Unexpected hanging paradox

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

by amelius • 8 years ago

77 80 8 years ago

en.wikipedia.org

Of all those paradoxical thought experiments, this is still my favorite, or really high up at least. Maybe because I've heard it at a young age and my little brain was all flabbergasted. The impression must have lasted.

mehrdadn• 8 years ago

Maybe because I've heard it at a young age and my little brain was all flabbergasted. The impression must have lasted.

That rhyme definitely wasn't something I would've forecasted...

EGreg2• 8 years ago

Martin Gardner popularized this one, I remember coming across it as a kid!

What, did you expect me to rhyme with the line that's last, Ted?

Cyphase• 8 years ago

I would add to this thread, but I don't want to get lambasted.

jancsika2• 8 years ago

The judge has underspecified the process.

Judge says "on a weekday," and that it will be a surprise to the person being executed. But the judge has not specified which weekdays are actually in play.

Now, let's attempt to specify it:

"Judge will schedule the execution for one of the following days in the upcoming week: Monday, Tuesday, Wednesday, Thursday, or Friday. Judge will pick one of these days at random. This random process will determine the day of the execution, such that the person being executed will not be able to use a process of elimination to predict the day of the execution."

Now we can see that there is no paradox, only a self-contradicting specification.

s-shellfish• 8 years ago

The surprise is a surprise in relation to the week. If we use the relation of truth value of surprise to week day, as the prisoner uses to reason, there's a chance it won't be a surprise. That's because the prisoner is treating the statement value as though the judge stated each day, that the hanging will be a surprise. I agree it's not a paradox.

MaxBarraclough• 8 years ago

So your real point is the tension between random process and process such that the person being executed will not be able to use a process of elimination to predict the day of the execution?

If we don't care about randomness, can't the judge just use the strategy of Always pick Wednesday?

If we do care about randomness, the job is to explore whether we're guaranteed to surprise the convict, and of course we're not: if the random process selects Friday, the convict won't be surprised.

jancsika2• 8 years ago

If we don't care about randomness, can't the judge just use the strategy of Always pick Wednesday?

The point is that because the process is underspecified the prisoner is using one interpretation while the judge is using another. (Otherwise the judge would have had to conclude that no choice could be made.) The reader switches between the two specs in an apparent paradox without considering that they are two different specs.

For example, the judge could simply be breaking the rule by considering Friday as a possible choice. That is a different spec from the prisoner, and if such a judge ended up settling on Wednesday then the prisoner would be surprised.

But here's another angle. If the prisoner logically concluded that the judge cannot pick any weekday, the judge can pick Friday and still satisfy the requirement that the prisoner be surprised.

I think that's cheating, though, and the important part is the underspecification.

tristanj2• 8 years ago

There is a simple resolution to this "paradox": it's only a paradox if you consider the judge's initial statement to be True. If the judge's initial statement is False (i.e. he's lying) then the whole situation is logically consistent. The truth table for modus ponens helps clarify, both F->T and F->F evaluate as T https://en.wikipedia.org/wiki/Modus_ponens#Justification_via...

The judge is saying he can predict how someone else will feel in the future (i.e. he can predict the future), which unless he is omnipotent is a false premise in my book.

mattdeboard3• 8 years ago

This was convincing at first. But this logic problem is predicated on the notion judges do not conceal their intentions (iow that his initial statement is True). There is just no point in even considering the question if the prisoner isn't actually going to be executed.

Gormisdomai• 8 years ago

The Judge could be lying even if the prisoner is actually executed - in particular he could be lying about the "and it will be a surprise" bit.

tristanj2• 8 years ago

I should clarify. The judge is not necessarily "lying" (i.e. making a false statement). He's making a statement which has no bearing on the future. The judge's statement could end up being True or False. Using that interpretation resolves the "paradox".

The question encourages you to start from a flawed premise to begin with. Why must the judge's first statement be True? A judge cannot guarantee how someone will feel the future. Further, why must the prisoner be surprised at the date of his execution? If the prisoner is not surprised at his date of execution, that is also a logically consistent situation.

The question tries to shoehorn you into an irreconcilable input state and output state. Yes, it is a paradox if you assume the question's implied start and end result. However if you remove the limitations implied by the question, in the bigger picture there is no paradox.

zbentley• 8 years ago

I think that misinterprets the term "surprise" in the paradox.

"Surprise" isn't meant primarily as "the prisoner will feel surprised" (though the ending is more pithy if it is put that way), but rather "the event will not be logically predictable for the prisoner".

paulddraper2• 8 years ago

I enjoy the math versions.

"Theorem" version:

Every integer is interesting. (Otherwise, one number would be the smallest uninteresting integer. And that would be of notable interest.)

"Paradox" version:

What is the largest number that cannot be described in fewer than 100 characters?

Or even the classic paradox by Russell https://en.wikipedia.org/wiki/Russell%27s_paradox

Does the set of all sets that do not include themselves include itself?

---

The essence of all these is the malformed self-referential definition. The surprise/interestingness/definition/inclusion is "defined" recursively in such a way as to be actually not well defined.

JadeNB• 8 years ago

the largest number that cannot be described in fewer than 100 characters

Almost certainly you mean the smallest number (= positive integer), no? (One doesn't automatically expect that the existence of some positive integer satisfying a property means that there is a largest such.)

paulddraper2• 8 years ago

Oops.

Yeah, smallest.

robocat2• 8 years ago

The prisoner introduces "not being hanged" as an outcome.

However the prisoner doesn't include that outcome in their analysis.

frv1032• 8 years ago

Can this concept be generalized to the point where NOTHING is a surprise? For example if a surprise party is planned for someone at some point in $TIME_PERIOD, it can't occur during the very last hour/minute/second of $TIME_PERIOD because it wouldn't be a surprise. The surprise party can't occur during $TIME_PERIOD - 1 because this would be known as well. Repeat this ad infinum just like the weekdays in the article.

mattdeboard3• 8 years ago

This sounds like a restatement of the Achilles & the tortoise paradox https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_an...

mcphage• 8 years ago

If the prisoner considers “I won’t be hung at all” from the start, his argument falls apart on its first day—on Friday he could still be surprised, since he doesn’t know if he’ll be hung at all. Thus getting hung would still be a surprise.

philippeterson• 8 years ago

Yes. He essentially opts out of his escape, because he's withdrawn from being safe by changing his expectations.

pieguy5• 8 years ago

Here's my take. The prisoner considers the statement "I will be hanged, and it will be a surprise", and after "proving" it false, takes its negation "Either I won't be hanged, or it won't be a surprise". But this is not correct. The prisoner actually disproves the statement "I will be hanged, and it will be a surprise no matter which day it happens". When you negate this statement, you obtain "Either I won't be hanged, or there is a day that it would not be a surprise", which is consistent with the outcome.

p1esk10• 8 years ago

The paradox is because the prisoner does indeed disprove the statement "I will be hanged, and it will be a surprise no matter which day it happens", but the statement turns out to be true (he is both hanged, and surprised no matter what day he's hanged).

pieguy5• 8 years ago

No, it would not have been a surprise if he were hanged on Friday. The prisoner disproves the existence of a strategy for the judge that guarantees surprise, but does not disprove the existence of a strategy that gives the possibility of surprise.

mkeyhani• 8 years ago

The backwards induction argument assumes the time of execution is a discrete variable.

However, at least as far as we know, time seems to be continuous.

frv1032• 8 years ago

What about a scenario that describes an event occurring during a discrete amount of time during a set length of time? For example: A teacher says that he will give the class a surprise pop quiz at the beginning of class some day next week. How does reverse induction fail here?

dooglius• 8 years ago

If the prisoner convinces himself on each day, “I will be hanged today,” then it is not be possible to surprise him. Paradoxes like this tend to make implicit use of assumptions regarding perfect rationality, deduction, and common knowledge, and I think the underlying problem here comes from a violation regarding these.

rwilson4• 8 years ago

It just seems to me it is impossible to be surprised by a bounded random variable in this context. If the death sentence were carried out x days from now, where x is drawn from an exponential distribution, that would fulfill the surprising criterion. The source of the paradox is that the “surprising” aspect is fundamentally incompatible with the bounded nature of the sentence.

throwaway2048• 8 years ago

I've never understood why the resolution to this paradox isn't because of his confidence he expected not to be hung, thus any hanging would be unexpected/a suprise.

Dosent seem terribly paradoxical to me.

colanderman• 8 years ago

Yes. The judge's statement is only valid because of the prisoner's short-sightedness (i.e., it is a prediction, not a logical statement). If the prisoner knows the punch line of the joke, the judge's pronouncement becomes undecidable and therefore meaningless.

It's no different than if the judge said, "you will be hung, and this statement is false", and the prisoner was surprised that they got hung.

Heck, even if the prisoner doesn't know the punch line, the judge's statement is invalid if they incorrectly judge the depth of the prisoner's thought. The prisoner could stop their logical reasoning after 2 days instead of 5, and be sure that they're being hanged Wednesday, making the judge's prediction false.

In summary: the judge made (intentionally or not) a prediction; the prisoner interpreted the prediction as a logical statement; the prisoner failed to see that the statement is undecidable, and drew a logical conclusion from it; the conclusion the prisoner drew happened to make the prediction true.

jzwinck• 8 years ago

There is a surprise on Thursday (if not before).

If the prisoner is alive on Thursday at noon he will either be surprised to find he is to be hanged on Thursday, or surprised to find it will be Friday. Before that he had 50% expectation for either, and there is bound to be a negative surprise with regard to one day and a positive surprise for the other.

everybodyknows• 8 years ago

Right, so if we suppose that the "punch-line" statement, describing the prisoner's state of mind, is true:

the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, was an utter surprise to him.

Then it follows there are two false epistemological claims in the text. First by the judge:

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

This is false, as you correctly reason, since a knock on Friday would come as no surprise (at the time it occurred).

The second false claim, this time by the narrator, reinforces the first:

Everything the judge said came true.

Not every claim the judge asserted was tested by the recounted physical events: There was no knock on Friday.

robocat2• 8 years ago

50% expectation for either

Incorrect. Not getting hanged is also a solution, which doesn't allow induction.

p1esk10• 8 years ago

he will either be surprised to find he is to be hanged on Thursday, or surprised to find it will be Friday

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 is right. Therefore, both the prisoner, and the judge are correct. Thus the paradox.

EGreg2• 8 years ago

I remember reading this as a teenager and concluding that the prisoner actually doesn't know for sure whether he will be hung on Tuesday or not, because he may be hung on Wednesday. Only on Thursday can he be reasonably sure he won't be hung on Friday, and thus he would know he is hung on Thursday. But any day before that, he wouldn't really know.

But also I think Martin Gardner concluded the same thing.

mattdeboard3• 8 years ago

tl;dr: This is a magic trick, not a formal system.

Reminds me of the definition of a magic trick from "The Prestige":

“Every great magic trick consists of three parts or acts. The first part is called "The Pledge". The magician shows you something ordinary: a deck of cards, a bird or a man. He shows you this object. Perhaps he asks you to inspect it to see if it is indeed real, unaltered, normal. But of course... it probably isn't. The second act is called "The Turn". The magician takes the ordinary something and makes it do something extraordinary. Now you're looking for the secret... but you won't find it, because of course you're not really looking. You don't really want to know. You want to be fooled. But you wouldn't clap yet. Because making something disappear isn't enough; you have to bring it back. That's why every magic trick has a third act, the hardest part, the part we call "The Prestige".”

Kind of abstract but it's like...

- The Pledge: Hello convict, here is your sentence.

- The Turn: Actually just kidding, look at this giant loophole in my sentencing. Wow! Expectations subverted!

- The Prestige: Haha, no, I'm just messing with you. There's no loophole but you lacked the context to discern that in The Turn phase.

edit: This is pretty interesting. As the wiki says, how one defines a "surprise" is the whole bugger of the thing. If you use the magic trick metaphor, "surprise" is reframed as "context."

The reason it's so murky trying to pick apart what actually happened is hard is because that's the point of the system. It's designed to be un-figure-out-able. I would love to know how you'd formalize a system whose output is "lack of context"

edit2: Also if you zoom out a level, giving someone a link to the Unexpected Hanging paradox page is a repetition of the same Pledge->Turn->Prestige pattern

- The Pledge: Here's a wikipedia page. You've looked at information on wikipedia pages. You share my confidence this info on this page will adhere to logical rules.

- The Turn: Oh but buh-bam, now you are busy trying to unscramble a paradox.

- The Prestige: You can't figure out paradoxes, that's their point. I will, however, confess this is a terrible magic trick.

whoopdedo• 8 years ago

Can the prisoner communicate with his jailor? He wakes up every morning and announces, "I expect I will be hanged today." Then the execution will not be a surprise if it occurs that day. In order to carry out the judge's order the execution must occur on a day that he is not expecting it. So rather than expect he will not be executed on any day, he should start every day as if he will be executed.

aekotra6• 8 years ago

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

aekotra6• 8 years ago

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.

p1esk10• 8 years ago

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...

incompatible• 8 years ago

"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.

aekotra6• 8 years ago

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

vignesh_m• 8 years ago

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.

aekotra6• 8 years ago

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".

kryogen1c• 8 years ago

This doesn't seem complicated. It is not logically possible to make a psychological statement, ie "you cannot surprise me".

panic2• 8 years ago

One way to think about this paradox is as a game with two players—a prisoner and an executioner. The game progresses over five turns, representing the days of the week.

On each turn, the prisoner secretly chooses whether or not they expect to be killed on that day. Then the executioner chooses whether or not to kill the prisoner. The game ends on the turn the executioner decides to kill the prisoner. If the prisoner wasn't expecting to die, the executioner wins; otherwise, the prisoner wins.

The important detail here (which is obscured by the logical perspective) is when the prisoner is allowed to expect to die. If they're allowed to expect their death on every single day, the prisoner can just do that and win automatically. If they can only expect to die once, there's a situation the usual paradoxical argument doesn't consider: that the prisoner has survived until Thursday but has already used their chance to expect to die. In this case, they know they're going to die on Friday, but there's nothing they can do about it.

LfLxfxxLxfxx• 8 years ago

or the prisoner could spread his expectations over the five days so that they sum to 1, and then lose 1 minus the expectation for the day of hanging

fanzhang• 8 years ago

In this view of the world (where the prisoner can play "I expect to die today" exactly once, there is actually not a Nash equilibrium where the executioner is guaranteed to win.

In other words the judge's sentence can't be guaranteed to be carried out.

bluecalm• 8 years ago

You almost always need mixed strategies for Nash Equilibrium to exist so that's not much of a surprise that it doesn't exist if the prisoner has to choose a pure one (expects to die exactly this day and not other days).

Fnoord3• 8 years ago

What I find interesting about the paradox is its relation to anxiety.

Quoting Wikipedia:

"Anxiety is an emotion characterized by an unpleasant state of inner turmoil, often accompanied by nervous behaviour such as pacing back and forth, somatic complaints, and rumination. It is the subjectively unpleasant feelings of dread over anticipated events, such as the feeling of imminent death." [1]

As well as -perhaps even more so- depression.

The expected outcome has a related effect on the psyche, but the fact is that in reality the prisoner has no effect on their death. Which means that the most useful way to deal with your last days (how many they are) is akin to the principles of carpe diem. [2] Which is something both anxiety and depression hamper.

[1] https://en.wikipedia.org/wiki/Anxiety

[2] https://en.wikipedia.org/wiki/Carpe_diem

zaroth• 8 years ago

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

I have placed a ball under one of five cups. You can check only sequentially if the ball is under each cup.

Do you know which cup the ball is under?

If you start turning over the cups one by one, will you be surprised when you find it?

You won’t be surprised after you’ve turned over all but one cup, if you still haven’t found it, since you know the ball is under at least one of them.

You would be surprised if the ball was found under any of the other cups though.

And so there are only 4 days on which you would be surprised. The executioner knocks on Thursday and provides more data than on the prior 3 days.

The judge makes a promise he cannot keep, that no matter what day he schedules the execution that it will be a surprise.

But how can you schedule something for the end of a finite series and still have it be a surprise?

You can’t. So then does it follow if the usable tail of my sequence is marked out of range, what then becomes of the 2nd to last item in the sequence?

This assumes a sequential cascade leads to the state where all possible days have been marked out-of-range.

The simple reality is that the judge can only promise you a surprise Monday thru Thursday. By Thursday at noon the surprise will be spoiled. And there’s a chance that you will sit for a day knowing exactly when you will die.

p1esk10• 8 years ago

The judge makes a promise that he cannot keep, but precisely because of that, he keeps it. Thus the paradox.

Fnoord3• 8 years ago

Your post describes the logical error that the prisoner can claim that "he is surprised" or "he is not surprised" (ie. "extected it"). He cannot, in honesty, claim he wasn't surprised on the first 4 days. He can pretend that he was surprised, sure, and if he could get away with it thereby winning: by all means, all the good to him. But logically, he cannot for certain claim that he isn't surprised" because he cannot dismiss the other possible options. If you follow that logical route, he's only allowed once to claim he's surprised.

The simple reality is that the judge can only promise you a surprise Monday thru Thursday.

Which means that the execution is on either monday/tuesday/wednesday/thursday but you don't know which one.

That'd result in a 1 in 4 (25%) chance of being right, if he's allowed to vote once. Which day he'd vote on, could be related to how much patience or how nervous he is, how eager he wants to live perhaps?

twhb• 8 years ago

Everybody talking about surprise should read the “Logical school” section, in particular this reformulation:

The prisoner will be hanged next week and its date will not be deducible the night before using this statement as an axiom.

This paradox isn’t a play on the definition of “surprise”, that’s just a weakness accidentally introduced by lax wording.

eximius• 8 years ago

This logic is essentially the solution to the XKCD Blue Eyes puzzle.

I quite like the puzzle and this paradox, but something always feels off about it.

beaner2• 8 years ago

My thought is that there is no paradox, "Friday" being the execution day can be a surprise. If the prisoner finds out that Friday is the execution day on Thursday because he is not executed on Thursday, the specific day is still a "surprise," just a day earlier than it happens.

kazinator3• 8 years ago

The prisoner is specifically told that the surprise will be such that he doesn't know until the knock on the door on at noon that he will be executed that day.

He cannot be hanged on Friday without that being a lie.

beaner2• 8 years ago

That makes sense.

Maybe an alternative solution is that he is never surprised because he is expecting it every day.

squirrelicus2• 8 years ago

For everyone saying the prisoner can just expect to be hanged every day and thus never be surprised.... Why do you say that? What reason, on Monday, does he have to believe that he would be executed on Monday and not the other days? He can say the words, but I'm not convinced he would be convinced.

panic2• 8 years ago

He has a completely valid argument—Friday can be excluded (that wouldn't be a surprise), Thursday can be excluded (since he already excluded Friday, it wouldn't be a surprise), and so on, until the last day left is Monday. So he can conclude the execution must be on that day.

(In fact, he has a valid argument for anything he might want to believe since the original statement made by the judge is self-contradictory.)

squirrelicus2• 8 years ago

This paradox generates so much banter, it's fun!

So now I'm thinking you're wrong because the logic that leads to excluding Tues-Fri can't also be used to exclude Mon. He can't expect execution on Monday and expect to be surprise executed on Monday.

When he inducts to Monday and then doesn't get executed Monday, he can't expect to use the same logic the induct Tuesday. Which means any day then becomes a surprise. He has a 1/3 chance (since we exlude Friday) of randomly choosing the right day, but no reason to not be surprised on any day except Friday.

air7• 8 years ago

I had a thought that this paradox is essentially a self-referencing paradox akin to "This sentence is false" (which is true if it is false, and false if it is true).

Consider Friday: The prisoner know that he will be hanged today, so he realizes he can't be hanged because it wouldn't be a surprise, which makes the hanging surprising... Or in other words, "I will be hanged" means "I won't be hanged" (because it won't be a surprise) and "I won't be hanged" means "I will be hanged" (because it will be a surprise)

Fnoord3• 8 years ago

Assuming the prisoner may only pick certainty of surprise once it seems the prisoner would need a trusted third party such as a notary to write down their solution of choice. Else, they can just reply "wrong, its next day, muwahaha" unless they picked Friday (in which case, the judge was wrong).

If the prisoner is allowed to pick a day every day, they'd pick the next day each time and win.

S_A_P2• 8 years ago

I understand the paradox but to me it really just highlights the ambiguity of the English language. The surprise in this case doesn’t literally mean he will be surprised, it really just means that he will not be given the information. The logic of hanging can’t be Friday because it’s not a surprise doesn’t really work for me. Maybe it’s just the Wikipedia example that’s flawed.

jjaredsimpson• 8 years ago

If I said, "you will be hung tomorrow, and it will be a surprise." That statement can not be true.

So the prisoner reasons: if it is thursday night, then being hung on a friday would not a be surprise. So I will not be hung on friday.

Then there is a kind of induction which further rules out each successive day. I think the thursday night argument is correct, but the induction argument is wrong.

S_A_P2• 8 years ago

I understand that, but I think the definition of surprise is too ambiguous to really make the inference. I think that surprise could just simply mean he doesnt know when it will happen- so it could happen at any day at noon. Does he have to be legitimately SURPRISED that he is being hung? It really just seems to me that the person on death row is delusional if he makes the inference that it cant be any day because it has to be a "surprise"

whack• 8 years ago

The prisoner's anticipated response seems entirely irrational here. There are a range of possible outcomes the prisoner faces:

- Will not be hung this week

- Will be hung on Monday

- Will be hung on Tuesday

- Will be hung on Wednesday

- Will be hung on Thursday

- Will be hung on Friday

You can use logic in order to determine that each of the above outcome is logically impossible. But once you've ruled out all possible outcomes as being "impossible", why would the prisoner stubbornly believe the very first outcome to be true. It too contradicts what the Judge had explicitly said. In such a situation, the prisoner should consider that all 6 possibilities have some significant likelihood of being true, since they all deviate from the Judge's promise in some way. And when dealing with such a rational prisoner, the Judge's promise is impossible to uphold.

Perhaps that is the true resolution to the paradox. The judge was able to make and deliver on his promise, only because he correctly anticipated the prisoner's irrational response. When dealing with a rational prisoner, it would be impossible to uphold such a promise.

kazinator3• 8 years ago

You've nailed it. The judge's sentence contains an inconsistency, and is therefore a falsehood; any conclusion whatsoever proceed from an inconsistency. The judge essentially said "you will and you will not be hanged".

kazinator3• 8 years ago

Straightforward contradiction spoken by the judge, from which anything whatsoever follows.

The statement "you will be hanged on a weekday next week" which we can express as

   H(weekday)

means precisely this:

   H(F) or H(Th) or H(W) or H(Tu) or H(M)

The statement "you will not know until the knock on the door whether you are hanged on that day" means exactly this:

   ~H(F) and ~H(Th) and ~H(W) and ~H(Tu) and ~H(M)

Well, no, not quite! It only has that meaning conditionally. That is to say:

   H(weekday) -> ~H(F) and ~H(Th) and ~H(W) and ~H(Tu) and ~H(M)

By De Morgan's, this right side rearranges to:

   H(weekday) -> ~(H(F) or H(Th) or H(W) or H(Tu) or H(M))

and that is of course

   H(weekday) -> ~H(weekday)

This is equivalent to ~H(weekday), since P -> ~P is ~P v ~P, which is just ~P. So, what the judge said in total is equivalent to the conjunction of these two propositions:

   H(weekday) ^ (H(weekday) -> ~H(weekday))

Where the right side reduces as above, leaving:

   H(weekday) ^ ~H(weekday)

A direct contradiction: you will be hanged and you won't be hanged.

The prisoner choose to believe that he won't be hanged.

Unexpected hanging paradox

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