The unexpected execution paradox ( Eng. Unexpected hanging paradox ) is a logical paradox , also known as the prisoner paradox , and in other formulations as the paradox of unexpected anxiety and the paradox with a surprise egg . First formulated and published in July 1948 by the philosopher of the University of Exeter D. J. O'Connor [1] . O'Connor’s wording included an officer announcing to his subordinates that “there should be an alarm next week that no one should know in advance until 6:00 p.m. of the day she’s assigned to.”
First of all, the paradox of the statement lies in the fact that through consistent logical thinking one can come to the conclusion that the “unexpected alarm” described in this way cannot take place - either it really will not be unexpected, or it should not take place next week, and each of these conclusions contradicts the original statement.
A much larger paradox is that, despite the logical conclusion, anxiety can still occur at the appointed week and really turn out to be unexpected, thereby in fact it will not contradict the original statement. This is clearly demonstrated in the unexpected execution problem.
Content
Unexpected Execution Problem Statement
One Sunday, the head of the prison called the criminal sentenced to death and told him:
- You will be executed at noon next week.
- The day of execution will come as a surprise to you; you will only learn about it when the executioner enters your cell at noon.
The head of the prison was an honest man and he never lied.
The prisoner thought over his words and smiled: “They cannot execute me on Sunday! After all, then already on Saturday night I will know about it. And, according to the chief, I will not know the day of my execution. Therefore, the last possible day of my execution is Saturday. But if they do not execute me on Friday, then I will know in advance that I will be executed on Saturday, which means that it can be expelled. ” By sequentially excluding Friday, Thursday, Wednesday, Tuesday, and Monday, the offender came to the conclusion that the boss could not execute him, having fulfilled all his words.
The following week, the executioner knocked on his door at noon on Wednesday — a complete surprise for him. Everything that the head of the prison said has come true. Where is the lack of reasoning of the prisoner?
Problem
Many scientific articles have been devoted to resolving this paradox. Martin Gardner, describing the paradox [2] , argues that the error is already in the first step of the argument. Suppose that the last alternative remains. The prisoner cannot make a logically flawless conclusion whether he will be executed tomorrow, since in any case this would lead to a contradiction of the two conditions. Therefore, tomorrow's event will be a surprise for him.
The fact that the prisoner's reasoning does not contain an error, but the conclusion he made turned out to be incorrect, suggests that the reason for the error lies in the incomplete examination of the situation by the prisoner, that is, the prisoner has lost sight of some details. Indeed, in order not to be unexpectedly executed on Monday , the prisoner must be sure that he will be executed on Monday , and he convinced himself that he will not be executed at all. Realizing that it is his conviction that makes the execution unexpected, he will lose this conviction that is unfavorable to him. And perhaps he will even understand that he must be sure every day that he will indeed be executed tomorrow . And then, when the day of execution comes, the execution will become fully expected for him and perhaps for this reason it will be canceled.
This leads to a simplified formulation of the paradox, called the “egg-surprise”.
Surprise Egg
To understand this paradox, it makes sense to consider its simpler form, where the number of days is reduced to one. In this version, the prison director reports that:
- You will be executed at noon next week on Friday;
- It will be a surprise to you.
The prisoner exclaims that both conditions cannot be fulfilled, since the execution cannot be unexpected if it has already been reported that it will happen on Friday and believes that the execution will not take place. The next Friday, the prisoner is executed. This comes as a surprise to him, as he convinced himself that they could not execute him. What was wrong in his reasoning? Or perhaps the condition “it will be a surprise to you” is false. If the prisoner is so confident in him that until the last seconds of his life he believes that the execution will be stopped in order to fulfill the condition of the chief.
A curious conclusion is that the prisoner himself will not be able to conclude that the sentence will be executed. Such a conclusion can only be reached by someone who has additional information that the prisoner is indeed executed on that day.
The equivalent wording will be as follows. Let someone Mr. Smith give the box and say: “Open it, and you will suddenly find an egg inside” [2] . Is it possible, reasoning logically, to come to some conclusion as to whether the egg is inside the box or is it not there? If Smith is telling the truth, then there must be an egg inside, but then this will not be a surprise. Therefore, Smith’s statement is false. If this convinces the addressee that this implies the absence of an egg (which, of course, cannot follow from this premise), then, opening the box and finding an egg there, the addressee will really experience surprise and surprise. Thus, Smith’s claim will be true.
Similarly, we can conclude that although Smith will keep his word, the addressee is not aware of this. Therefore, the addressee cannot make a logical conclusion from Smith's words, and they simply do not carry information .
The egg paradox is a paradox only under the condition that Smith's interlocutor is a thinking creature and is able to draw logical conclusions, and is also able to have expectations. Otherwise, the contradiction ceases to exist.
See also
- The paradox of the satanic bottle of Stevenson
- Other Prisoner Paradoxes: The Three Prisoners Challenge , Prisoner Dilemma
Notes
- ↑ DJ O'Connor. "Pragmatic Paradoxes" // Mind 1948, Vol. 57, pp. 358-9.
- ↑ 1 2 Martin Gardner . Well guess it! = Aha! Gotcha. Paradoxes to puzzle and delight. - M .: Mir , 1984. - S. 33-36. - 213 p.