eight times used the same method and in seven cases received an incorrect result. It is easy to understand that the egg may be in any box, including at the very last.
Even after you opened 9 empty boxes, the question of whether one can logically come to the conclusion about the whereabouts of the egg (whether it is in a box of 10 or less), remains open. Having only one assumption ( "One of the boxes is sure to contain egg"), you are, of course, will have the right to assert, without entering into conflict with the laws of logic, the egg is in the box 10. In this case, the detection of eggs in box 10 - the event, predictable in advance, and the statement that if it can not be predicted, false. Taking another assumption (that your friend is telling the truth when he says that "coordinate" the eggs, that is, number of boxes of eggs, it is impossible to predict in advance), you are depriving themselves of the opportunity to make any logical conclusions, because, according to the first assumption, the egg It should be in the box 10 (and you can claim it in advance), and the second - you need to find the egg suddenly for himself. Because to come to any conclusion can not detect eggs in box 10 should be considered an unpredictable event in advance, and both assumptions - correct but their "rehabilitation" does not come before you open the last box and find an egg in it.

Let us examine once again the solution of the paradox by giving it this time form of the paradox of a man sentenced to be hanged. Now we know that the judge sentence correctly formulated and the Prisoner of reasoning is wrong. It is wrong is the first step in his argument when he believed, if he can not hang on the last day of the week. In fact, the convict there is no reason to do what whatever conclusions about his fate, even in the evening before the execution (the situation here is the same as in the paradox of the egg, when the last one remains closed box). This idea plays a decisive role in the well-known logic of Quine, which he wrote in 1953.
Quine says, as if he thought the prisoner on the spot. We must distinguish four cases: the first - I will hang tomorrow afternoon, and I know about it now (but actually I do not know); Second - I did not hang tomorrow afternoon, and I know about it now (but actually I do not know); third - I did not hang tomorrow afternoon, but now I do not know about and, finally, the fourth - hang me tomorrow afternoon, but now I do not know about.
The last two cases are possible, the latter of which would mean a reduction of sentence. In such a situation there is no need to think ahead and catch the judge in the controversy. We can only wait, hoping for the best.

Scottish mathematician Thomas G. O'Beirne article with somewhat paradoxical title "Can the unexpected ever happen?" (The New Scientist, May 25, 1961) provides an excellent analysis of the discussion of the paradox. As O'Beirne shows the key to the paradox lies in the awareness of a relatively simple things: one person has information that allow it to be considered the correct prediction of a future event, one can not say anything about the accuracy of the predictions as long as it is event does not occur. It is easy to give a simple example, supporting the idea O'Beirne. Let someone handing you a box, she says: "Open it - inside the egg." He knows that his prediction is correct, you do not know this until, until you open the box.

The same can be said about this paradox, and the judge, and the person laying an egg in one of the boxes, and our friend with thirteen cards - each of them knows that his prediction to be fulfilled. However, their words to the prediction can not serve as the basis for the reasoning chain, leading eventually to the refutation of the prediction. Именно здесь кроется то бесконечное блуждание по кругу, которое, подобно фразе на лицевой стороне карточки из парадокса Журдена, обрекает на неудачу все попытки доказать ошибочность предсказания.

Суть нашего парадокса станет особенно ясной, если воспользоваться одной идеей, высказанной в статье Скривена. Предположим, что муж говорит своей жене:
"Я сделаю тебе ко дню рождения сюрприз. Ты ни за что не догадаешься, какой подарок тебя ожидает. Это тот самый золотой браслет, который ты видела на прошлой неделе в витрине ювелирного магазина".
Что же теперь делать его несчастной жене? С одной стороны, она знает, что муж никогда не лжет и всегда выполняет свои обещания. Однако если он все же подарит ей золотой браслет, то это уже не будет сюрпризом и тогда обещание окажется невыполненным, то есть муж сказал ей неправду. А если это так, то к каким выводам может она прийти, рассуждая логически? Не исключено, что муж сдержит слово и подарит ей браслет, нарушив обещание удивить ее неожиданным подарком. С другой стороны, он может сдержать свое слово, что подарок будет неожиданным, но нарушить второе обещание и вместо золотого браслета подарит ей, например, новый пылесос. Поскольку муж своим утверждением сам себе противоречит, у нее нет никаких разумных оснований предпочесть одну из этих возможностей другой, следовательно, у нее нет оснований надеяться на золотой браслет. Нетрудно догадаться, что будет дальше: когда. в день рождения муж преподнесет ей браслет, подарок мужа окажется для нее приятным сюрпризом, поскольку его нельзя предсказать заранее никакими логическими рассуждениями. Муж все время знал, что может сдержать слово и сдержит его. Жена же этого не знала до тех пор, пока обещанное событие не произошло. Утверждение мужа, которое еще вчера казалось ей чепухой и ввергло ее в запутаннейший клубок логических противоречий, сегодня вдруг стало абсолютно правильным и непротиворечивым благодаря появлению долгожданного золотого браслета.

На примере рассмотренных парадоксов мы ясно ощутили волшебную силу слова (или, точнее, если воспользоваться выражением Бурбаки, силу "вольности речи"). Она-то и делает парадоксы столь сложными и вместе с тем столь привлекательными.

Очень многие читатели сообщили о весьма остроумных попытках решения парадокса об осужденном, которого должны повесить в не предсказуемый заранее день недели. Некоторые из них даже посвятили решению парадокса целые статьи в серьезных журналах.

Л. Экбом, преподаватель математики из Стокгольма, сообщил нам историю, которая вполне могла послужить поводом для формулировки парадокса о неожиданной казни. Как-то раз в 1943 или 1944 году шведское радио сообщило о том, что на следующей неделе намечено объявить учебную воздушную тревогу. Чтобы проверить готовность войск ПВО, учения решено провести внезапно, так что даже утром в день тревоги ни один человек не сможет предугадать, в котором часу она будет объявлена. Автор письма усмотрел а этом логический парадокс и обсудил его со своими студентами. В 1947 году один из этих студентов, будучи в Принстоне, услышал какой-то из вариантов того же парадокса из уст известного математика и логика Курта Гёделя. Далее автор пишет, что сначала он никак не связывал происхождение обсуждаемого парадокса со случаем объявления тревоги но шведскому радио, но это событие вполне могло быть источником парадокса, поскольку Куайн впервые узнал об этом парадоксе в начале сороковых годов.

Ниже вы прочтете два письма, авторы которых вовсе не пытаются разрешить парадокс, но приводят ряд весьма забавных (и запутанных) рассуждений.

Уважаемая редакция!

При чтении статьи о парадоксе с яйцом-сюрпризом создается впечатление, будто автор, логически доказав, что яйцо не может лежать ни в одной из коробок, был несколько удивлен, обнаружив его в коробке с номером 5. На первый взгляд это и в самом деле удивительно, но после тщательного анализа задачи можно доказать, что яйцо всегда будет находиться в коробке 5.

Доказательство проводится следующим образом.

Пусть S — множество всех утверждений, а Т — множество всех правильных (истинных) утверждений. Любой элемент множества (то есть любое утверждение) может принадлежать либо множеству Т, либо множеству С = S - Т. то есть дополнению множества Т, но не может принадлежать тому и другому множеству одновременно. Рассмотрим следующие два утверждения:
1. Каждое утверждение, написанное в этом прямоугольнике, принадлежит множеству С.
2. Яйцо всегда должно лежать и коробке 5.
Утверждение 1 принадлежит либо множеству Т, либо множеству C, но не тому и другому одновременно.
Если утверждение 1 принадлежит множеству Т, то оно истинно. Но если оно истинно, то любое утверждение, написанное в прямоугольной рамке — в том числе и утверждение 1, — принадлежит множеству С. Таким образом, предположив, что утверждение 1 принадлежит множеству Т, мы получим, что оно принадлежит множеству С, то есть придем к противоречию.
Предположим теперь, что утверждение 1 принадлежит множеству С. Тогда нам придется рассмотреть два случая:
случай, когда утверждение 2 принадлежит множеству С, и случай, когда утверждение 2 принадлежит множеству Т.
Пусть утверждение 2 принадлежит множеству С, тогда утверждения 1 и 2, то есть оба утверждения, обведенные прямоугольной рамкой, принадлежат множеству С. Именно в этом и состоит утверждение 1; следовательно, оно истинно и должно принадлежать множеству Т. Таким образом, предположив, что оба утверждения 1 и 2 принадлежат множеству С, мы получили, что утверждение I принадлежит множеству Т, то есть опять пришли к противоречию.
Если же утверждение 2 принадлежит множеству Т (а утверждение 1 — множеству С), то утверждение I, смысл которого сводится к тому, что каждое из утверждений, заключенных в прямоугольную рамку, принадлежит множеству С, противоречит тому, что утверждение 2 есть элемент множества Т. Следовательно, утверждение 1 ложно и должно принадлежать множеству С в полном соответствии со сказанным выше.
Таким образом, существует единственный непротиворечивый случай: когда утверждение 1 принадлежит множеству С, а утверждение 2 — множеству Т. Последнее означает, что утверждение 2 истинно.
Следовательно, яйцо будет всегда лежать в коробке 5.
Как видите, особенно удивляться, обнаружив яйцо в коробке 5. не стоит.

Дж. Вэриэн
Д. С. Беркс
Станфордский университет, штат Калифорния.

Мильтон Р. Сэйлер
Уортингтон, штат Огайо.