Making sense of the Unexpected Hanging Paradox

In the Unexpected Hanging Paradox, the prisoner is sentenced to be hung the following week, but promised that he will be surprised of the execution date. He reasons that he cannot be surprised on any date and so concludes that he cannot be hung. To his surprise, he gets hung on the following Wednesday. A variation of this paradox is presented in this video and it is argued that in order for the prisoner's story to remain consistent, there must be a day for which his state is unknown. Analogies are drawn to quantum mechanics, relating the paradox to schrödinger's cat, delayed-choice, quantum erasers, and the Principle of Complementarity.

♦♦♦ LINKS ♦♦♦
Minute Physics video entitled "Schrödinger's Cat":
PBS Space Time video entitled "The Quantum Experiment that Broke Reality":
PBS Space Time video entitled "How the Quantum Eraser Rewrites the Past":
Physics Videos by Eugene Khutoryansky video entitled "Delayed Choice Quantum Eraser":

Hashtags: #UnexpectedHangingParadox #SchrödingersCat

♦♦♦ ATTRIBUTION ♦♦♦
The image of John Wheeler was provided by Emielke and is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. No changes were made to the image.

[keystoneperspectives]

*RYAN’S NOTES*
Please keep in mind that this series is composed of short and raw videos containing many original ideas. The videos are not textbooks. Let’s use these videos as a starting point for a conversation. Whether you agree or disagree with the ideas in the videos, leave your thoughts in the comments and we’ll talk it through.

I will be posting comments, corrections, and FAQs here once I get some feedback on this video.

[rprevolv]

If the prisoner is alive on Thursday night, he knows he will be executed on Friday and thus realizes he has been lied to.

[virusoiul]

Good explanation man, thank you for the video.

[HCHI]

I have been searching YouTube on this paradox and this is by far the best entry imo!

I think the main point is that the logical statements inside the prisoner's head are inconsistent/contradictory. Now the question is, what the reaction to this contradiction should be. There is no right or wrong answer, as this is not in any way specified. It depends on how we model the prisoner's mind mathematically. For example, he could believe in contradictory statements, which allows him to believe to be executed on every day at the same time. Then, if we interpret "surprise" as "The prisoner does not believe he is executed on the day of the execution", the judge was technically lying.

[yav]

they put the calendar back or forward??

[elmehdifetouaki]

pfff still don't get it, it just feels like the assumption are wrong from the beginning

[jamestagge]

this Pinocchio paradox of yours is really, really clever and i have not beat it yet, but i know it has to be beatable. give me a little time to try. Full marks on this though. Be back to you soon.

[Omlet]

So what happens in the quantum model if on Thursday hes still alive?

[theodriggers]

What if the box gets confused and the poison happens on Saturday?

[chrisg]

I hope the following version of the paradox will both entertain and reveal its flaw.

In the prison there's a courtyard paved with flagstones. The prisoner's told that under one of them, but nobody knows which, there's a hole leading to a tunnel he can escape through. He only has to look under each stone till he finds it. He sneaks into the courtyard at night with a crowbar and decides to start in one corner and continue with one stone after the other till he reaches the far corner. He reasons that it can't be under the last stone in his plan because by the time he gets to it he'll know that must be where the hole is, but he's been told nobody knows, so it can't be. But that means it can't be the second to last either, because by the time he gets to it he'll know that one's where the hole is, but again it can't be since no-one knows. And so on back to the very first stone. So it's not under any stone at all and he therefore decides he'll never get out that way at all. Later he learns to his mortification that a fellow prisoner escaped by a hole under a stone in the courtyard, and he could have as well.

[ishaan-dg]

It's like a quantum state, if you try to analyze some particular state it collapse and here if we'll observe the cat on any particular day, it might be dead or alive.?

[jamestagge]

Though you might find this interestig but it deals with the hangman of all, many who consider possible solutions to the hangman’s paradox attempt to resolve it by rewriting it or second guessing the author’s intent, the latter very visible in presentation. The hangman’s paradox, as all are, was formulated as it was deliberately to create the paradoxical function. In the hangman’s paradox, that which the judge instructed must be believed as true and proper or there is no paradox, that aspect its entire purpose. That the prisoner’s understanding of his instructions might be found to be inadequate but would have to be considered as logical in the second order (perhaps arising from a false premise) though not necessarily true (because it cannot be true for there are no real paradoxes). If not then the paradox fails. Written in formal logic, its conclusion drawn in formal logic and it presented in formal logic, it must be critiqued in formal logic. “That” is logical. One cannot apply some other logical system to try to generate a remedy and consider it valid, such as the nonsense of dialethism, which would then violate or defy all of the work of Frege, Wittgenstein, et. al., on meaning. One cannot have it both ways.

So, the conclusion the prisoner came to that the hangman would not come for him on Friday has a logical structure to it but, obviously is in error. One could calculate that he could be taken any day of the week, including Friday, each day included as a possibility by the judge as per the paradox and the prisoner would in fact be surprised in the context of his theory that he would not be taken. The paradox does not claim that the prisoner’s theory is necessarily correct. It is only required that it be logical and it demonstrated a proper measure of that. His theory was the conclusion drawn from, and secondary to what we are obliged to accept as the truth of the judge’s instruction.

The consideration of the above is that the prisoner’s theory addresses the days of the week backwards (each day effecting the previous, not that subsequent) which is not how they progress, i.e., they progress in the other direction in time. As per the paradox, the prisoner calculated from Friday to Monday that the hangman could not take him, for once Friday was eliminated, Thursday became the new end of the week then Wednesday, etc. But he moved forward in time through that week. The key to the resolution of this paradox then is that the judge said quite specifically that he would be taken at “noon” (on whichever day) and the prisoner calculated that if he were not taken by 12:00 pm that day, he could not be taken the next day for it being the new end of the week as per his calculation, he would not be surprised. The error in his logic is simple to see. It is only by his presence in Thursday not Wednesday that Friday could be eliminated as a possible execution day and he is not yet there. Also, he could not eliminate Thursday because Friday could not be eliminated prior to Thursday’s arrival and on Wednesday he is two days away. This applies to Monday and Tuesday as well.

In other words, the week would have to pass to Thursday before Friday could be eliminated, beginning the chain of eliminated days and he could be taken any time before that.


What do you think?

[jamestagge]

I may wind up with egg on my face, but I would say that this supposed paradox fails as such and seems more ridiculous than Russell’s paradox, which is not a paradox either. I will refer to it as one for the sake of expediency. So…I think that the resolution to this is the following; if they came for him on any day, before noon he would not know if they would and would be thus surprised, making the time of day as significant as the day itself in his calculations. Also, most significantly, in the paradox it is stated clearly that “IF” the man were not hanged on Thursday, he could not be on Friday for then it would not be a surprise. It is only for that reason that Friday is excluded from the possible days for the hanging. The meaning of this is clear in the definition of the paradox and carries a significant measure of weight in terms of the logic and structure. The “IF” conveys that the execution “COULD” take place on Thursday and the entire paradox collapses (this extends to each preceding day of the week). Additionally, that the paradox states that the hangman came on Wednesday, demonstrates an aspect to the logical structure which seems to be ignored by all who have contemplated the resolution.

So, “IF” the man were still alive on Wednesday, the possibility of Thursday still existed for the execution as stated above. I think that the hangman coming for him on Wednesday is because he was putting a buffer zone (day) between the date of the hanging and Friday, that day of the week under which circumstance it could not take place by definition. This in effect, broke the chain used in the defining of the paradox.

What do you all think? I am still working on this but I don’t see it as the big deal it is purported to be on the videos. Please comment. It’s a fun study.

[bobbysnobby]

The cleanest method of dispelling the paradox is in its imprecise language in particular "Surprise" Surprise is not the same thing as unpredictable surprise carries other connotations. No day can be ruled out of being a surprise this is because as the more confident you are in any day being excluded the more likely it is to be the day you are executed. There is no paradox because no day can be ruled out he can be executed on any day and it would be a surprise.

[philjamieson]

NO! This is not a paradox. If the prisoner believes he WILL NOT be hanged on any of the 5 days then any of those 5 days will be a surprise for the prisoner.

[jamestagge]

I would like your opinion on this given that it has come up in our discussions on paradoxes. You seem to have great insight on this sort of thing.
So, Schroedinger’s cat…..the same box, the same emitter, the same counter, the same hammer and the same vile, but rather than be filled with poison, it is filled with acid the fumes of which would still kill the cat. Now for the only change…..the vile is mounted on a mechanism which if the cat were to fall over to the floor of the box, would cause the vile to tip over and still its contents on the floor. I guess you should also consider spikes of some sort to prevent the cat from voluntarily lying down.

So, after a time, we see the acid leaking out of the box from where it ate through the bottom meaning- the vile was broken by the hammer, the cat died and fell over, the mechanism turned the vile over and the acid fell to the floor and ate through it. Here we would have a string of deterministic events which we know were completed for we see the acid leaking from the box meaning that the cat had to be dead and only dead or that would not have happened. We did not open the box or have to. No wave form of the cat being alive and dead at the same time, just dead. What do you think? Have I missed something?

[Gen_]

Ah, I think I'm starting to get this, I was leaning to the same conclusions but needed someone to explain it clearly. If it's currently Sunday, the only day we can exclude with certainty is Friday, because we know with 100% certainy it's not going to be Sat/Sun. That's facts. But do we know for sure if it's going to be Monday? No. The entire idea is that we can't make certain 100% future predictions based on POTENTIAL events that didn't happen yet. Thus simply we can't exclude Thursday the same way we did with Friday. And if it's Wednesday night and we are still alive only and only then we have to assume it's Thursday and in that case Judge lied to us, because there wasn't any surprise. If we make it alive to Thursday night, the judge still lied to us, since we know it's Friday for sure. Uhh, am I getting confused again?

[jamestagge]

do you see the comment now? Let me know. i reposted

[dsadf]

This argument cannot be correct. In your construction you claim there is a 1/5 chance of being alive after noon on Thursday. That contradicts the premise which is that the execution will come as a surprise, because in those 1/5 of cases the execution on Friday would not be a surprise. That is the heart of the issue and cannot be sidestepped by thinking probabilistically or drawing quantum mechanical analogies.


The same mistake occurs even earlier when you are talking about "ruling in" the cases again. The case of living on Thursday afternoon cannot ever be ruled in.

[kevinciviclord]

God once said to human, you will die within a hundred year, but I will not let you know which date to come, otherwise you will become eternity. So all human live happily