More crazy, but logical, people

  • #1
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Main Question or Discussion Point

Okay, here's a follow-up of sorts to my earlier post about green-eyed people who must kill themselves.

It turns out that the stranger who caused all this trouble was later caught by a fishing boat and brought back to the town to be tried for his offense. He was, of course, sentenced to death, but the judge took pity on him and said, "You will hanged by the neck till dead at dawn on one of the next seven mornings, but so that you do not ever have to go to bed knowing that you will die the next day, I will not tell you what morning it will be."

The stranger, to everyone's surprise, started hooting and hollering and generally whooping it up, obviously quite pleased to hear this sentence. When the bailiff asked him why, he answered, "Well, I clearly can't be executed on the seventh morning, because then I'd know on the sixth night that I was going to die the next day, which the judge said he wouldn't allow. But if the seventh day isn't allowed, then the sixth day is the last day I can be executed, but on the 5th night I would then know that I was to die the next day, so the sixth day is out, too. The same reasoning works for the 4th, 3rd, 2nd, and 1st days - I'm off the hook!!"

The stranger went off to his cell happy in the strength of his reasoning -- and was executed on the 4th morning, to his complete surprise. :eek:

What was wrong with his reasoning?
 

Answers and Replies

  • #2
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What was wrong with his reasoning?
He assumed that his captors would not violate their own logic, and that they would rather NOT kill him at all rather than kill him and in the process violate their own logic. Unfortunately, by doing this, he VALIDATED their logic, because his assumption prevented him from knowing that he'd die the next day.

DaveE
 
  • #3
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This doesn't entirely convince me, although I agree that the expectation that the captors might refrain from fulfilling the sentence by not killing him is key.
What if the court guarantees with complete certainty that 1. he will be killed on one of the seven following mornings, and 2. he will not be able to determine that he will die on the next morning following any night? It seems that they can make this assertion, simply by planning to kill him on, say, the third morning. There's no way he could predict this on the second night, but it's difficult to see how his reasoning fails. It is true - I think - that there is no way they could choose to kill him on the seventh morning, so they must kill him on the sixth or earlier morning. The rest follows ... I think.
So let me ask - could they kill him on the sixth morning, or would he know that on the fifth night?
 
  • #4
-Job-
Science Advisor
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The same reasoning doesn't work for night 3 and before. On the 3rd night he might be killed on morning 4 or morning 5, so he's not sure whether tomorrow's the day.
 
  • #5
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The same reasoning doesn't work for night 3 and before. On the 3rd night he might be killed on morning 4 or morning 5, so he's not sure whether tomorrow's the day.
That's not correct. If morning 6 is disallowed, then morning 5 is the last available day for the execution, so he would know on evening 4 that he was to be executed the next day. Therefore morning 5 is also disallowed. The point is that the same reasoning applies equally to all mornings, since the last available morning of any set of allegedly acceptable mornings automatically becomes unavailable, so the set is reduced by one morning.

I think davee123 is onto the key by mentioning the inconsistency of the two main requirements. One or the other requirement would have to be violated on the last evening, so really the court would have to state which one takes priority. If it's the requirement that he be executed on one of the seven mornings, then they could do it on morning 7, violating the other requirement. If the requirement that he cannot ever know with certainty that he'll be executed on the next morning takes priority, then they'd let him live if he made it to morning 7.

What I'm wondering now is what his reasoning would be if the court explicitly stated that the the requirement that he cannot know he'll die the next day will take precedence over the requirement that he be executed on one of the next 7 mornings. Wouldn't that just strengthen his argument? On the other hand, they could then execute him any morning, since he would be convinced that they weren't going to ...
 
  • #6
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One or the other requirement would have to be violated on the last evening, so really the court would have to state which one takes priority. If it's the requirement that he be executed on one of the seven mornings, then they could do it on morning 7, violating the other requirement. If the requirement that he cannot ever know with certainty that he'll be executed on the next morning takes priority, then they'd let him live if he made it to morning 7.
Exactly. Of course, going strictly by the wording, they could simply prevent him from "going to bed" on the 3rd night and then they're off the hook.

What I'm wondering now is what his reasoning would be if the court explicitly stated that the the requirement that he cannot know he'll die the next day will take precedence over the requirement that he be executed on one of the next 7 mornings. Wouldn't that just strengthen his argument? On the other hand, they could then execute him any morning, since he would be convinced that they weren't going to ...
The problem lies in "knowing the future". If you ever profess to "know" the future in advance, the future can arguably always be changed, thus meaning you didn't "know". In essence, he'll never REALLY know whether or not he'll die the next day, because who knows, maybe he'll escape from captivity with the help of a sympathetic tribe member. Or maybe a wild animal will kill the whole village, but not him. Or maybe the whole thing was just an elaborate joke to teach him a lesson about interfering. Or whatever.

Thus, knowing the future is a logical tool we can use, but in a practical sense, it's useless because it's not absolute. If knowledge of the future really WERE an absolute, the tribe really COULDN'T kill him in the 7 day span because he really COULD deduce that on every day, he would DEFINITELY have to die on the following day, thus, could not be killed. But in reality, you can't ever deductively "know" the future, and so it turns this problem into an interesting quasi-paradox.

DaveE
 
  • #7
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...
"Well, I clearly can't be executed on the seventh morning, because then I'd know on the sixth night that I was going to die the next day, which the judge said he wouldn't allow."
...
What was wrong with his reasoning?

Since he guesses he won't die on the seventh morning, then it will be a surprise if he is executed on the seventh morning. So he can die, even on the last morning.

In fact, he can be executed any morning for the same reason.

:smile:

(edit) The point is:

He has no ways to conclude anything, even on the last night (if he thinks he will be executed, then he could conclude he wouldn't. But in this case, he would. So he can't conclude anything).

So, he can be executed.
 
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  • #8
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I think his biggest mistake was that he announced to the judge his thoughts that he was certain he could not be killed on any of the 7 mornings, thereby being satisfied that he could not be killed. As the condition was that he would not know if he was to be executed the next morning, he had just signed his own death. Hence they had permission to kill him on any morning since he would not be expecting it.
 
  • #9
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He reasoned that he wouldn't be killed any of the 7 days. So he went to bed thinking he WASN'T going to die the next day. Thus the judge has him killed ultimatly without him know which morning. Whether the prisoner had reasoning or not
 
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  • #10
DaveC426913
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The same reasoning doesn't work for night 3 and before. On the 3rd night he might be killed on morning 4 or morning 5, so he's not sure whether tomorrow's the day.
Yes, the flaw in his reasoning is in the claim that any given day is disallowed. It is not.

As Job points out, on day 3, he could be executed the next day or the following day, without knowing it beforehand.
 
  • #11
lisab
Staff Emeritus
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I highly recommend "The Unexpected Hanging," by Martin Gardner.

Btw, when I was in college and had a half hour or so to spend between classes, I would go to the archives in the library, to the Scientific American section, and read Martin Gardner. Wow...probably some of the best spent minutes of my education.
 
  • #12
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The flaw in his reasoning:

He reasons that if he is to be executed then he cannot be executed on morning 7,

he follows the cascade and reasons that the probability of him being executed on any morning is the same = zero
so that the sum of all 7 mornings probabilties is also zero

but his reasoning depends on the assumption that he will be executed.
so that the sum of all mornings probabilities is 1. contradiction.

another way to thinks of it:
if he reasons that he cannot be executed on any morning they he will sleep each night believing he is safe, thus his captors are free to execute him on any morning including the 7th.

i'm not sure i've got it, i havnt even convinced myself, its a great teaser! and each time i imagine myself as the inmate, i cant help but believe in his reasoning!!:D
 
  • #13
DaveC426913
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but his reasoning depends on the assumption that he will be executed.
so that the sum of all mornings probabilities is 1. contradiction.
Well yes, but the prisoner's argument is that it is the judge's logic that leads to contradiction, not his (thus he would be set free).

No, we must show that it is the prisoner's logic that is flawed (thus the judge's is not).
 
  • #14
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if he reasons that he cannot be executed on any morning they he will sleep each night believing he is safe, thus his captors are free to execute him on any morning including the 7th.
Now I agree with the above statement so far as the answer goes, but what about this:
Given the judge assigns a seven day period over which he is to be executed (lets say Monday to Sunday), that is a definite time frame, one which you would reference independently of the actual day of death (Sunday would always be the last day, regardless of when the prisoner was executed). So although his logic says "no matter what day I am put to execution it will be the last and so cannot be executed", if you take it as a fixed period of seven days, where the condition I set above is in force, the only day he cannot die is the seventh, this would mean that any day in between he could be executed.

Now that may seem a little confusing, but it seems a fairly logical solution to myself. Just because counting so far as the prisoner goes finishes on his execution, the order given by the judge still has the other days within it, they just become redundant. I honestly do not know how to word this properly.
No, done in by my own logic on that attempt at a solution there, that doesn't make sense. It works only if the prisoner is assuming he can't be executed and as such leasds you back to a previous argument someone gave.

I'm trying to think 'outside the box' a bit more often, clearly not working.
 
  • #15
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The judge only said that he would not TELL him which morning he was going to die, he didn't assure the prisoner that he couldn't figure it out for himself.
 
  • #16
DaveC426913
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The judge only said that he would not TELL him which morning he was going to die, he didn't assure the prisoner that he couldn't figure it out for himself.
You're right. The original riddle must be re-worded to accommodate this, otherwise there's no riddle.
 
  • #17
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On the other hand, let's assume that the riddle intends that he will be spared if he can deduce beyond a doubt when he is scheduled to be executed. The fallacy in his thinking may be that he does not take into account that he must be alive on evening 6 in order to escape being killed on the 7th. If every day is equally likely for the execution, he will only be alive 1/64 of the time that evening 6 arrives and he will indeed escape execution that one time in 64. The other 63 times day 7 arrives he will already be dead. So his assumption begins with a dependency that he assumes is 1, when it is actually 1/64.
 
  • #18
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Just clue me in where you get the 1/64 from? Perhaps I'll see your logic there.

Personally, I agree that as long as he believes he cannot be executed because of his own logic, he will always go to bed accepting he will not be executed the next day. As such, he could be executed any day.
 
  • #19
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Just clue me in where you get the 1/64 from? Perhaps I'll see your logic there.
My logic wasn't that great... the 1/64 came from thinking if the judge flipped a coin each day to see if the execution happens, then his chance of surviving until day 7 was 1 in 2^6. On the other hand, if the judge throws a dart (metaphorically speaking), then the chance of surviving until day 7 would be 1/7. Either way, the criminal doesn't take into account that dependency when calculating his chances. He is correct that he can't be executed on day 7, but neglects that most of the reason (63/64 or 6/7, take your pick) is because he will already be dead.
 

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