[aekotra]

The prisoners reasoning is wrong.

> 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.

This belief is FALSE. The moment of surprise for a Friday hanging takes place at 12pm on Thursday. Of course, this also happens to be the moment of surprise for a Thursday hanging. In other words, at 11:59am on Thursday he will not know if his hanging takes place on Thursday or Friday. At 12pm he will be surprised to know for certain which one it is: it will be Thursday if he hears a knock and it will be Friday if there is no knock.

The prisoner holds this belief because he makes the FALSE assumption that his moment of surprise MUST occur at the sound of the knock ON THE DAY of the hanging. The prisoner correctly understands that there cannot be a moment of surprise on Friday but, due to this assumption, incorrectly reasons that a Friday hanging cannot be a surprise. It will be a surprise, but he will experience that moment on Thursday, not Friday.

The prisoner pictures himself at 12:01pm on Thursday wondering how he could possibly be surprised for a Friday hanging. He doesn't realize he has ALREADY been surprised by the news of his Friday hanging. That moment occurred one minute earlier (12:00pm)!

The remainder of the "paradox" text is rendered nonsensical because the reasoning is based on the apparent impossibility of being surprised by a Friday hanging.

TLDR

The prisoner makes the following faulty reasoning:

1) The moment of surprise must occur on the day of the hanging (FALSE)

2) There cannot be a moment of surprise on Friday (TRUE)

-> Therefore, a Friday hanging cannot elicit surprise

-> Therefore, Friday cannot be chosen as the hanging day

-> Therefore, Thursday cannot be chosen as the hanging day, etc

#1 is assumed to be TRUE by the prisoner. Unlike the other days, the moment of surprise for a Friday hanging occurs the day before when he does NOT hear a knock at 12pm. Therefore, #1 is false and leads to the false deductions

[incompatible]

"He will not know the day of the hanging until the executioner knocks on his cell door at noon that day."

The wording of the problem is everything.

[aekotra]

I am very aware of that line. I've reworded most of the OP, to explain more clearly!

[vignesh_m]

The main issue is the definition of surprise for perfectly rational entities. Your argument doesn't really add much as you can just fix that time of day when all executions occur, and you've just restated the paradox.

[aekotra]

The main issue is not the definition of surprise. The main issue is given to you plainly: the prisoner's reasoning is wrong.

I encourage you to give an example where the execution times are changed and I guarantee you it will suffer from the same assumptive traps as the original "paradox".

[aekotra]

> Despite significant academic interest, there is no consensus on its precise nature and consequently a final correct resolution has not yet been established.

This is, frankly, embarrassing.

[p1esk]

He will not know the day of the hanging until the executioner knocks on his cell door at noon

Did you miss that line?

Embarrassing indeed...

[aekotra]

I don't follow. Can you please explain? I am very aware of that line.

[p1esk]

The judge says the prisoner would be surprised exactly when the knock on the door happens (not before). As a consequence, the prisoner is right to believe he won't be hanged at all (following his reasoning), but as a consequence of that, he is surprised when the knock happens. In turn, as a consequence of that, the judge turns out to be right. Therefore, both the prisoner, and the judge are correct. Thus the paradox.

[aekotra]

My post is all about how the prisoner's reasoning is WRONG. My mistake: I obfuscated it by having the reader take the perspective of the prisoner. I reworded the OP, but I can't edit it anymore. I have a simpler revised version that I'll include below:

. . .

The prisoners reasoning is wrong.

> 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.

What the prisoner doesn't realize is that, by noon on Thursday, he would have ALREADY been surprised by the news of his Friday hanging. That moment occurred when he did not hear the knock!

The moment of surprise for a Thursday hanging or Friday hanging happen at the SAME point in the future. In other words, at 11:59am on Thursday he will not know if his hanging takes place on Thursday or Friday. At 12pm he will be surprised to know for certain which one it is: it will be Thursday if he hears a knock and it will be Friday if he does not.

"If he hasn't been hanged by Thursday" (ie. has not heard the knock) IS, ITSELF, the surprise of a Friday hanging. He wrongly assumes that his moment of surprise can ONLY happen following a knock. He doesn't realize that this assumption is true for every scenario EXCEPT a Friday hanging, where his surprise will happen in the ABSENCE of a knock on Thursday.

Because he cannot envision himself being surprised by a Friday knock (correct) and assumes he can ONLY be surprised by a knock (incorrect), he wrongly concludes that a Friday hanging is an impossibility, and subsequently concludes the same for a Thursday hanging, etc.

[p1esk]

Here's that line again: "He will not know the day of the hanging until the executioner knocks on his cell door at noon that day".

The judge not only said that the prisoner will be surprised, he also said precisely when the prisoner will be surprised, which is noon on Friday, for Friday execution.

It does not matter if he gets surprised by the absence of the knock on Thursday. All that matters is that he would not be surprised when the knock happens on Friday.