Hangman Paradox

Manuel

I've been interested in this paradox for a while. I can't seem to find a satisfactory answer. Maybe it's a problem with language, or some bad logical reasoning or something else.

Wikipedia gives a better summary of the issue than I could write, without making any mistakes:

"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 noon, 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."

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

I'm not good with formal logic, so if possible, I'd prefer answers to be stated simply.

Just to throw one idea out here, one problem I see is that the day of the week which the judge tells the prisoner his sentence is not given. So I don't know how the prisoner can even begin eliminating any days. Maybe I'm wrong here.

Any thoughts on this issue?

T Clark

Joyfully he retires to his cell confident that the hanging will not occur at all. — Manuel

Solution #1 - Given that he is confident he won't be hanged, he'll be surprised whichever day he actually is.

Solution #2 - The executioner comes to the cell on Wednesday at noon. The prisoner says, "Hey, that's not fair. The judge said ..." Then the executioner laughs, says "surprise," takes him to the gallows, and hangs him.

Solution #3 - It's noon on Friday and the executioner hasn't shown up. The prisoner heaves a sigh of relief. Then, at 12:10, the executioner comes in. "Sorry I'm late" he says. Then he takes the surprised prisoner to the gallows and hangs him.

Manuel

Solution #2 — T Clark

Solution #3 — T Clark

:rofl:

That's pretty good I must say.

I'd only add one obscene word after "surprise". :wink:

Solution #1 - Given that he is confident he won't be hanged, he'll be surprised whichever day he actually is. — T Clark

What you point out here is actually an interesting component. He'd be surprised no matter what. Unless he doesn't killed next week.

unenlightened

I blame the judge. As the responsible adult, she should know better than to give warning of surprise. The prediction might turn out to be false, because no one knows the future in such detail. Indeed, as the prisoner reasons, the prediction must turn out to be false because it is a contradictory speech act. Unfortunately, in this case his logic rescues the judge from her contradiction, because he concludes that the execution cannot take place, rather than that the judge is irrational.

Fooloso4

If the executioner shows up Friday he will be surprised because he has already ruled it out. The same goes for every other day.

Manuel

Indeed, as the prisoner reasons, the prediction must turn out to be false because it is a contradictory speech act. Unfortunately, in this case his logic rescues the judge from her contradiction, because he concludes that the execution cannot take place, rather than that the judge is irrational. — unenlightened

Interesting perspective of focusing mostly on the judge. Yet, given what happens in the paradox, he is surprised and gets killed.

So let me play devil's advocate here and ask: what's wrong with the prisoners reasoning specifically? Since the paradox turns out to be true, his reasoning must be faulty or so it appears to me.

Manuel

↪Fooloso4

He can rule out Tuesday if he isn't killed by Monday night, because he would be expecting it.

If he expects to be killed on any given day, he won't be surprised.

....Or I think. I'm stretching my logic here. :sweat:

unenlightened

his reasoning must be faulty — Manuel

It is. That the judge's words constrain reality is an unwarranted assumption. The judge could have said anything, true or false, sensical or not. Never believe a judge!

Fooloso4

When I came home I expected a surprise and there was no surprise for me, so of course, I was surprised. — Wittgenstein

RogueAI

If you look at it in a Bayesian sense, then the probability of getting hanged on any particular day is mildly surprising, since absent any other information it's a 1/5 chance.

Manuel

If you look at it in a Bayesian sense, then the probability of getting hanged on any particular day is mildly surprising, since absent any other information it's a 1/7 chance. — RogueAI

Sure. Except that in this case, we are led to believe weekends don't count. So it would be a 1/5 chance.

So the problem starts out on Monday. He can only rule out every other day if he is not killed by noon on any specific day.

I wonder if there would be a situation in which he would not be surprised.

RogueAI

↪Manuel

Yeah, I edited it to reflect the true odds.

It's a tricky case. I think the problem stems from the prisoner looking at Friday in isolation when he should be looking at a disjunction of possibly getting hanged on "Monday OR Tuesday OR...", which is captured by using a Bayesian model of surprise. It's a good paradox. I'm terrible at solving these sorts of things.

Manuel

It is. That the judge's words constrain reality is an unwarranted assumption. The judge could have said anything, true or false, sensical or not. Never believe a judge! — unenlightened

But the prediction was true. If he were not surprised the day he was going to be killed or if he knew the day, then she would be wrong.

But in general, I'll grant you the not trusting a judge thing. :wink:

Manuel

It's a good paradox. I'm terrible at solving these sorts of things. — RogueAI

Goodness gracious I'm not the only one!

I'm horrified of a logician coming here and saying, "pss, that's easy: take X to be the function of the prisoners state, then 5/12 for the possible hours he could be killed and divide by the variable y/h and you'll see!"

Fooloso4

Upon reconsideration: Friday can be eliminated. Thursday would be a surprise only if he lived past Wednesday. Monday, Tuesday, and Wednesday would all be a surprise.

Manuel

Upon reconsideration: Friday can be eliminated. Thursday would be a surprise only if he lived past Wednesday. Monday, Tuesday, and Wednesday would all be a surprise. — Fooloso4

Why would Monday be a surprise? He can only consider any other day after if he lives through Monday. He starts with Friday and goes backward, but the week hasn't begun.

How can he maneuver Monday? If the week begins and he gets killed Monday, he might not be surprised and the judge would be wrong.

Unless I'm missing something, which is very likely.

Fooloso4

He may be confident he will not be executed, but despite his confidence, if the judge is true to her word he will be executed. He has no way of knowing whether it would be Monday. It cannot be eliminated ahead of time.

matt

If it hasn't happened by noon Thursday, the prisoner should not be surprised it happens on Friday. But he doesn't know that it WON'T happen Monday-Thursday in advance. You can only eliminate Friday after it not having happened Monday-Thursday. You can't work backwards assuming it won't happen and then eliminating each day to fit that assumption.

Manuel

↪matt

So let's just say he makes it to Thursday night, he's pretty confident he can't be killed Friday, because he would know about it.

But actually he can be killed on Friday, so he would be surprised if by Friday at noon, they come to get him. So no matter what, he's going to be surprised … (?)

SolarWind

Strange that it should be unsolved. The solution is the impossibility of practical execution. How could the prisoner prove that he is not surprised? He could claim it, but he could do that every day. If he claims it on Monday and is not killed, may he claim it again on Tuesday?

TheMadFool

↪Manuel

I'm rather poor in logic but I love paradoxes despite not being able to understand them, forget about solving them.

That out of the way, what matters most in the hangman's paradox seems to be the word "surprise" - duh!. It means the prisoner on death row, given how the judge informed him about the particulars of his day of execution, must always expect to be strung up. He can't/shouldn't do anything that would give him a reason not to expect execution. In other words, the prisoner's fatal, literally, mistake was/is that he, rather foolishly in my humble opinion, convinced himself that he couldn't be taken to the gallows which, unbeknownst to him or so it seems, made the execution a surprise and thus something that can be carried out.

Manuel

↪TheMadFool

Many thanks for your kind and thoughtful reply. I enjoy certain paradoxes a lot too, this one caught my attention, I'm not sure why. I guess it has to do with the fact that it is presented quite well.

Yes, this seems to be the main theme, the word "surprise." After all the judge could have told the prisoner "You'll be hanged next week, but you won't know which day." He could do the same thought experiment and conclude that he can't be killed any day, but he would still be killed and then the puzzle is less interesting.

Doing very bad logic, suppose we only had a one day week and a two day weekend. In this case, he'd conclude he cannot be killed on the weekday, because he'd know for certain that this was the day he would be killed and he could not be surprised. Or so he thinks. But he would still be surprised when they come to get him.

It looks like a "this is liar" type situation.

Interesting... :chin:

bongo fury

we only had a one day week and a ~~two~~ one day weekend. — Manuel

The judge tells K on Sunday afternoon that he, K, will be hanged the following noon and will remain ignorant of the fact till the intervening morning. It would be like K to protest at this point that the judge was contradicting himself. And it would be like the hangman to intrude upon K's complacency at 11.55 next morning, thus showing that the judge had said nothing more self-contradictory than the simple truth. If K had reasoned correctly, Sunday afternoon, he would have reasoned as follows. "We must distinguish four cases: first, that I shall be hanged tomorrow noon and I know it now (but I do not); second, that I shall be unhanged tomorrow noon and know it now (but I do not); third, that I shall be unhanged tomorrow noon and do not know it now; and fourth, that I shall be hanged tomorrow noon and do not know it now. The latter two alternatives are the open possibilities, and the last of all would fulfil the decree. Rather than charging the judge with self-contradiction, therefore, let me suspend judgment and hope for the best." — Quine - On a so-called paradox

if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise [he'll know it already] if he's hanged on Friday. — Manuel/Wiki

to confuse two things; (i) a hypothesis, by K at t, that the decree will be fulfilled, and (ii) a hypothesis, by K at t, that K will know at t + n - 1 that the decree will be fulfilled. Actually hypothesis (i), even as a hypothesis made by K, admits of two sub-cases: K's hypothetical ignorance and K's hypothetical awareness of the hypothetical fact. — Quine - On a so-called paradox

Manuel

↪bongo fury

Oh logic! How I wish I could follow as I do reading words. :lol:

that I shall be hanged tomorrow noon and do not know it now. — Quine - On a so-called paradox

OK.

But if the judge says you will be killed tomorrow, then how can he not know he's going to get killed, the judge said so now.

Logic aside, the only way out would be to add in the extra possibility that the executioner couldn't show up tomorrow for some strange reason, maybe an accident or he gets very sick or something like that.

bongo fury

But if the judge says you will be killed tomorrow, then how can he not know he's going to get killed, the judge said so now. — Manuel

It is notable that K acquiesces in the conclusion (wrong, according to the fable of the [eventual] hanging) that the decree will not be fulfilled. If this is a conclusion which he is prepared to accept (though wrongly) in the end as a certainty, it is an alternative which he should have been prepared to take into consideration from the beginning as a possibility. — Quine - On a so-called paradox

Manuel

it is an alternative which he should have been prepared to take into consideration from the beginning as a possibility. — Quine - On a so-called paradox

Yes, this much is clear. His attitude of confidence should not have arisen, irrespective of the judges pronouncement.

Hangman Paradox

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