Unexpected hanging paradox
Encyclopedia
The unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or prediction paradox is a paradox
about a person's expectations about the timing of a future event (e.g. a prisoner's hanging, or a school test) which he is told will occur at an unexpected time.
Despite significant academic interest, there is no consensus on its precise nature and consequently a final 'correct' resolution has not yet been established. One approach, offered by the logic
al school of thought, 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 of thought, suggests the unexpected hanging paradox is an example of an epistemic paradox because it turns on our concept of knowledge
. Even though it is apparently simple, the paradox's underlying complexities have even led to it being called a "significant problem" for philosophy.
Other versions of the paradox replace the death sentence with a surprise fire drill, examination, or lion behind a door or when the bin will be emptied.
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.
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 that a last-day hanging would 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:
Some authors have claimed that the self-referential nature of this statement is the source of the paradox. Fitch has shown that this statement can still be expressed in formal 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.
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.
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.
In other words, his reasoning is incorrect, as if the hanging was on Friday, he will have found it unexpected because he would have expected no hanging. It would be true even if the judge said: "You will unexpectedly be hanged today."
Chow (1998) provides a detailed analysis of a version of the paradox in which a surprise examination is to take place on one of two days. Applying Chow's analysis to the case of the unexpected hanging (again with the week shortened to two days for simplicity), we start with the observation that the judge's announcement seems to affirm three things:
As a first step, the prisoner reasons that a scenario in which the hanging occurs on Tuesday is impossible because it leads to a contradiction: on the one hand, by S3, the prisoner would not be able to predict the Tuesday hanging on Monday evening; but on the other hand, by S1 and process of elimination, the prisoner would be able to predict the Tuesday hanging on Monday evening.
Chow's analysis points to a subtle flaw in the prisoner's reasoning. What is impossible is not a Tuesday hanging. Rather, what is impossible is a situation in which the hanging occurs on Tuesday despite the prisoner knowing on Monday evening that the judge's assertions S1, S2, and S3 are all true.
The prisoner's reasoning, which gives rise to the paradox, is able to get off the ground because the prisoner tacitly assumes that on Monday evening, he will (if he is still alive) know S1, S2, and S3 to be true. This assumption seems unwarranted on several different grounds. It may be argued that the judge's pronouncement that something is true can never be sufficient grounds for the prisoner knowing that it is true. Further, even if the prisoner knows something to be true in the present moment, unknown psychological factors may erase this knowledge in the future. Finally, Chow suggests that because the statement which the prisoner is supposed to "know" to be true is a statement about his inability to "know" certain things, there is reason to believe 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.
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
about a person's expectations about the timing of a future event (e.g. a prisoner's hanging, or a school test) which he is told will occur at an unexpected time.
Despite significant academic interest, there is no consensus on its precise nature and consequently a final 'correct' resolution has not yet been established. One approach, offered by the logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
al school of thought, 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 of thought, suggests the unexpected hanging paradox is an example of an epistemic paradox because it turns on our concept of knowledge
Knowledge
Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...
. Even though it is apparently simple, the paradox's underlying complexities have even led to it being called a "significant problem" for philosophy.
Description of the paradox
The paradox has been described as follows:Other versions of the paradox replace the death sentence with a surprise fire drill, examination, or lion behind a door or when the bin will be emptied.
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
Objectivity (philosophy)
Objectivity is a central philosophical concept which has been variously defined by sources. A proposition is generally considered to be objectively true when its truth conditions are met and are "mind-independent"—that is, not met by the judgment of a conscious entity or subject.- Objectivism...
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 formal logic is made difficult by the vague meaning of the word "surprise". An attempt at formulation might be:- The prisoner will be hanged next week and the date (of the hanging) will not be deducible in advance from the assumption that the hanging will occur 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 that a last-day hanging would 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. Fitch has shown that this statement can still be expressed in formal 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.
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 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 Friday will be the latest possible day for the hanging" does little to reassure the condemned man. The prisoner's argument in any case carries the seeds of its own destruction because 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.
In other words, his reasoning is incorrect, as if the hanging was on Friday, he will have found it unexpected because he would have expected no hanging. It would be true even if the judge said: "You will unexpectedly be hanged today."
The epistemological school
Various epistemological formulations have been proposed that show that the prisoner's tacit assumptions about what he will know in the future, together with several plausible assumptions about knowledge, are inconsistent.Chow (1998) provides a detailed analysis of a version of the paradox in which a surprise examination is to take place on one of two days. Applying Chow's analysis to the case of the unexpected hanging (again with the week shortened to two days for simplicity), we start with the observation that the judge's announcement seems to affirm three things:
- S1: The hanging will occur on Monday or Tuesday.
- S2: If the hanging occurs on Monday, then the prisoner will not know on Sunday evening that it will occur on Monday.
- S3: If the hanging occurs on Tuesday, then the prisoner will not know on Monday evening that it will occur on Tuesday.
As a first step, the prisoner reasons that a scenario in which the hanging occurs on Tuesday is impossible because it leads to a contradiction: on the one hand, by S3, the prisoner would not be able to predict the Tuesday hanging on Monday evening; but on the other hand, by S1 and process of elimination, the prisoner would be able to predict the Tuesday hanging on Monday evening.
Chow's analysis points to a subtle flaw in the prisoner's reasoning. What is impossible is not a Tuesday hanging. Rather, what is impossible is a situation in which the hanging occurs on Tuesday despite the prisoner knowing on Monday evening that the judge's assertions S1, S2, and S3 are all true.
The prisoner's reasoning, which gives rise to the paradox, is able to get off the ground because the prisoner tacitly assumes that on Monday evening, he will (if he is still alive) know S1, S2, and S3 to be true. This assumption seems unwarranted on several different grounds. It may be argued that the judge's pronouncement that something is true can never be sufficient grounds for the prisoner knowing that it is true. Further, even if the prisoner knows something to be true in the present moment, unknown psychological factors may erase this knowledge in the future. Finally, Chow suggests that because the statement which the prisoner is supposed to "know" to be true is a statement about his inability to "know" certain things, there is reason to believe that the unexpected hanging paradox is simply a more intricate version of Moore's paradox
Moore's paradox
Moore's paradox concerns the putative absurdity involved in asserting a first-person present-tense sentence such as 'It's raining but I don't believe that it is raining' or 'It's raining but I believe that it is not raining'. The first author to note this apparent absurdity was G.E. Moore...
. 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.
See also
- Centipede gameCentipede gameIn game theory, the centipede game, first introduced by Rosenthal , is an extensive form game in which two players take turns choosing either to take a slightly larger share of a slowly increasing pot, or to pass the pot to the other player...
, the Nash equilibriumNash equilibriumIn game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
of which uses a similar mechanism as its proof. - Interesting number paradoxInteresting number paradoxThe interesting number paradox is a semi-humorous paradox that arises from attempting to classify natural numbers as "interesting" or "dull". The paradox states that all natural numbers are interesting...
- Crocodile DilemmaCrocodile DilemmaThe Crocodile Dilemma is an unsolvable problem in logic. The premise states that a crocodile who has stolen a child promises the father that his son will be returned if and only if he can correctly predict whether or not the crocodile will return the child....
- List of paradoxes