The Unexpected Execution -- You Bet Your Life
What, if anything, is wrong with this reasoning?
Quote: jfalkYou have committed a hanging offense in a small, but highly logical, town. You are convicted and sentenced on Friday. The Judge pronounces the date of your execution: "You will be hanged one day next week: Monday through Friday. In order to make you suffer a bit more, I guarantee you will not be able to deduce the day of your execution the day before." "Great," you respond, "then I'm free." I can't be executed on Friday, because then I would know it on Thursday. But that means I can't be executed on Thursday, because on Wednesday I'd know it would have to be Thursday since it can't be Friday. But if it can't be Thursday, it can't be Wednesday, and if it can't be Wednesday it can't be Tuesday. But that means it can't be Monday either. That exhausts the possibilities, so you cannot execute me next week."
What, if anything, is wrong with this reasoning?
Hanging is no longer a permissible means of execution?
.... And you know it today - THREE days beforehand.Quote: MoscaYou're gonna get hanged on Monday.
The probolem is, there is nothing prolonged about the suffering.
Quote: jfalkI guarantee you will not be able to deduce the day of your execution the day before.
I'm changing my answer.
You'll be hanged Thursday afternoon.
The problem does NOT say that you WILL figure it out the day before, just that the day before, you still won't know.
Therefore, on Thursday, it could still be that day or the next day, but the longer the day goes on, the higher the likelihood that you'll figure it's gonna be Friday.
Quote: jfalkIn order to make you suffer a bit more, I guarantee you will not be able to deduce the day of your execution the day before." "Great," you respond, "then I'm free. I can't be executed on Friday, because then I would know it on Thursday. But that means I can't be executed on Thursday, because on Wednesday I'd know it would have to be Thursday since it can't be Friday. But if it can't be Thursday, it can't be Wednesday, and if it can't be Wednesday it can't be Tuesday. But that means it can't be Monday either. That exhausts the possibilities, so you cannot execute me next week."
What, if anything, is wrong with this reasoning?
It assumes the Judge is correct. That's not necessarily true, especially if his intent is to cause suffering. As a result, it's entirely possible for the you to be hung on Friday. On Friday, you can exult in the knowledge that the Judge was wrong about your powerful deductive abilities ... as the hangman slips the noose over your head.
But there is something manifestly wrong with the one day at a time reasoning, as for example it could be tuesday you will be hanged, and monday you wouldnt know that, even though it seems you could make a logical sequence working back from Friday, but it is some kind of false logic.
Any day of the week is fair game, except for Friday (if you are still alive Thursday night, then you'll have figured it out).
IF you are alive at the end of Thursday, THEN you can deduce you will be executed on Friday, because it's Friday. However, if it is still Thursday, you cannot correctly deduce that you will be hanged on Friday, since you could still be hanged today. Therefore, the Judge's second statement "I guarantee you will not be able to deduce the day of your execution the day before" is always true, and it is only via improper interpretation and logic that you reached the conclusion you did. You can be hanged on Friday because by the time Friday rolls around and you know you're gonna die, Thursday is over. Therefore, even if you are hanged on Friday, you did not deduce it "the day before", on Thursday.
Quote: MathExtremistit is only via improper interpretation and logic that you reached the conclusion you did.
What "improper interpretation and logic" do you have in mind? you say
Quote: MathExtremistHowever, if it is still Thursday, you cannot correctly deduce that you will be hanged on Friday, since you could still be hanged today.
My question is: how? I know that I can't be executed on Friday. It's impossible under the stated logical rules. Do you agree? But then if that's true, how can you say that I could be hanged on Thursday if I got to Thursday? Wednesday night wouldn't I have have to know it had to be Thursday since Friday is impossible?
For those hung up on the hours and the continuity of time (if indeed it is continuous), let's say that every execution is announced at 11 am and carried out at noon, so we won't have any problems with what happens as we approach midnight.
Quote: jfalk... For those hung up on the hours and the continuity of time (if indeed it is continuous), let's say that every execution is announced at 11 am and carried out at noon, so we won't have any problems with what happens as we approach midnight.
In that last statement, you made a material change in the problem. Under those conditions, I don't think ME's specific proof holds.
With that revised problem, I think that you can, indeed, deduce the execution is scheduled for Friday as of 11:01 A.M. on Thursday (assuming the execution is not announced for Thursday.) However, I don't think that you have basis to rule out either Thursday or Friday prior to that time. Both are still possible until that moment. Anticipating that you will be able to rule out Thursday (and deduce Friday) when 11:01 Thursday arrives does not mean that you can rule out Thursday any earlier, and certainly not way back on Wednesday.
The usual simple proof that there is an error in the logic is that once you conclude that you cannot be executed on any day, then the executioner shows up one day completely to your surprise -- you didn't see that coming a day ahead of time!
Quote: jfalkMy question is: how? I know that I can't be executed on Friday. It's impossible under the stated logical rules. Do you agree? But then if that's true, how can you say that I could be hanged on Thursday if I got to Thursday? Wednesday night wouldn't I have have to know it had to be Thursday since Friday is impossible?
Of course you can be executed on Friday. If it's Thursday, you don't know whether you're going to die today or tomorrow, so you can only deduce the following truth: "If I make it until Friday, I'll die on Friday". That follows from the first proposition (you'll die some day this week) and is unrelated to the second. It is impossible for you to deduce the day of your death "a day before" under any event.
Quote:For those hung up on the hours and the continuity of time (if indeed it is continuous), let's say that every execution is announced at 11 am and carried out at noon, so we won't have any problems with what happens as we approach midnight.
Okay then: it's Thursday at 11am. The judge announces you're going to be executed in one hour. Did you deduce that on Wednesday? No, of course you didn't - your logic said you couldn't be hanged at all. The Judge's statements are both correct, so your logic is flawed.
Here's an analogy that might help:
I tell you (a) I'm going to count, starting at 0 and going up to a number between 1 and 5, and (b) you cannot deduce where I will stop. (The more accurate phrasing would be "you cannot deduce that the next number is the last based on the previous number", but this form is even stronger.) I also assert that both (a) and (b) are true statements, to eliminate any question of deliberate falsehood.
Suppose I've just said "four". Will I say "five"? I say you have no logical basis for deducing either way from the rules given, just as in the original scenario.
If there is a flaw in the analogy itself, what is it?
Quote: EvenBobSome of us have been hung all our lives..
I'd like to see you try to get out of your sentence by telling the Judge that.
Suppose the judge can only say "Today you live" or "Today you die.". On Thursday he said "Today you live.". We now havve two startemts from the judge. 1) ONe day he will say you will die. 2Y The only day left is Friday. What's wrong with tthe deduction: Tomorrow the judge will say LYou will dieL. Doesn't that set up a contradiction?
Quote: jfalkMathextremist -- it's not an analogy -- it's the same problem. I say once you say four, you can't say 5 without violating predictibility. Thus, once you get to 4 you must stop....
Well, did you predict "5", or did you already predict another number, or did you say he can't stop on any number? Unless you got it right, he must not have violated predictability.
I agree that if the judge gets to the point of saying on Thursday, "Today you live," then, and only then, can you decide it will be Friday. But you cannot rule out Thursday being the day to die until he says that very day, "Today you live." If on Wednesday you were to decide that Thursday definitely has to be the day, then you are mistaken -- at that point it could still be planned for Friday.
Unless you make the statement each and every day, "Tomorrow is the day for the execution," you can't be sure of getting it right, and I would consider that to be outside the rules, because you wouldn't really have deduced the answer at all.
Quote: jfalkMathextremist -- it's not an analogy -- it's the same problem. I say once you say four, you can't say 5 without violating predictibility. Thus, once you get to 4 you must stop.
Perfect, if it's the same problem, let's use it instead. It's simpler. Remember the two propositions were:
1) I'm going to count from 0 up to a number between 1 and 5, and
2) You cannot deduce when I will stop.
If you agree that's the same problem, let's make it even simpler:
1) I'm either going to say "1 2" or just "1"
2) You cannot deduce which I will say.
Presumably you agree this reduction is also equivalent. If so, I'm guaranteed to say "1". Do you believe you can tell whether I'm going to say "2" after I've said "1"? If so, how?
Quote: weaselmanSince you already figured out you can't be hung on Monday, that means ... that you actually CAN. Or Tuesday ...
Any day of the week is fair game, except for Friday (if you are still alive Thursday night, then you'll have figured it out).
That's my answer, why I said you'll be hung on Monday. Once you're certain, it's over.
One and only of the five cards is red. It must be placed in such a way that, knowing this rule, an observer can not determine its location by seeing all the black cards before it.
Then it becomes clear that such a placement is impossible: red card's location will be revealed by seeing all the black cards before it.
If that's not clear enough as an analogy, imagine a game. A places the card and B must guess if the next card is red or black. If B says it's red and the next card is red, B wins. If B says it's black and the next card is red, B loses. Only the turn on which the next card is red counts. It's clear that such a game can be always won by B, because decisions on previous turns do not affect the outcome.
What if there are multiple execution times during a day? Then each day can be split into multiple cards (say 24 per row). Now we come to the question of what constitutes "the day before".
If player B must guess 24+ cards in advance, he loses, because in the middle of a day he still has no information if he will run into the red card this row or the next row. But the wording mentioned "the day before", not "a day before".
If player B must guess during the previous row, it's more interesting. Row 0 (Sunday) is guaranteed black, so he says row 1 is red. On row 1, he can just turn over all cards in that row at once. If any is red, he wins his row 0 guess. In case all are black, he must play with a guess that row 2 has the red card.
This once again branches into two variants. If he must mark his guess during the previous row, but can mark a new one during the new row, player B always wins using the strategy above. If he must mark his guess and maintain it, he loses.
This comes down to the wording, then. "You will not be able to deduce the day of your execution the day before" can be read one of two ways. Common sense dictates that the judge's intent was to make sure the prisoner would not be able to count hours he has left and prepare. But that would have to be formulated as "You will not be able to maintain knowledge of the day of your execution from the day before and onward". The judge only referred to a single act of deducing the day, which set the rules of the game such that there is a guaranteed winning strategy.
Thus, Prisoner's original reasoning was correct. If he was to be executed, Judge's intended torment would have been administered, but Judge's conditions as stated would not be met.
Quote: jfalkI my have spoken slightly too hastily when I said they were the same problem. There are actually two statements: "Today you live" and "Today you die." You cannot collapse these two into the single statement "1." But I grant that the paradox doesn't require 5 days, just two. If, on day 1, the judge says" "Today you live," then his hand is forced tomorrow. He MUST say "Today you die" tomorrow. Otherwise, he has failed to do what he said he will do. But if his hand is forced, I can predict it. I don't think there's anything controversial about this. The controversy comes on the first day (of two). I say the judge cannot say "You live" on day one. If he does, then we get to the situation above. Thus, he must say "You die" on day one. Once again, his hand is forced, so once again it is predictible. Your question, and it's a good one, is now that I hve deduced he cannot say either "You die" or "You live" On day one, what does he say? That is the (to my mind) as yet not entirely resolved paradox.
I think it's still a misinterpretation of what the judge meant by "you will not be able to deduce". If his simple statement has successfully gotten you so twisted about that you think you will both be executed and spared every day of the week, then you are clearly "not able to deduce" the day of your execution beforehand. In fact, whatever the judge says on the first day will be surprising to you - you didn't deduce it beforehand.
Edit: I found this book on the Internet, with a much better version of the paradox (by Martin Gardner):
A man tells his wife that he's getting her an unexpected gift for her birthday, and that it will be a gold watch. She thinks, "he wouldn't lie to me, so he's not getting me a gold watch." And then when she gets the gold watch, it's unexpected, and he didn't lie after all.
The premise of that section of the book, which is similar to what I've been poorly attempting to convey, is that deduction doesn't work when one party is privy to knowledge that the other party is not. In my number example, you had absolutely no way of deducing whether I'd say "2" after I said "1"; therefore, my statement that you could not deduce it was/is always true. You simply don't have the information required to make a proper deduction. The paradox, if there is one, is that you think you do -- but as we've seen, the reasoning that flows from that false sufficiency of information leads to a contradiction.
Quote: P90
If that's not clear enough as an analogy, imagine a game. A places the card and B must guess if the next card is red or black. If B says it's red and the next card is red, B wins. If B says it's black and the next card is red, B loses. Only the turn on which the next card is red counts. It's clear that such a game can be always won by B, because decisions on previous turns do not affect the outcome.
How can it always be won? Imagine, that you are playing this game. You turned two cards over, and both were black. What is your decision on the third one?
Quote: weaselmanHow can it always be won? Imagine, that you are playing this game. You turned two cards over, and both were black. What is your decision on the third one?
"The next card is red". Since only the decision on the turn where the next card is red is rated ("the day before the execution"), if the next card was black, it doesn't matter.
Quote: P90"The next card is red". Since only the decision on the turn where the next card is red is rated ("the day before the execution"), if the next card was black, it doesn't matter.
I see. I think, it's a different game then (at least, it's different from how I understand the original question). Under these rules, obviously, you can just keep saying "red" every day, and the judge's promise is impossible to keep, there is no paradox or even an ambiguity.
In the game I had in mind, once you say "red", the game is over, and you either win or lose, depending on what the next card is.
Quote: weaselmanIn the game I had in mind, once you say "red", the game is over, and you either win or lose, depending on what the next card is.
That could be different, but it was not the original set of conditions. Under the original set of conditions, there was no requirement that the Prisoner not make a mistaken guess before the day before; only that he can not make a correct guess the day before.
Quote: P90That could be different, but it was not the original set of conditions. Under the original set of conditions, there was no requirement that the Prisoner not make a mistaken guess before the day before; only that he can not make a correct guess the day before.
No, "making a correct guess" is different than deducing something. I can correctly guess heads 50% of the time on a coin flip, but I cannot ever *deduce* heads.
Quote: P90... Under the original set of conditions, there was no requirement that the Prisoner not make a mistaken guess before the day before; only that he can not make a correct guess the day before.
Then I guess you and I must have read the original post differently.
Quote: jfalk (in the original post of this thread)The Judge pronounces the date of your execution: "... I guarantee you will not be able to deduce the day of your execution the day before." ..
I don't think it is reasonable to equate "making an erroneous guess every day until one day by chance it happens to be true" with "deducing the answer." If that were the case, we could easily play a game where you reveal the cards of a shuffled deck one at a time and I am so clever that I can "deduce" and tell you in advance exactly when the next card will be the Ace of Spades. Of course, I might get it right on first try or I might make as many as 51 errors first, but that would be OK, I suppose. Not.
Edit: Sorry, ME, you're ahead of me again. Just gotta learn to compose faster.
Quote: DocI don't think it is reasonable to equate "making an erroneous guess every day until one day by chance it happens to be true" with "deducing the answer."
But if he is to be executed Friday, he will correctly deduce on Thursday than it's Friday. If it's Thursday, he will deduce on Wednesday that it's Thursday because it can't be Friday. That counts as deduction, and not merely guessing.
Ultimately, his deductions could be proven wrong, or even considered inadmissible if they can be changed, but they won't be, because under the stated conditions he can't be executed at all, and he made that deduction.
If you just have to put it that way, it would be "making a guess every day that is the closest to correct for that day, unless the rules are changed". The conditions don't say he must be able to have the correct answer at all times, merely on the day before his execution.
Quote: P90But if he is to be executed Friday, he will correctly deduce on Thursday than it's Friday. If it's Thursday, he will deduce on Wednesday that it's Thursday because it can't be Friday. That counts as deduction, and not merely guessing.
It's not a deduction, because it is based on a contradiction. The statement "It is Thursday because it cannot be Friday" is self-contradictory, because it uses the premise that you cannot deduce the day to ... deduce the day. Either the premise is false, and then you cannot use it to deduce anything (you can "deduce" any statement from a false premise), or the premise is true, and then your deduction cannot be.
Quote: P90... But if he is to be executed Friday, he will correctly deduce on Thursday than it's Friday. If it's Thursday, he will deduce on Wednesday that it's Thursday because it can't be Friday. That counts as deduction, and not merely guessing. ...
I think I covered this a few pages back, but I'll try again.
Your claim that, "... he will deduce on Wednesday that it's Thursday because it can't be Friday," is in error -- it can be Friday. Back on Wednesday it could still be either Thursday or Friday, or even Wednesday.
On Thursday, if he is still alive, he can deduce that the execution will be on Friday -- but only after some point during the day when it is no longer possible to be executed on Thursday. He cannot rule out a Thursday execution (and deduce "Friday") until that point. The fact that he might still be alive at that time on Thursday to make that deduction does not mean that earlier on Thursday, or earlier in the week, he could be certain to still be alive later to be deducing anything at all about Friday.
Making the mistake of believing that on Wednesday he could deduce "Thursday" because he anticipates that at some time late on Thursday he will be alive and able to deduce "Friday" is what leads to the false conclusion that he cannot be executed at all -- and the complete surprise on whatever unanticipated day the executioner actually shows up at his cell.
He can only be certain of guessing correctly if he guesses every day, which I have discounted, since it is quite different from a logical deduction.
This may be equivalent to the same thing that weaselman said, but I'm not certain.
Quote: DocHe can only be certain of guessing correctly if he guesses every day, which I have discounted, since it is quite different from a logical deduction.
The angle I was approaching it from is that it counts even if based on imperfect reasoning, as long as it's more than guesswork and the result is correct.
Quote: P90The angle I was approaching it from is that it counts even if based on imperfect reasoning, as long as it's more than guesswork and the result is correct.
The thing is, you can deduce *anything* "based on imperfect reasoning".
For example. Suppose, A is any false statement, that we are going to accept as an axiom. Then "A or B" is a theorem for any statement B. Now "A or B" is equivalent to "If not A then B", which is therefore also a theorem. Since A is false, "not A" is true. And if "not A" is true, B must also be true, because of the last theorem.
As you can see, *any* statement is provable if one of the premises is wrong, which basically means that prove/reasoning doesn't mean anything in this case.
Quote: P90The angle I was approaching it from is that it counts even if based on imperfect reasoning, as long as it's more than guesswork and the result is correct.
Perhaps I missed something in one of your earlier posts. Otherwise, I have to disagree with your, "... and the result is correct." I thought that you had decided to agree with the prisoner's original conclusion: "... That exhausts the possibilities, so you cannot execute me next week."
Quote: P90 on page 3Thus, Prisoner's original reasoning was correct. If he was to be executed, Judge's intended torment would have been administered, but Judge's conditions as stated would not be met.
The prisoner's conclusion/result is incorrect -- he can indeed be executed and would not have deduced the day. In fact, since he falsely concluded that he couldn't be executed "next week", the executioner could show up any day -- including Friday -- and execute him without the day having been deduced. That's why it is called "The Unexpected Execution."
Quote: P90The angle I was approaching it from is that it counts even if based on imperfect reasoning, as long as it's more than guesswork and the result is correct.
I disagree. I think the premise of the problem is that the Judge's statements are logically valid, and that valid logic is required to address the problem. Suggesting that you can address the problem with invalid logic is to throw out the entire basis for discussion. I could argue that you'll be hung on Wednesday because the highway is flat. It's a total non sequitur, but if you assign validity to an implication based on a non sequitur then all bets are off.
If his basis for deriving the conclusions is not examined, his logic would be correct.
Quote: P90This becomes rather semantic... OK, I have to agree, the term "deduce" perhaps restricts his options in deriving the conclusions. However, with slightly different semantics, like "predict", "foresee" or some other term, the situation could readily fall on the prisoner's side.
In the same sense in which every human can be said to be able to "foresee" his own death (or anything else for that matter) - all he needs to do is to say "today, I die" every morning when he wakes up.
I agree this is semantics (don't know why everybody assumes that semantics is unimportant BTW) , but I can't agree, that this is "foresight".
Quote:If his basis for deriving the conclusions is not examined, his logic would be correct.
If the basis is not examined, then ANY logic is correct.
Quote: weaselmanIn the same sense in which every human can be said to be able to "foresee" his own death (or anything else for that matter) - all he needs to do is to say "today, I die" every morning when he wakes up.
"It's Thursday, today's executions are over, it's gonna happen this week, Friday is the only day left" is rather different.
But w/e, as said, the "paradox" is simply in conflicting interpretations.
Quote: P90"It's Thursday, today's executions are over, it's gonna happen this week, Friday is the only day left" is rather different.
How about Wednesday? How is it different?
I know, I am going to die some day this century. Let's also accept a premise that I am not clairvoyant, or otherwise exceptionally gifted with foresight, and therefore, cannot predict the day I die.
By the same logic as yours, every day I know it cannot happen tomorrow, and therefore I conclude that it must happen today.
Quote:But w/e, as said, the "paradox" is simply in conflicting interpretations.
I don't think so. I think the paradox is caused by the contradiction in the reasoning system. You simply cannot use an "axiom" stating "foo" is false to prove that "foo" is true.
Quote: P90This becomes rather semantic... OK, I have to agree, the term "deduce" perhaps restricts his options in deriving the conclusions. However, with slightly different semantics, like "predict", "foresee" or some other term, the situation could readily fall on the prisoner's side.
If his basis for deriving the conclusions is not examined, his logic would be correct.
Not really. The original problem statement was taken from a factual announcement by the Swedish civil authorities. They simply said "A civil defense exercise will be held this week. In order to make sure that the civil defense units are properly prepared, no one will know in advance on what day this exercise will take place".
There's a limit to what you can deduce or predict if you simply don't know something. In my counting-up-to-5 example, you simply cannot deduce whether I'll keep counting or whether I'll stop until after I've stopped. By then it's too late.
