“And in the mornin’… I’m makin’ WAFFLES!” - Donkey
Excellent… All philosphical resolutions waffle anyway, so… 
I think Joe’s explanation is right here: the rules make a Friday execution day indeterminate, since no matter what is picked, it violates the rules. You might just as well argue that the prisoner should reason “I can’t still be alive by Friday, because then the rules would be broken.”
What roadfood and BillH described is the game theory version of this, not the riddle version. The riddle version is essentially from the perspective of a captor, promising these two things to his prisoner. It isn’t about either of them trying to “beat” the other in a game, but simply to acheive the two conditions, given that both the prisoner can reason perfectly, and the captor can predict the reasoning of the prisoner. You can even think about the captor trying to be humane about it: he has to kill the prisoner, but he doesn’t want the prisoner to know the day before. The problem is in letting the prisoner know that we wants it to be a surprise: if the prisoner didn’t know that the captor had that intention, there would be no problem at all.
If he lives 'til Thursday night, he absolutely will know he’s being killed on Friday, it’s the only day left. This is why we know the king would never pick Friday (the last possible execution day) to be killed because that violates his own rule.
You just contradicted yourself again. Well, of course you did. This is a paradox after all. Here are the two things you said:
[ol][li]If the guy lives until Thurday night, then he knows that he’ll be killed on Friday.[/li][li]The king cannot choose Friday as an execution day.[/ol]However, if the king cannot choose Friday, then if the prisoner lives until Thursday night, he knows that he cannot be executed on Friday, which contradicts your first statement.[/li]
In other words, (1) states that, should the prisoner live until Thurdsay night, he must be killed on Friday. (2) States that if the prisoner lives until Thursday night, he cannot be killed on Friday. That is a logical contradiction.
The problem is that you keep ignoring the paradox and assuming that the king cannot choose Friday.
Let me focus your attention on the heart of the matter: If the prisoner knows that he will be executed, then he cannot be executed. That is a logical paradox. The prisoner can only know that he will be executed if he is going to be executed. As soon as you decide that he can’t be executed, the above statement is no longer in effect, and he can once again be executed.
The king cannot choose Friday.
The king can choose Friday.
Both of those statements are true, and both of those statements are false. This is the nature of the paradox.
Fortunately (for the king), the execution can go on as planned, because the conditions require the prisoner to know when he will be executed, and on Sunday, he does not know if he will be executed or not on Monday, and thus Monday is a perfectly valid day to execute him.
It was at this exact point that the paradox unwound. The reason the prisoner cannot be executed on a Friday is because on Friday there is only one possible day to execute him. However this condition only exists on Friday. The “logic” does not apply to the other days of the week. On every previous day, the executioner has a choice of days and therefore the prisoner cannot prove that today isn’t the day of his execution.
Let’s recast the puzzle. Imagine you put a feather in a box. You then shuffle the box together with six other identical boxes. Can you predict which box holds the feather before you open it? If you open six of the boxes and don’t find the feather, you obviously can now predict that it is inside the unopened box. But that doesn’t mean you could have made any predictions while two or more boxes were still unopened.
You’re not following the prisoner’s reasoning far enough to see the paradox. Here’s his full thought process:
“Well, it’s Thursday night and I’m still alive. Friday is the only day left, therefore I know I’m getting executed tomorrow. Now wait a minute, since I know I’m getting executed tomorrow, and the king obviously knows that I know, then I must not be getting executed tomorrow, according to the 2nd rule the king gave me. Hmmmm, now since I don’t know anymore if I’m going to be executed, then I guess the king could kill me after all, and Friday’s the only day left, so now I know that I will get executed tomorrow. But since I know, … HEAD EXPLODES”.
I think the error that you’re making is that you are giving the first rule (prisoner must be executed on M-F) primacy over the second rule (prisoner can’t know about it the day before). If you make the two rules equally important, then we have a paradox.
There is no contradiction here. All I am saying is if he lived until Thursday he’ll know he’s dying Friday, which is why the king will never pick Friday. In other words, he can never live until Thursday night. The king wouldn’t allow it. The if scenario was just to show why it couldn’t happen.
I don’t think this should be called a paradox because I think there must be a solution. Here’s why: Instead of looking at Monday through Friday, let’s assume he’s going to be killed anywhere from April 1st to April 30 (all of the old rules apply). Well of course the king wouldn’t schedule his execution for the 30th, because on the night of the 29th he would know he is dying on the 30th and the king doesn’t allow anyone to ever know a day in advance when one is being killed. We know from the OP that every day eventually gets eliminated using the same logic. This is what really gets me. I don’t see why he can’t just be executed, say, on the 15th. But at the same time I’m sure he can’t be killed on the 30th. And then not on the 29th… etc. Where is the cut off date?
For what it’s worth, I’ll add my two serious bits in here. I don’t buy this argument.
You’ve already stated that the prisoner ‘knows’ that Friday is impossible execution day on Thursday. In fact, he can ‘know’ that if he gets to Thursday, Friday is an impossible execution day. In fact, it can be the previous Sunday, and he knows Friday will be an impossible execution day. He can sit there, in his cell one day, two days, or a week prior thinking “the day set for my execution cannot be Friday because on Thursday I will know, thus breaking the Judge’s rule”.
Let me be more succinct: At any point, the prisoner knows that the judge cannot have already chosen Friday, because eventually the prisoner will be at a point in time which is Thursday, and the prisoner can know this at any point prior.
But even if the Judge hasn’t chosen it to be Friday at some point prior to Thursday, the Judge knows he can’t choose Friday at any point in the week either for exactly the same reasons.
Even worse, the Prisoner can perfectly logically deduce that the Judge knows he cannot choose Friday at any point in the week.
This means Friday is absolutely excluded no matter what the current time is.
But this means Thursday cannot be chosen either for exactly the same reasons. It is the next possible day that could be chosen, but the Judge cannot choose it because the Judge is aware that the Prisoner is aware (by simple logic) that Friday cannot be chosen, and so Thursday also cannot be chosen. The Prisoner can be certain on Wednesday night that he will not be executed Thursday because Friday has already been excluded, and Thursday is the next remaining day.
Again, to put it more succinctly, both parties can logically deduce that Friday is out, so at some point Wednesday must come along and that will exclude Thursday as a day of execution.
This process extends backward in time forever.
Therefore there is no day on which the Judge can choose a day of execution that does not break the rules of which both the Judge and prisoner are logically aware.
I fail to see how this is anything but a paradox constructed out of a mathematical (logical) recurrence relation.
You can philosophize about ways around it, true, and that’s where the interest in the puzzle lies, but this has nothing to do with human fallibility or choice or lies.
The only pseudo-logical answer that I can personally come to is the one that I originally stated, however flippantly. Because both parties are aware of the lack of ability for the Judge to choose any day of the week, then if the Judge executes the man on a particular day, he won’t be expecting it, satisfying all of the conditions. But this answer depends on ‘expectation’ instead of ‘knowing’ (by deduction), which is a slight bending of the meaning of the puzzle.
In order for this to reflect the original puzzle, you have to add the condition that if you know the box you are about to open contains the feather, then it won’t be there.
Therein lies the conundrum.
OK, consider the following statement (bear with me):
You cannot correctly believe that this statement is true
Say you believe this statement to be true. “If it really is true,” you might reason, “then my belief that it is true is correct, which contradicts the statement itself. So either it isn’t true, or I am deluding myself that I believe it.” Whatever reasoning you use, you cannot “correctly believe” the statement to be true, which is just what the statement says, ironically proving to anybody but you that the statement is true, with no contradiction.
How is this relevant? Well, the statement above puts you in the same position as the prisoner in the OP. The warden’s second statement, that the prisoner will not know the day of his execution in advance, is just a variation on the statement above - known to be true by the warden (and anyone, except the prisoner, who cares to consider it), unknowable by the person mentioned in the statement itself, the prisoner.
…
BTW I agree with apos that Roadfood’s restatement of the problem is quite different to the original, because in Roadfood’s version there is a chance that the prisoner won’t be executed at all, in which case the warden’s first statement is simply false, with no paradox. The point of the original version is that the prisoner does get executed, despite apparently disproving the warden’s statements, which all turn out to be true.
The story involves two premises or initial conditions:
- The guy will be hanged on one day next week.
- The hanging will always be unexpected i.e. he will never know in advance which day it is.
To resolve the paradox, you have to realise that although the two premises seem reasonable and plausible, they are logically inconsistent.
It is easy to see that there is no paradox unless you, or the man in the story, believes both conditions are true. However, suppose he has not been executed on Thursday. At this point, if he still subscribes to the view that both premises are true, then it follows that:
a. he will be hanged the next day (Friday)
b. he will not be hanged the next day (because it must be unexpected)
In other words, the hanging is both expected and unexpected, which is logically inconsistent. This inconsistency can only be resolved by declaring that one or both premises must be false, in which case there is no paradox.
refusal wrote:
Not as far as I can tell. Given original proposition #1 and your re-worded proposition #2, the prisoner still only has the ability to “logically deduce” when he won’t die, not when he will.
And I put “logically deduce” in quotes because, as so many here have already stated, given the two rules, the prisoner can, if he so chooses, use some reasoning right after he’s told the “rules” to deduce that he cannot be executed on any of the five days (or 30 days, in x-ray vision’s extended version).
However, once he comes to that conclusion, the prisoner should think some more to see that the only possibilities are:
A) at least one of the two propositions is a lie,
B) his reasoning is invalid, or
C) a paradox exists, from which no amount of intelligence will help him escape.
No matter which is really true, he’s screwed. There’s no way he can tell which is true, without more information. Therefore, there’s no way he can deduce (or ‘know’) whether or not he’ll be executed on Friday, even if he lives through Thursday.
On Friday morning:
Prisoner: But, I knew last night that I’d be executed today!!!
Hangman: Oh? Did you really?
Prisoner: Well, last Sunday I proved to myself that I couldn’t be executed today…
Hangman: So did you know last night or not?
Prisoner: Look, my head hurts already. Just get it over with.
First a reality check. Remember the conditions of the OP.
The prisoner, using psuedologic, reasons that he cannot be executed Friday, then works backwards through the week and “proves” to his satisfaction that he can’t be executed on any day.
The prisoner wakes up Tuesday morning and is told he is being executed. He didn’t know that would be the day of his execution.
So now it’s clear that the premises of the OP can be fulfilled. Therefore, any logical argument that says they can’t must be false. At this point, all that remains is to figure out specifically how the logical argument went wrong. For that, refer to my previous post.
The difference between this riddle and the one we’re dealing with is that in the one we’re dealing with, the days of the week aren’t equal and the boxes are. We know we can eliminate Friday right from the start, but we can’t eliminate one of the boxes from the start.
I agree, the premise of the OP can be fulfilled. That’s why I don’t think it should be labeled as a paradox. It only seems as if it’s a paradox. My question is, what is the last possible day he can be executed and still have it be a surprise? Obviously not Friday for reasons we’ve stated about a hundred times.
Sure you can. First of all, you have to add the condition (as I mentioned above) that if you KNOW the feather is going to be in the box you are about to open, then it WON’T be there. Without this condition, it is not a true reflection of the prisoner’s scenario.
Once you have that condition, you can definitely eliminate the last box, right? Because you know it would be in there, right? But if you know that it’s in there, then by definition it is not in there, therefore we have a paradox.
The real confusion comes from whether or not this paradox affects the second-to-last box/day. My gut feeling is that it does not, but it is difficult to explain why… but I’ll give it a shot anyway. The paradox’s result is that the king both can and cannot schedule the execution for Friday. The reason he can is because he knows that the prisoner would remember rule #2 and would therefore not expect to be executed Friday. But the reason he cannot is because he knows the prisoner would remember rule #1 and would therefore expect to be executed Friday. So is Friday eliminated as a possible day? Both yes and no. Is Thursday eliminated? We don’t know. Since we can’t answer the first question because of the paradox, the rest of the questions become pointless. We can’t know whether or not the other days are eliminated because the first piece of logic we need to reason through is a contradiction. So I guess the answer to: “Does the paradox affect the previous days?” is yes and no.
You just made the same mistake as x-ray vision, you didn’t take the prisoner’s reasoning far enough. Let’s say he does “prove” to himself that he can’t be executed any day, so he’s not expecting to be executed on any day. Since he’s not expecting it, now he can be executed, as your example went on to demonstrate. The problem is that our prisoner is no dummy. He realizes that if he knows he can’t be executed on any day, then he could be executed on any day after all and the whole train wreck of logic starts over.
The reasoning doesn’t affect the outcome. Imagine there’s three co-defendents; Archie, Dilton, and Moose. The judge finds all three guilty and gives them the same speech.
Archie’s pretty smart and figures out he can’t be executed on Friday without violating the Judge’s decree. Archie then figures that using the same logic he can eliminate Thursday as well, then Wednesday, Tuesday, and Monday. Archie’s now confident he’s safe.
Dilton uses his superior intellect and keen logic and also deduces that he cannot be executed on Friday. But he realizes that Friday is a special case and the same logic says nothing about other days of the week.
Meanwhile, Moose goes back to cell and decides to read Garfield comics until his execution. He vaguely remembers the judge said something about what day he was going to be executed but he wasn’t really paying attention.
On Tuesday morning, Archie, Dilton and Moose are told they’re going to be executed that day. None of them predicted Tuesday as their final day and all were equally surprised.
Meanwhile, Reggie takes off to Brazil with Betty, Veronica, and all the loot.
Again, like I said, look at it as if the prisoner’s reasoning goes the other way, focusing on the fact that he can’t still be alive by Thursday, because then the rules would be broken by the paradox. Now the prisoners logic would lead him to believe that he can’t live to see any day of the week.
I think Joe Random is on the right track. The logic should not extend beyond Friday because the paradox does not allow Friday to be resolved.
What if we extend it to a year? Judge say you will be executed one day this coming year and it will be a surprise. Prisoner reasons it can’t be the last day of the year or he’d know. therefore it can’t be the second last day either, or the third, etc etc. Prisoner was still surprised however, when he was executed on the 25th of April.
If the prisoner was to live till the second last day of the year, then he would not be surprised to be killed on the last day. If however he does not live that long, then he is free to be surprised on any other day. If he does live that long, then he is still surprised, because he has convinced himself he can’t be executed on the last day.
Think of it this way. On the Thursday, the prisoner is certain that he will be killed on Friday. On Sunday he is not really sure what day he will be killed on. As each day passes his certainty grows stronger that he will be killed tomorrow, but he is not totally certain, ie he does not know until Thursday.