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The unexpected hanging paradox is an alleged paradox about a prisoner's response to an unusual death sentence. It is alternatively known as the hangman paradox, the fire drill paradox, or the unexpected exam (or "pop quiz") paradox.
Despite significant academic interest, no consensus on its correct resolution has yet been establishedCite error: Closing
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<ref> tag. Even though it is apparently simple, the paradox's underlying complexities have even led to its being called a "significant problem" for philosophy.
Formalizing the paradox
The paradox runs as follows: A judge tells a condemned prisoner that he will be hanged at noon on one day in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that if the hanging were on Friday then it would not be a surprise, since he would know by Thursday night that he was to be hanged the following day, as it would be the only day left. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.
He then reasons that the hanging cannot be on Thursday either, because that day would also not be a surprise. On Wednesday night he would know that, with two days left (one of which he already knows cannot be execution day), the hanging should be expected on the following day.
By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
The next week, the executioner knocks on the prisoner's door at noon on Wednesday — an utter surprise to him. Everything the judge said has come true. Other versions of the paradox replace the death sentence with a surprise fire drill, examination, or a lion behind a door. The informal nature of everyday language allows for multiple interpretations of the paradox. In the extreme case, a prisoner who is paranoid might feel certain in his knowledge that the executioner will arrive at noon on Monday, then certain that he will come on Tuesday and so forth, thus ensuring that every day really is a "surprise" to him. But even without adding this element to the story, the vagueness of the account prohibits one from being objectively clear about which formalization truly captures its essence. There has been considerable debate between the logical school, which uses mathematical language, and the epistemological school, which employs concepts such as knowledge, belief and memory, over which formulation is correct.
The logical school
Formulation of the judge's announcement into mathematical logic is made difficult by the vague meaning of the word "surprise". A first stab at formulation might be:
- The prisoner will be hanged next week and its date will not be deducible from the assumption that the hanging will occur sometime during the week (A)
Given this announcement the prisoner can deduce that the hanging will not occur on the last day of the week. However, in order to reproduce the next stage of the argument, which eliminates the penultimate day of the week, the prisoner must argue that his ability to deduce, from statement (A), that the hanging will not occur on the last day, implies a last-day hanging will not be surprising. But since the meaning of "surprising" has been restricted to not deducible from the assumption that the hanging will occur during the week instead of not deducible from statement (A), the argument is blocked.
This suggests that a better formulation would in fact be:
- The prisoner will be hanged next week and its date will not be deducible in advance using this statement as an axiom (B)
Some authors have claimed that the self-referential nature of this statement is the source of the paradox. However, Fitch has shown that this statement can still be composed in mathematical logic. Using an equivalent form of the paradox which reduces the length of the week to just two days, he proved that although self-reference is not illegitimate in all circumstances, it is in this case because the statement is self-contradictory.
The first objection often raised to the logical school's approach is that it fails to explain how the judge's announcement appears to be vindicated after the fact. If the judge's statement is self-contradictory, how does he manage to be right all along? This objection rests on an understanding of the conclusion to be that the judge's statement is self-contradictory and therefore the source of the paradox. However, the conclusion is more precisely that in order for the prisoner to carry out his argument that the judge's sentence cannot be fulfilled, he must interpret the judge's announcement as (B). A reasonable assumption would be that the judge did not intend (B) but that the prisoner misinterprets his words to reach his paradoxical conclusion. The judge's sentence appears to be vindicated afterwards but the statement which is actually shown to be true is that "the prisoner will be psychologically surprised by the hanging". This statement in formal logic would not allow the prisoner's argument to be carried out.
A related objection is that the paradox only occurs because the judge tells the prisoner his sentence (rather than keeping it secret) — which suggests that the act of declaring the sentence is important. Some have argued that since this action is missing from the logical school's approach, it must be an incomplete analysis. But the action is included implicitly. The public utterance of the sentence and its context changes the judge's meaning to something like "there will be a surprise hanging despite my having told you that there will be a surprise hanging". The logical school's approach does implicitly take this into account.
Leaky Inductive Argument
The argument that first excludes Friday, and then excludes the last remaining day of the week is an inductive one. The prisoner assumes that by Thursday he will know the hanging is due on Friday, but he does not know that before Thursday. By trying to carry an inductive argument backward in time based on a fact known only by Thursday the prisoner may be making an error. The conditional statement "If I reach Thursday afternoon alive then Thursday will be the latest possible date for the hanging" looks about as appealing as last week's lottery ticket at half price. The prisoner's argument in any case carries the seeds of its own destruction (execution?) as if he is right then he is wrong, and can be hanged any day including Friday.
The counter-argument to this is that in order to claim that a statement will not be a surprise, it is not necessary to predict the truth or falsity of the statement at the time the claim is made, but only to show that such a prediction will become possible in the interim period. It is indeed true that the prisoner does not know on Monday that he will be hanged on Friday, nor that he will still be alive on Thursday. However, he does know on Monday, that if the hangman as it turns out knocks on his door on Friday, he will have already have expected that (and been alive to do so) since Thursday night - and thus, if the hanging occurs on Friday then it will certainly have ceased to be a surprise at some point in the interim period between Monday and Friday. The fact that it has not yet ceased to be a surprise at the moment the claim is made is not relevant. This works for the inductive case too. When the prisoner wakes up on any given day, on which the last possible hanging day is tomorrow, the prisoner will indeed not know for certain that he will survive to see tomorrow. However, he does know that if he does survive today, he will then know for certain that he must be hanged tomorrow, and thus by the time he is actually hanged tomorrow it will have ceased to be a surprise. This removes the leak from the argument.
Additivity of surprise
A further objection raised by some commentators is that the property of being a surprise may not be additive over cosmophases. For example, the event of "a person's house burning down" would probably be a surprise to them, but the event of "a person's house either burning down or not burning down" would certainly not be a surprise, as one of these must always happen, and thus it is absolutely predictable that the combined event will happen. Which particular one of the combined events actually happens can still be a surprise. By this argument, the prisoner's arguments that each day cannot be a surprise do not follow the regular pattern of induction, because adding extra "non-surprise" days only dilutes the argument rather than strengthening it. By the end, all he has proven is that he will not be surprised to be hanged sometime during the week - but he would not have been anyway, as the judge already told him this in statement (A).
The epistemological school
Various epistemological formulations have been proposed which show that the assumption that the prisoner knows that he will know the content of his sentence throughout the week, together with several plausible assumptions about knowledge, are inconsistent. Several commentators have in particular targeted the assumption that after hearing the sentence, the prisoner "knows" its content. Since the statement which the prisoner is supposed to "know" is a statement about his inability to "know" certain things, this suggests that the unexpected hanging paradox is simply a more intricate version of Moore's paradox. A suitable analogy can be reached by reducing the length of the week to just one day. Then the judge's sentence becomes: You will be hanged tomorrow, but you do not know that.
Centipede game, the Nash equilibrium of which uses a similar mechanism as its proof.