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The following is the unexpected hanging paradox:

A judge tells a condemned prisoner that he will be hanged at noon on one weekday 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 the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on Friday. 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 surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. 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 — which, despite all the above, was an utter surprise to him. Everything the judge said came true.

The solution given is the following:

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 the night before 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.

My question:

Why should the prisoner 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? I can't understand the solution.

Can anyone kindly explain to me please?

user21820
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user10024395
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  • The judge's order only implies that the hanging can't be on a Thursday. It is the slippery-slope fallacy to then say that it applies to preceding days as the condemned criminal reasoned. – Fomalhaut Apr 09 '23 at 05:16

2 Answers2

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The day of the hanging can be either Monday, Tuesday or Wednesday for it to be a surprise. I define Monday, Tuesday, Wednesday and Thursday as "flexible options" while Friday is a "non flexible option" as its conclusion is apparent without any further data. The solution is to have more than one flexible option to have doubt.

It can't be Friday as past noon on Thursday if he is not hanged he knows the day for the hanging to be Friday thus he can't be surprised on Friday.

It can't be Thursday as past noon Wednesday if he is not hanged he can't be surprised on Thursday knowing that being surprised on Friday is not an option, whatever be the outcome on Thursday. Friday isn't an option in any scenario from the beginning as it leaves the inevitable result obvious after noon Thursday. Therefore after noon Wednesday if he is not hanged he cannot be surprised as Thursday is the only flexible option left.

But before noon on Wednesday he has two flexible options Wednesday and Thursday on which he can't be sure on. Noon on Wednesday is the last possible time to surprise the prisoner after that he can't be surprised as described above.

Viv
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I have a minority viewpoint of the Unexpected Hanging paradox.

My viewpoint is that the prisoner's logic is valid. The way that the paradox is normally presented, the prisoner is told on Saturday that he will be hanged on one of the next $~7~$ mornings, and that at the time that he is brought out to be hanged, he will be surprised.

So, as discussed in this posting and in other postings, the prisoner deduces that it is impossible. Then, on Wednesday morning, the prisoner is brought out to be hanged, and the prisoner is surprised.

So, the supposed paradox is that the prisoner deduced that the premises were contradictory. Then, supposedly, it was established that the premises were not contradictory, because the premises were all satisfied.

In Math, this is a reasonable viewpoint. If you prove that a set of premises lead to a contradiction, it is generally regarded as a proof that you can not have all of the premises simultaneously be true.

My minority viewpoint is that the analysis in the previous paragraph does not necessarily hold, when one of the premises represents a meta-statement that regards someone's state of mind, when that state of mind is vulnerable to the analysis of the premises themselves.

Therefore, I see no paradox. Just because the premises are contradictory, which they are, does not imply that the premises can not all be simultaneously true.

user2661923
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