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Unexpected hanging paradox

The unexpected hanging paradox is a paradox involving logic and provability.

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The Paradox A judge tells a condemned prisoner two statements:

  • You will be hanged at noon one day next week, Monday through Friday.
  • The choice of day will be a surprise to you. You won't know the day of the hanging until the executioner knocks on your cell door that morning.
The prisoner thinks about that for a while, then smiles. If the hanging were on Friday, then it wouldn't be a surprise. He would wake up Friday morning expecting it, since it's the last day. Therefore the hanging absolutely can't be on Friday.

Now that he knows it can't be Friday, what will he think if he wakes up Thursday morning? Since it can't be Friday, it must be today. So a Thursday hanging wouldn't be unexpected either. Therefore a Thursday hanging is impossible too. Continuing the reasoning, the prisoner concludes that logically, he can't be hanged any day next week. He returns to his cell confident in his safety.

The next week, he is hanged on Wednesday, which is a complete surprise. Everything the judge said turned out to be true, so where is the flaw in the prisoner's reasoning? The flaw is the reasoning itself. Since the prisoner had reasoned that the hanging could not occur on any day, he was not expecting it to ever happen. Therefore, on Wednesday, he was suprised that he was hanged.

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. Moved by pity for the poor man's confusion, the judge grants a full pardon.

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 solution has been 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 "judge" is seen as an all-honest authority by some naive 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. While prisoner has every reason to believe he will be hanged - it is in the power of authorities, independent of a prisoner - the claim that he will be surprised is something judge can't know. This is a self-referential and dubious claim. The simple logical conclusion is that prisoner can not know that he will be surprised, if he knows that he will be hanged next week. So, if the first statement of a judge is true and prisoner knows that, the prisoner then cannot know the truth of a 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 can not be determined from the original situation, but it depends on the rest of the story too. Suppose prisoner was in fact hanged on Friday, and was not surprised. The judge proved to be a liar, was dismissed and executed few days later for this and numerous other abuses of power. Some people prefer this ending, as a convict gets to live longer, and repressive state aparatus gets justly punished too.

One other solution has been to take into account human nature. In the first form of the paradox, the prisoner would probably wake up every morning convinced that this is the day. Therefore, he wouldn't be surprised, no matter what day the hanging happened to be. When the judge claimed that he would be surprised, the judge was simply wrong.

Using that definition of "surprise", the paradox is hardly very interesting. A more interesting definition of surprise would be that the prisoner 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. The contradictions only arose when the prisoner tried to prove something from the axioms.

This paradox is sometimes formulated as the unexpected examination paradox in which students know that they are to have an exam sometime during the week but will not know in advance which day it will be on. The underlying logic is quite clearly the same as for the unexpected hanging paradox. Many students would argue that there is very little difference between the two situations. This paradox was used as a joke in Sideways Stories from Wayside School[?] by Louis Sachar. In the end of the chapter, the test simply wasn't given.

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.



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