Unexpected hanging paradox
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The unexpected hanging paradox is a paradox involving logic and probability. 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 choice of day will be a surprise to you, in that 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, then it wouldn't be a surprise in the sense announced. For he would know by Thursday night that he 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 then the hanging can't be on Thursday either. If it were, then it wouldn't be a surprise either. For he would know on Wednesday night that he was going to be hanged on Thursday, since no hanging had yet occurred and only two days were left, one of which (Friday) he already knows is impossible. 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.
The next week, the executioner knocks on his door at noon on Wednesday - an utter surprise. Everything the judge said has come true, but 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. The judge tells the condemned these two statements:
- You will be hanged at noon next week on Friday.
- The choice of day will be a surprise to you.
The prisoner exclaims, "How can it be a surprise, if you've already told me it will be on Friday? That's contradictory! I can't logically conclude anything rational from such an inconsistent system. This is just nonsense. Since I obviously can't trust you, I don't know when I'll be hanged, or even whether I will be hanged at all. I've learned nothing from your words!"
The next Friday, the prisoner is hanged. He wasn't at all sure that would happen, so it was a surprise. Everything the judge said turned out to be true, even though the prisoner had "proved" that the judge was contradicting himself. What was wrong with his reasoning?
The simplest form of the paradox
The judge is tired of hanging people, so he just tells the prisoner one statement:
- The prisoner cannot know that this statement is true.
The prisoner thinks about that statement: "I'll need to reason logically about this statement. Suppose I logically prove that it's true. Then I will know it is true. But then it must be wrong when it said I couldn't know that. So then it must not be true. If I logically decide that it isn't true, then the prediction in that statement was accurate, so it must be true. Either way I see contradictions. Therefore, I conclude that this statement is as self-contradictory and meaningless as the claim 'this statement is false'".
The judge and the other people in the courtroom listen to this speech. They see the prisoner's confusion, and see that he doesn't end up knowing the statement is true. That's exactly what the statement predicted, so they all see that the judge was correct once again. There is no contradiction, despite the prisoner's conclusion that there is.
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.
The unexpected hanging paradox is similar to the liar's paradox in that the axioms are self-referential. The statement is talking about itself. The unexpected hanging differs in that it adds one more element. The axioms refer to a particular person who might be doing the proof. The word "surprise" is essentially an axiom stating that the prisoner cannot perform certain proofs, but anyone else is free to perform those proofs. If the prisoner tries to do it, then there is a contradiction, and the system collapses. The "solution" to the paradox is that there's really no problem here. We can prove something the prisoner cannot prove, as a result of the odd way that the axioms refer to the prover.
It is interesting that Gödel's incompleteness theorem can be thought of as a way to translate the liar's paradox into formal mathematics. He found a formal way to let axioms refer to themselves. No such construction can be done for the unexpected hanging paradox. Formal axioms cannot refer to a specific prover in that way.he:פרדוקס בוחן הפתע es:Paradoja del examen sorpresa