Unexpected hanging paradox

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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 paradox.

Despite significant academic interest, no consensus on its correct resolution has yet been established^[1]. One approach, offered by the logical school, suggests that the problem arises in a self-contradictory self-referencing statement at the heart of the judge's sentence. Another approach, offered by the epistemological school, suggests the unexpected hanging paradox is an example of an epistemic paradox because it turns on our concept of knowledge^[2]. Even though it is apparently simple, the paradox's underlying complexities have even led it to being called a "significant problem" for philosophy^[3].

Contents 1 Formalizing the paradox 2 The logical school 2.1 Objections 3 The epistemological school 4 See also 5 References

[edit] 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 or examination.

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 everyday 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.

[edit] 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^[4] 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.

[edit] Objections

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 in spite of the fact that I am telling you there will be a surprise hanging". The logical school's approach does implicitly take this into account.

[edit] 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.

[edit] See also

Centipede game, the Nash equilibrium of which uses a similar mechanism as its proof.

[edit] References

1. ^ T. Y. Chow, "The surprise examination or unexpected hanging paradox," The American Mathematical Monthly Jan 1998 [1]
2. ^ Stanford Encyclopedia discussion of hanging paradox together with other epistemic paradoxes
3. ^ R. A. Sorensen, Blindspots, Clarendon Press, Oxford (1988)
4. ^ Fitch, F., A Goedelized formulation of the prediction paradox, Amer. Phil. Quart 1 (1964), 161–164

• 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.
• 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.
• 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.
• T. Y. Chow, "The surprise examination or unexpected hanging paradox," The American Mathematical Monthly Jan 1998 [2]
• M. Gardner, "The Paradox of the Unexpected Hanging", The Unexpected Hanging and Other Mathematical Diversions 1969. Completely analyzes the paradox and introduces other situations with similar logic.