Unexpected hanging paradox
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The unexpected hanging paradox is a paradox involving logic. It is alternatively known as the hangman paradox, the fire drill paradox, or the unexpected exam paradox.
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The paradox
A judge makes two statements to a condemned prisoner:
- You will be hanged at noon one day next week, Monday through Friday.
- The execution will be a surprise to you: you won't know the day of the hanging until the executioner knocks on your cell door at noon that day.
The prisoner reflects on these statements, and then smiles. "If the hanging were on Friday," he thinks, "then it wouldn't be a surprise; for I would know by Thursday night that I was going to be hanged on Friday, since no hanging had yet occurred and only one day was left. So the hanging can't be on Friday."
"But," he continues, "then the hanging can't be on Thursday either. If it were, then it wouldn't be a surprise either. For I would know on Wednesday night that I was going to be hanged on Thursday, since no hanging had yet occurred and only two days were left, one of which (Friday) I already know cannot be execution day. So a Thursday hanging is impossible too."
Similar reasoning shows that the hanging can't be on Wednesday, Tuesday, or even Monday! He returns to his cell confident in his safety: "I have been sentenced not just to a hanging, but to a surprising hanging and one that must occur next week; but no hanging meeting both conditions can be carried out."
The next week, the executioner knocks on his door at noon on Wednesday - an utter surprise. Everything the judge said has come true. Where is the flaw in the prisoner's reasoning?
A simpler form of the paradox
To gain some insight into this problem, it's helpful to look at a simpler form of the paradox, which reduces the number of days to one (Friday). In this version, the judge's sentence is:
- You will be hanged at noon next week on Friday.
- The execution will be a surprise to you.
The prisoner exclaims, "No hanging can meet both conditions of the sentence. How can it be a surprise, if you've already told me it will be on Friday? That's contradictory! Therefore, the sentence as you have made it, your honor, cannot be carried out. The executioner cannot hang me in a way that is consistent with your sentence."
The next Friday, the prisoner is hanged. This comes as a surprise to him, since he had convinced himself that the sentence could not be carried out. What was wrong with his reasoning?
Discussion
This paradox is unsettling because the prisoner seems to show that the judge is being self-contradictory, yet in the end the judge ends up being perfectly correct in every statement. Several solutions have been suggested for this paradox.
One possible solution is to note the difference between the truth of a statement and knowledge about this truth. The judge's statements might be true, but the prisoner can't know that they are true. He has no reason to assume they are true, other than to believe what the judge says. Because judges are seen as completely honest by some people, this point is obscured. If it were a thief instead of a judge making these statements, it would not be much of a paradox. (Note that in some versions of the paradox, it's the executioner, not the judge, making these statements. For this reason, the paradox is also called the hangman paradox.) While the prisoner has every reason to believe he will be hanged - it is in the power of authorities, independent of the prisoner - the claim that he will be surprised is something the judge can't know. This is a self-referential and dubious claim. The simple logical conclusion is that the prisoner cannot know that he will be surprised, if he knows that he will be hanged next week. So, if the first statement of the judge is true and the prisoner knows that, the prisoner then cannot know the truth of the second statement. The fact that the judge tells him does not mean that he knows it, since judges in general do not have access to absolute truth despite being seen as such by some, not to mention their possible dishonesty. It can turn out that judge was right. But it can turn out that the judge was wrong too - if a prisoner is hanged on Friday, he will not be surprised, if he believes in the first statement. So, in some cases, second statement is true, in some other it is not - the truth of the second statement cannot be determined from the original situation, it depends on the rest of the story too. Had the prisoner somehow guessed he would be hanged Wednesday, the judge would indeed have been wrong.
One could also argue that it is unclear what the prisoner is "allowed to expect" and when he is thought to be surprised. Consider a paranoid prisoner who decides every evening he would be hanged the following day. Obviously he cannot be surprised then and the paradox basically disappears. However, when contemplating the paradox we tend not to "grant" the prisoner the freedom to redecide. Yet, in his own reasoning, the prisoner grants himself this freedom: "If I am not hanged Thursday I will decide to expect Friday, so Wednesday evening I will actually decide to expect Thursday..." and so on.
Another possible solution is to contemplate the prisoner's view of the judge's statement versus everyone else's. We'll say that the prisoner is "surprised" if he can't logically and consistently prove what will happen, using the judge's statements as axioms. In that case, the prisoner truly is surprised by the hanging. Although the prisoner couldn't prove what would happen, everyone else could (as described above). The contradictions only arose when the prisoner tried to prove something from the axioms.
It has been alleged, by Shaw among others, that if the prisoner's premises were made completely explicit, then they would be revealed as self-referring. If this is true, then the unexpected hanging paradox is similar to the liar's paradox. Kirkham and Wright & Sudbury, however, have both argued that the prisoner's reasoning does not involve self-referring premises.
Annotated Bibliography
C. S. Chihara, "Olin, Quine, and the Surprise Examination" Philosophical Studies 1985, vol. 47, pp. 19-26. The author claims that the prisoner assumes, falsely, that if he knows some proposition, then he also knows that he knows it.
R. Kirkham, "On Paradoxes and a Surprise Exam," Philosophia 1991, vol. 21, pp. 31-51. The author defends and extends Wright and Sudbury's solution. He also updates the history and bibliography of Margalit and Bar-Hillel up to 1991.
A. Margalit and M. Bar-Hillel, "Expecting the Unexpected", Philosophia 1983, vol. 13, pp. 337-44. A history and bibliography of writings on the paradox up to 1983.
D. J. O'Connor, "Pragmatic Paradoxes", Mind 1948, Vol. 57, pp. 358-9. The first appearance of the paradox in print. The author claims that certain contingent future tense statements cannot come true.
M. Scriven, "Paradoxical Announcements", Mind 1951, vol. 60, pp. 403-7. The author critiques O'Connor and discovers the paradox as we know it today.
R. Shaw, "The Unexpected Examination" Mind 1958, vol. 67, pp. 382-4. The author claims that the prisoner's premises are self-referring.
C. Wright and A. Sudbury, "the Paradox of the Unexpected Examination," Australasian Journal of Philosophy, 1977, vol. 55, pp. 41-58. The first complete formalization of the paradox, and a proposed solution to it.
