Unexpected hanging paradox

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.

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.

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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.

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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.

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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.

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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.

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