Paradox of the Unexpected Execution

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The Unexpected Execution Paradox (also known as the Unexpected Examination Paradox, Darkening or Inspection, or Executioner's Paradox ) is an epistemological paradox in which an antinomy appears to occur in anticipation of something unexpected.

The paradox presumably dates from the 20th century and expresses a false rationalization of living conditions with a constant existential threat. In certain interpretations there are points of contact with Newcomb's problem .

There is still no agreement on the solution in the academic debate and new papers are still emerging on the subject. In 1998 TYChow listed over 200 works and books dealing with this topic.

Classic representations

The paradox is first mentioned in writing in the July 1948 issue of the English philosophical journal Mind . The variant there is: A military commander announced a total blackout ("Class A blackout") in the coming week, and those affected should only find out about it after six o'clock on the corresponding day.

The paradox has been circulating orally since 1943 at the latest. Swedish radio reportedly had announced an air raid drill in 1943 or 1944 that would take place the following week. It was added that no one could predict when it would take place, even on the morning of the day of the exercise. Lennart Ekbom , professor of mathematics at Östermalms College in Stockholm , had become aware of the logical difficulties involved.

Michael Scriven , professor of scientific logic at the University of Indiana , discussed the paradox as a "new and powerful paradox" in 1951, also in the mind.

In classic representations, the paradox is depicted using the example of a man sentenced to death. Defused versions replace the execution of the prisoner with a surprise test announced to students for the near future.

Executioner's paradox

A prisoner is sentenced to be executed within a week (Monday to Sunday). Executions always take place at exactly noon. He is not told the day of the execution to keep him in fearful anticipation. He is also told that the date is completely unexpected for him. However, he thinks: “If I survive midday on the penultimate day of the week, I will have to be executed at midday on the last day, but that would not be unexpected. So the last possible date can be excluded. If I am still alive at noon before the penultimate appointment, the execution could be scheduled for the last or penultimate appointment, but I have already excluded the last one, so only the penultimate one remains; however, that would not be unexpected. And so on: If I am still alive at noon before the penultimate appointment ... - I cannot be executed at all. ”This conclusion in particular leads to the fact that it is completely unexpected for him when he is led to the execution block on one of the days.

Unexpected test

A teacher says to her class: “In the next week you will write a completely surprising test on this topic!” One of the children thinks that this is impossible. She says, “The class has this subject on Mondays, Thursdays, and Fridays. If the test is written on Friday, it is not surprising, but rather predictable on Thursday after the lesson. Does the test take place on Thursday? No, because I have already excluded Friday and Monday is already over and can also be excluded. So the test has to be on Monday and then it wouldn't be a surprise. ”Can the teacher still make her announcement true?

Knowledge paradox

According to Kaplan and Montague, the paradox can be reduced to the so-called "knower paradox" ( knowledge paradox ), which consists of the following sentence: "It is known that this sentence is wrong."

Analyzes

In addition to the resolution of the paradox, the question arises where the error lies in the logic of the prisoner who assumes that he will survive.

1. Analysis: The prisoner's mistake lies in carrying out an induction step at all after recognizing a contradiction. Basically everything can be deduced from something wrong, including (inapplicable) survival. If the prisoner is still alive on Sunday morning, then he knows that one of the two statements of the guard (“You will be executed by Sunday at the latest” and “You will not know the day before”) was wrong. But because he does not know which of the two statements was wrong, he cannot draw any further conclusions.

Of course, the prisoner can draw the conclusion: "If both statements of the guard are true, then I will not live to see Sunday."

On Saturday morning there are the following options: “Either the hangman is coming today or the guard lied.” The prisoner does not know which of the two statements about the “or” is true. Ergo, the executioner can come “surprisingly” on Saturday. And so, of course, especially on Friday, Thursday, Wednesday, Tuesday or Monday.

2. Analysis: Let us assume that the prisoner is still alive on Saturday evening: could he predict with 100% certainty that he will be executed on Sunday? The paradox comes about by answering yes to this question , but the correct answer is no . The prisoner assumes that the statement that he will be unexpectedly executed in the next week is true; but if he presupposes an unexpected execution , even on Saturday evening he cannot assume that he will be executed on Sunday, as this would contradict his own assumption . Ergo, the prisoner can be executed surprisingly even on Sunday, which would refute his argument .

Analogous case: I will give you the book you have wished for and my present will be a surprise. At first glance, only one of the two promises can be kept. But if the other person assumes that my statement is correct, it is impossible for them to predict that I will give them the corresponding book because, from that person's point of view, the two partial statements contradict each other, which makes a prediction impossible. So I can give the person the book they have wanted as a surprise.

The logical mistake that makes both cases paradoxes is the assumption that an unambiguous prediction can be made based on the facts . This is not true for the simple reason that both times the statement is made that a prediction is impossible. Since one must assume that this statement is truthful, no day can be excluded (based on the prisoner's situation), since an exclusion is also a clear prediction, which contradicts the surprise statement and therefore cannot be accepted. In other words, the statement The execution is surprisingly automatic means that the execution can take place any day of the week; therefore even Sunday cannot be excluded.

3. Analysis: The prisoner's logical reasoning takes place as a backward induction, i.e. In other words, he begins his line of argument with the approach: "If I am still alive Saturday evening ..." However, the argument can no longer be started if he no longer lives on Saturday because he was executed beforehand. His reasoning tacitly assumes that he will still be alive to be sure or surprised. Or to put it another way: From the prerequisite that the execution did not take place up to and including Saturday, it is correct to conclude that Sunday is also canceled as the date of the execution. The further conclusions are then derived from this first conclusion, namely that Saturday, then Friday, then Thursday, etc. should also be excluded. Since these conclusions build on each other and thus ultimately on the first, they all only apply if the requirement for the first conclusion is met, namely that the execution did not take place by Saturday. The train of thought only proves that the delinquent cannot be executed if he survived until Saturday evening. Otherwise the conclusions simply repeat their premise.

literature

  • TY Chow: The Surprise Examination or Unexpected Hanging Paradox . In: The American Mathematical Monthly , January 1998; Copy including a comprehensive list of literature (PDF; 165 kB)
  • Martin Gardner : Logic under the gallows . Vieweg, Braunschweig 1971
  • Roy A. Sorensen: Blindspots . Oxford University Press, 1988, ISBN 0-19-824981-0 , pp. 257 ff.
  • Avishai Margalit, Maya Bar-Hillel: Expecting the unexpected . In: Philosophia , 13 / 3-4, 1983, pp. 263-288.
  • RM Sainsbury: Paradoxes . 4th edition. Reclam, Stuttgart 2010, pp. 208 ff.
  • D. Kaplan, Richard Montague : A paradox regained . In: Notre Dame Journal of Formal Logic , 1/3, 1960, pp. 79-90.

Web links

Individual evidence

  1. ^ Gerhard Vollmer : Paradoxes and Antinomies. Stumbling blocks on the way to truth . In: Roland Hagenbüchle, Paul Geyer: Das Paradoxon . 2nd Edition. Würzburg 2002, pp. 159–195, here p. 180
  2. ^ TY Chow: The Surprise Examination or Unexpected Hanging Paradox . In: The American Mathematical Monthly , January 1998; Copy including a comprehensive list of literature (PDF; 165 kB)
  3. ↑ Donald J. O'Connor: Pragmatic Paradoxes . In: Mind , New Series 57/227, July 1948, pp. 358-359
  4. ^ So WV Quine : On a So-called Paradox . In: Mind , 62, 1953, pp. 65-66
  5. ↑ See e.g. B. Bryan H. Bunch, Robert Ascher, Marcia Ascher: Mathematical Fallacies and Paradoxes . Dover Publications, 1997, p. 34 f. However, there is no evidence for the radio message in question.
  6. ↑ Michael Scriven: Paradoxical Announcements . In: Mind , 60/239, 1951, pp. 403-407.
  7. ^ Roy Sorensen:  Epistemic Paradoxes. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . - Description there as a surprise test paradox .
  8. ^ D. Kaplan, Richard Montague: A paradox regained . In: Notre Dame Journal of Formal Logic , 1/3, 1960, pp. 79-90