Unexpected Hanging Paradox

Devans99

Despite significant efforts by academia, this is remains an unsolved paradox:

The judge tells a prisoner he will be hanged on a weekday next week at noon and it will be a surprise. The prisoner thinks; if he has not been hanged by Thursday then Friday would not be a surprise so he cannot be hanged on Friday. If he has not been hanged by Wednesday, since he cannot be hanged on Friday, he cannot be hanged on Thursday either because it would not be a surprise. And so on to the conclusion that he will not be hanged. But he is hanged. On Wednesday. And it is a surprise.

https://en.wikipedia.org/wiki/Unexpected_hanging_paradox

First Thoughts - Its not a Paradox

I first thought through the prisoner’s reasoning and found it valid. So maybe the judge is right - the reason it is a surprise is that the prisoner logically should not be hanged. IE the judge went through the same reasoning the prisoner did, realised that it was not logically possible to hang the prisoner next week by surprise, and decided to hang him anyway (so it comes as a surprise because being surprised is logically impossible).

Second Thoughts - a Possible Resolution?

1. On Thursday night, the prisoner would know he will not be hanged on Friday
2. But [1] above becomes 'knowledge' only on Thursday night
3. So on Wednesday night, [1] above is not knowledge
4. So on Wednesday night, the prisoner cannot determine if he will be hanged on Thursday or Friday - he does not know ‘not Friday’ until Thursday night
5. So the judge is right, the hanging will be a surprise

Shamshir

Consider the following: If the prisoner is told he will be hanged, he will expect to be hanged. The date turns irrelevant.

If he is expecting to be hanged, he cannot be hanged, because it will not be a surprise.

He will hang and he will not hang, leaving him 'hanging' as to when he will hang.
Perhaps this little metaphor is the surprise?

Terrapin Station

One problem with this is that "surprise" seems to be used in two rather incompatible ways. One is the sense of an emotional reaction that someone has based on not expecting something. That doesn't necessarily have anything at all to do with logic. An individual can expect (and so not be surprised) or not expect (and so be surprised) any arbitrary thing, for any reason imaginable, or even for no reason--it can just be purely intuitive. Of course, in this sense of surprise, we can't predict, with any certainty, whether anyone will be surprised by anything or not.

"Surprise" also seems to be used here in the sense of whether a better-than-chance prediction is possible using some logical metric.

The problem with the supposed conundrum on this end is that the ONLY situation where we could make a better-than-chance prediction is once noon has passed on Thursday.

For example, say that whether the prisoner is going to be hanged on any given day (M-Th at least, since otherwise Friday is set) is determined by an "ideal coin," so that the flips are really random. Well, on every day, Monday through Thursday, there's no way to predict, better than chance, whether the prisoner will be hanged, because on every one of those days, the answer is literally determined by chance. The only way we can make a prediction better than chance at some point in that process is once noon has passed on Thursday.

Devans99

One problem with this is that "surprise" seems to be used in two rather incompatible ways — Terrapin Station

Yes, it is a poor choice of word.
Rather than:
'the hanging will be a surprise to the prisoner'
Better to say:
'prisoner will not be able to deduce the time of hanging'

Devans99

On Thursday evening the prisoner can deduce that he will be hanged on Friday (so he cannot be hanged)

On Wednesday evening though, the prisoner cannot deduce that he will be hanged on Friday (so he could be hanged on Thursday or Friday).

TheMadFool

↪Devans99

The paradox if rephrased is this: Not a single day will be a surprise hanging as far as the prisoner is concerned. His logic is perfect. However, thus reasoning he reassures himself that he'll NOT be hanged. There lies the prisoner's error since then the hanging will surprise him, just as the judge predicted. Oh! It seems there really is no paradox. Just a foolish prisoner and a shrewd judge!

Devans99

↪TheMadFool

I concur that is a possible solution to the original paradox.

With the more specific definition: 'prisoner will not be able to deduce the time of hanging' though, I think the prisoner cannot deduce the time of hanging:

https://thephilosophyforum.com/discussion/comment/282788

ZhouBoTong

Like most hypothetical situations, a lack of information leaves options. First off, I think The Mad Fool's explanation is more clever (and in line with the point of the scenario), but here is a way that the prisoner could be surprised on any day, including Friday:

On Thursday night the judge walks in and says that you (the prisoner) have been pardoned by the governor who will be here in the morning to sign the release papers. Then in the morning, PSYCHE, it's a hangin'.

I do not think that was what was intended by this scenario, but no reason it doesn't work based on the parameters (notice we could come up with a bunch of other reasons where new information could lead to the surprise).

andrewk

.

Yes, it is a poor choice of word.
Rather than:
'the hanging will be a surprise to the prisoner'
Better to say:
'prisoner will not be able to deduce the time of hanging' — Devans99

If we take that approach then the statement of the judge is false because whatever day it occurs, the prisoner will be able to prove it must happen on that day, because the system is inconsistent. Here's why:

We take the judge's statement as an axiom, and then go through the process that for each day we can deduce that the hanging does not occur on that day. From that we deduce that the hanging does not occur. We put that together with the axiom stated by the judge and, using AND elimination, we get the deduced statement:

'The hanging will occur next week and it will not occur next week'.

which is a perfect contradiction.

By the principle of explosion, from a contradiction, any well-formed proposition can be deduced. So for every day X we can deduce the proposition:

'The hanging will occur on day X'

Hence, on whatever day the hanging occurs, it could be deduced beforehand that it would occur on that day (as well as on every other day!).

That then forces us back to reading 'surprise' as surprise, rather than as 'could not have been deduced beforehand'. But in that case it's simply a statement about the prisoner's state of mind, and that could be anything. Even if it happens on Friday it can be a surprise, as the prisoner may by then believe that the judge was mistaken and it will not happen at all.

Janus

↪Devans99

Given that the prisoner believes he will hang the next week at noon on one of five days, on first examination the only day that could be a surprise is Friday and so Friday must be ruled out tout court. If Friday is ruled out tout court, then all the other days must also be ruled out; which means that the prisoner cannot be hanged on any of the days if the prisoner must also be surprised at being hanged on that day. If the prisoner realizes this and deduces that he therefore cannot be hanged on any day that satisfies the condition that he will also be surprised to be hung on that day, he will leave himself open to being surprised whatever day he is hung.

Devans99

↪Janus

↪andrewk

↪ZhouBoTong

This is making my head spin. Probably good exercise for the mind then.

Surprise may mean:

1. Unable to deduce the time of hanging
2. Was able to deduce when he would not be hanged, so was surprised when hanged anyway

I think the first definition is most in spirit with the intent of the paradox. Because of the vagueness of the original, I think it is better to focus on the paradox with the first definition.

I think the problem with the 'prisoner will not be able to deduce the time of hanging' version is the timing of deductions:

On Thursday evening the prisoner can deduce that he will be hanged on Friday (so he cannot be hanged), but he can’t make that deduction on Wednesday evening because ‘can’t be hanged on Friday’ is only true if it is Thursday evening.

On Wednesday evening, there is the possibility of being hung on Thursday. The prisoner can only deduce that he will not be hung on Friday On Thursday evening, so on Wednesday evening, the prisoner cannot deduce between being hung on Thursday or Friday?

Devans99

I think you have to be very specific - any deduction requires you to specify when the deduction takes place, because what is known about the situation is changing with time.

So we must accompany ‘Unable to deduce the time of hanging’ with a statement of when he will be unable to deduce the time of hanging. Returning to the original definition of the 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.’
https://en.wikipedia.org/wiki/Unexpected_hanging_paradox

So we have to conclude that the meaning is that the execution is not deducible just before the execution takes place. That leads to:

‘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 not be deducible just before the execution takes place’

So on Friday, the prisoner can deduce he will be executed, so he can’t be executed.

On Thursday, the prisoner cannot tell if he will be executed on Thursday or Friday so he can be executed.

Devans99

On Thursday the prisoner can only deduce 'if I make it to Friday, I will not be executed'. He has not made it to Friday yet...

Unexpected Hanging Paradox

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