What Is the Unexpected Hanging Paradox?
The unexpected hanging paradox is a problem relating to the prediction of surprise future events. A similar paradox was written about in 1948 by the philosopher DJ O'Connor and the better known example described in this article was first brought to public attention by the mathematics writer Martin Gardner in Scientific American in 1963.
The paradox is as follows:
The paradox is as follows:
The Unexpected Hanging Paradox
A prisoner has been found guilty of a heinous crime and has been sentenced to death by hanging. The judge informs the prisoner that the sentence will be carried out one day next week, but that the day will be a surprise to the prisoner. He will not know that it is the day of his hanging until he is collected by the executioner.
The prisoner thinks about the judge's words and works out by careful logic that he will actually escape execution. He deduces that he can't be executed on the Friday as this is the last day of the week. If he hasn't been hanged by the end of Thursday, he will wake up on Friday knowing that it is his execution day and it therefore wouldn't be a surprise.
He further surmises that he can't be hanged on the Thursday either. With Friday now eliminated as a possible date, if Wednesday passes without his execution happening, then it would have to happen on the Thursday and again, not be a surprise.
By the same reasoning, he works out that he can't be executed on Wednesday, Tuesday or Monday either, therefore he won't actually be executed.
Imagine the prisoner's surprise when the executioner comes to collect him on Wednesday morning despite the prisoner's deductions. The judge's words actually came true.
The prisoner thinks about the judge's words and works out by careful logic that he will actually escape execution. He deduces that he can't be executed on the Friday as this is the last day of the week. If he hasn't been hanged by the end of Thursday, he will wake up on Friday knowing that it is his execution day and it therefore wouldn't be a surprise.
He further surmises that he can't be hanged on the Thursday either. With Friday now eliminated as a possible date, if Wednesday passes without his execution happening, then it would have to happen on the Thursday and again, not be a surprise.
By the same reasoning, he works out that he can't be executed on Wednesday, Tuesday or Monday either, therefore he won't actually be executed.
Imagine the prisoner's surprise when the executioner comes to collect him on Wednesday morning despite the prisoner's deductions. The judge's words actually came true.
Other Versions of the Unexpected Hanging Paradox
The version of the story told by DJ O'Connor in an article entitled 'Pragmatic Paradoxes' featured a military commander informing his troops that there would be a surprise 'Class A Blackout' in the coming week. Unlike in our example above, O'Connor's example ends with the decision that the blackout cannot actually take place.
Another version very similar to our example replaces the judge with a teacher and the execution with a surprise exam that the students decide can't actually take place, but does anyway much to their surprise.
Another version very similar to our example replaces the judge with a teacher and the execution with a surprise exam that the students decide can't actually take place, but does anyway much to their surprise.
Solving the Paradox
So how do we resolve this paradox? O'Connor notes in his work that this paradox is not caused by a fault of logic, but rather by a fault in the set-up of the problem. The judge's wording does not allow for the execution to happen.
A better way of wording the sentence is discussed by Timothy Chow in American Maths Monthly (1998) where, using the teacher and surprise exam version of the paradox, he adds that the event 'will not be deducible in advance from the assumption that the examination will occur some time during the week'.
By adding this phrasing to the end, it can still be deduced as before that the exam cannot happen on the Friday. However when we attempt to then continue the logic throughout the rest of the week, we can't. By altering our vague definition of 'surprise' to 'something that is not deducible in advance' the previous logical argument is blocked, hence the surprise test can still occur.
A better way of wording the sentence is discussed by Timothy Chow in American Maths Monthly (1998) where, using the teacher and surprise exam version of the paradox, he adds that the event 'will not be deducible in advance from the assumption that the examination will occur some time during the week'.
By adding this phrasing to the end, it can still be deduced as before that the exam cannot happen on the Friday. However when we attempt to then continue the logic throughout the rest of the week, we can't. By altering our vague definition of 'surprise' to 'something that is not deducible in advance' the previous logical argument is blocked, hence the surprise test can still occur.
Robert Louis Stevenson - Creator of the Bottle Imp Paradox
A Similar Paradox - The Bottle Imp
A paradox that works in a very similar way was created by Robert Louis Stevenson.
He wrote about having the opportunity to buy, for whatever price you wish, a bottle containing a genie who grants wishes. However you must then resell the bottle for a lower price than you paid or be subject to endless torment for the rest of your life.
Stevenson reasoned that you couldn't buy the bottle for 1¢ as you would then need to give the bottle away for free, and the next person wouldn't want it as they would then be unable to pass it on. Likewise you couldn't buy it for 2¢ as, by our reasoning above, you wouldn't be able to find anybody to buy it for 1¢. This logic continues forever through ever increasing amounts of money. However, if you look at the problem from the other end, there must be a larger amount of money that exists which is so large that it will always be possible to find a buyer, contradicting our earlier reasoning. Hence a paradox exists again.
He wrote about having the opportunity to buy, for whatever price you wish, a bottle containing a genie who grants wishes. However you must then resell the bottle for a lower price than you paid or be subject to endless torment for the rest of your life.
Stevenson reasoned that you couldn't buy the bottle for 1¢ as you would then need to give the bottle away for free, and the next person wouldn't want it as they would then be unable to pass it on. Likewise you couldn't buy it for 2¢ as, by our reasoning above, you wouldn't be able to find anybody to buy it for 1¢. This logic continues forever through ever increasing amounts of money. However, if you look at the problem from the other end, there must be a larger amount of money that exists which is so large that it will always be possible to find a buyer, contradicting our earlier reasoning. Hence a paradox exists again.
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