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.

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 execution will be a surprise to you: 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," he thinks, "then it wouldn't be a surprise; for I would know by Thursday night that I 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," he continues, "then the hanging can't be on Thursday either. If it were, then it wouldn't be a surprise either. For I would know on Wednesday night that I was going to be hanged on Thursday, since no hanging had yet occurred and only two days were left, one of which (Friday) I already know cannot be execution day. 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: "I have been sentenced not just to a hanging, but to a surprising hanging and one that must occur next week; but no hanging meeting both conditions can be carried out."

The next week, the executioner knocks on his door at noon on Wednesday - an utter surprise. Everything the judge said has come true. 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, which reduces the number of days to one (Friday). In this version, the judge's sentence is:

You will be hanged at noon next week on Friday.
The execution will be a surprise to you.

The prisoner exclaims, "No hanging can meet both conditions of the sentence. How can it be a surprise, if you've already told me it will be on Friday? That's contradictory! Therefore, the sentence as you have made it, your honor, cannot be carried out. The executioner cannot hang me in a way that is consistent with your sentence."

The next Friday, the prisoner is hanged. This comes as a surprise to him, since he had convinced himself that the sentence could not be carried out. What was wrong with his reasoning? Or, perhaps, the statement "this comes as a suprise to him" made above is false, if the prisoner is so sure of his logic that he assumes, until the last minute of his consciousness, that the hanging is going to be stopped just before his death, so that the officials can make it not a true hanging, and thereby avoid violating either of the judge's mandates.

Discussion

This paradox is unsettling because the prisoner seems to demonstrate, with valid reasoning from true premises, a conclusion that turns out to be false. Thus, as Kirkham points out, the paradox is a prima facie counterexample to standard logic. Resolving the paradox means defending logic by finding an invalid step in the prisoner's reasoning, by showing that he relies on a false premise, or by showing he misinterprets the judge's possibly ambiguous statements.

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.

Representation using modal logic

If the paradox is represented using modal logic it can be shown that the prisoner's belief system will be inconsistent. The arguments given above require a selective use of reductio ad absurdum in order to escape the inconsistency.

However if the judge's second statement is interpreted to mean you will never expect to be hanged so long as you remain consistent, it can be shown that believing the judge's statement does not result in an inconsistent belief system, although the prisoner will believe that he is inconsistent.

Modal system K is sufficient for these purposes.

Explicitly defining what 'expect' means

If we explicitly write out the judge's statements in the form:

1: You will be hung on friday.
2: It is not possible to deduce from these axioms you will be hung on friday.

Then they are obviously contradictory, and so one must be false. Whilst rewriting them as:

1: You will be hung on friday.
2: It is not possible to deduce you will be hung on friday, given no information.

It is possible that they both are true.

Knowledge that the conjunction of the first pair of statements is false, does not let us rule out the possibility that both of the second pair may be true.

That would be an invalid use of reductio ad absurdum.

Problems with The Paradox

A key problem in the paradox is then a conflict in given/assumed information. Based on the judge's statements, it is given/assumed that if the prisoner knows that the hanging will be the next day, then, assuming that the officials who carry out the execution understand that the prisoner would no longer be surprised, it will not be carried out -- a fact that, under the aforementioned assumption, the prisoner would know. Therefore, we assume that the prisoner knows that the hanging will be the next day and that he knows that the hanging will not be the next day. Likewise, if the judge and executioners, meeting in their chambers without the prisoner, feel/know that they must execute on Friday, then they must conclude that they cannot execute on Friday. This is akin to saying, "if X must be true, then X cannot be true," but it is not a paradox. It could be part of a proof by contradition that "X must be true" is not a true statement. Importantly for the present paradox, concluding that "X must be true" is a false statement is not the same as concluding "X is false" (see note on Modal logic, above). In fact, neither "if X is true, then X is false" nor "if X is false, then X is true" is, individually a paradox, but is rather a proof by contradition which refutes the assumption; Russell's paradox is a paradox by virtue of both of these two parts being present simultaneously.

Another problem is that, like many paradoxes, the informal nature of (natural, human) language allow for multiple interpretations. For example, is the prisoner is completely paranoid, and is certain that the executioner will come on Monday at noon, then immediately after 12:00pm on Monday, is absolutely certain that the executioner will knock on his door at noon on Tuesday, and so forth, without ever questioning his own logic or even noticing he was previously mistaken, then the judge's statement that "you won't know the day of the hanging until the executioner knocks on your cell door at noon that day" is never met, since the prisoner, at all times, does in fact "know" of the day of the hanging. Some scholars, like Chow, have argued more broadly that part of the confusion lies in a lack of understanding, or common understanding, of what "resolving the paradox" means, a process which involves formalization of a paradox whose initial form was informal, natural human language.

Regarding the original informally phrase paradox, Chow further argues that "there is a simple but important point here that is often overlooked: the question of whether or not a particular formalization of a paradox 'captures its essence' is to some extent a matter of opinion. Given two formalizations of the paradox, one person may think that the first captures the essence better but another may prefer the second. One cannot say who is objectively right, since there is always some vagueness in the original informal account."

References

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 [1]