This, IMHO, is not at all a paradox. (I knew a guy who wrote a nice review article on this problem, pro-paradox. Lot’s of stuff going there.) It is merely a simple statement of 2 true facts:
I think the problem here comes from the (frankly, rather silly) notion that a surprise isn’t a surprise, unless you never find out. A month later, does the man know whether he’s been executed? Of course! It’s not a surprise any more, and nobody is troubled by the fact that it’s not a surprise any more. Eventually, the cat’s going to be let out of the bag. When the warden says that they day of execution will be a surprise, all he means is that the prisoner doesn’t know right now when it’ll be. He does not mean that the prisoner will never know.
Ok, so you’re saying he can’t be executed on Friday because Thursday night he’ll know, right? Then don’t you see how he can’t be executed on Thursay either because Wednesday night he’ll know that since he can’t be executed on Friday, he must be getting executed Thursday?
That doesn’t follow. He doesn’t know what his captors are thinking, so he doesn’t know if they know that he can’t be executed on Friday. So if he “knows” that he will be executed on Thursday, and he isn’t, then he didn’t actually know, did he?
Here is what the whole “paradox” boils down to: The prisoner knows that he cannot be executed on day X. However, that knowledge makes it possible to execute him on day X, since he no longer knows that he will be executed. If he’s smart enough to figure that out, then he will know that he will be executed on day X, which brings us back to square one.
The resolution is to ask him on Monday if he knows on what day he will be executed. If he says no, then execute him right then. If he says yes, ask him which day he knows he will be executed on, and then execute him on a different day. If he refuses to answer, execute him immediately.
This solution is based on the fact that the prisoner never actually knows when he will be executed, exept for the last day, which results in a one-day paradox. The paradox does not spread to consume the rest of the week, because, even though logical analysis shows that Friday becomes invalid for executions, it is still listed in the initial conditions as being a valid execution day.
Thus, before the week starts, he simply cannot know when on which day he’ll be executed. He can only guess, and that’s not the same thing.
Sorry, but I have to get redundant here. Again, you’re saying he definitely can’t be executed on Friday. Then if he lives to see Wednesday night, he will know he’s getting executed Thursday since Friday has definitely been ruled out. Now Thursday night is also out of the question.
We are assuming as part of the riddle that his captors know he is intelligent enough to figure out that he can’t be executed on Friday, and that the captors are intelligent enough to pick a day that he can’t possibly foretell.
There’s no paradox. It boils down to what the prisoner can know with absolute certainty. Just because someone tells him he’s going to be executed next week, it doesn’t necessarily mean it’s true. In fact, the hangman, the judge and the prison could just be figments of his imagination. However, one thing he can be absolutely sure about is the state of his own knowledge, including such things as whether or not he knows that he is going to be executed. Since we have just established that he doesn’t know this, he can be absolutely sure that Statement 2 is true. He cannot be absolutely sure that Statement 1 is true, so if he does get executed it will indeed be a surprise, even on Friday.
You can simplify the “paradox” down to this:
I am going to hand you a playing card, face down.
Say I do actually give you the four of diamonds. Will you know, with 100% certainty, that it is the four of diamonds before you turn it over? Of course not, I could have been lying.
Okay, lets look at this from a different perspective.
Assume that I am the captor, and you are the prisoner. I will execute you next week. Now tell me, on what day will you be executed?
You’ve got to be kidding. This is a logic puzzle; we’re to assume we are told the truth about the puzzle itself and everything is as if it were real.
BTW, your paradox is a contradiction in terms, not the same as the riddle we’re dealing with here.
True. This has to be handled from a purely logical or mathematical basis, if possible. I fail to see how ‘human nature’ even enters into the original puzzle at all. Classic style of logic problem, only this one purports to have a logical or mathematical paradox.
The resolution may delve slightly into more abstract reasoning in some sense… something a little more philosphical than pure logic, but analyses that depend on human nature would seem right out.
At least in terms of this puzzle. Redefine it in terms of human nature if you want a different puzzle.
Wont work Joe, you’ll just get the same working-backwards-from-Friday routine.
I see an image of heads beating against brick walls…
Well, definitely not on Friday, because then on Thursday night I’d know it. Definitely not on Thursday, because then on Wednesday night… I don’t know! I just know it’s an amazing question that there has to be an answer to, but I don’t think anyone here has gotten it yet.
Both the statements could be true, so how is it a contradiction in terms? Anyway, how about I reword it:
Same as the original problem. You can add all the red herrings about Monday, Tuesday, Friday if you want, doesn’t make any difference apart from distracting people.
Sure it’ll work. The paradox is entirely mental. The reality of the situation is completely different.
If the prisoner is not able to correctly identify the day he will be executed with 100% surity, then he can still be executed. The paradox simply acts to confuse the prisoner and insure his demise.
The key concept here is that the prisoner must know the day of execution before the actual day. All the mental gymnastics of working backward from Friday do nothing to increase the prisoner’s actual knowledge.
It’s quite simple, really. Does the prisoner know before the week starts that he will be executed on, say, Monday? If he knows that he will be executed, he can’t be executed. So, of course, he now knows that he can’t be executed, so he can be executed…
Exactly. For the prisoner.
However, all the paradox does is make it so that the prisoner cannot have perfect knowledge of his execution date, which means that all days are valid, since the prisoner will be so confused by the paradox that he can never be certain of which day he will be executed on.
In practical terms, the following must take place.
[Sunday]
Prisoner: “I know that I will be executed on Monday.” (He must say this to avoid execution on Monday)
Captor: “Damn, I guess I can’t execute you on Monday.”
[Monday]
Prisoner: “I know that I will be executed on Tuesday.”
Captor: “You knew that you would be executed today, and you won’t be. Therefore, you are lying. You don’t know that you will be executed tomorrow. You’re just guessing to save your own ass.”
[Tuesday]
Captor: “You’re going to be hung today.”
Prisoner: …
See how easy that is. The whole trick is that the prisoner must know instead of just guessing. For him to know, he can only pick one day. After all, you can’t know you’re going to be executed on multiple days; that’s just not possible. However, since he “knows” he’ll be executed on a particular day, all other days are valid for execution. The prisoner is not allowed to guess just to save his own ass.
Ok Joe, let me ask you this: Since we agree he can’t be executed on Friday, can he be executed on Thursday?
Both statements can’t be true. You can’t tell me my card will be a four of diamonds and then also tell me I won’t know what my card is. Same problem with your new scenario.
Even if it is the four of diamonds, you won’t know that until you turn it over. Even if the prisoner is executed on Friday, he won’t know that until he is summoned for execution. Both statements can be true, but you/the prisoner can never know whether Statement 1 is true until it’s too late.
Yes. Actually, to say that he can’t be executed on Friday is false. To say that he can be executed on Friday is also false.
Assume for a moment that it is Thursday, and he hasn’t been executed yet. He now “knows” that he will be executed on Friday. That means that, according to the rules, he can’t be executed on Friday. But then, since he can’t be executed on Friday, and he knows that he can’t be executed on Friday, the rules once again allow him to be executed on Friday. Begin infinite loop.
It’s a paradox alright. It is impossible to determine if he can be executed or not on Friday. However, you can’t even bring Thursday into the equation because you haven’t yet resolved whether he can be executed or not on Friday
Your argument that he cannot be executed on Thursday assumes that he cannot be executed on Friday, yet the paradox prevents you from making that assumption; The state of Friday remains unknown. Thus the prisoner can be executed on any day other than Friday.
The problem is that there is not enough info. With a concrete set of rules, there is no paradox.
What is missing:
a) what is the process for the prisoner to declare his prediction for which day he will be executed? will he say before the week begins, “I think it’ll be wednesday”, or does he each morning declare, “I think it will be today”?
b) when will the execution date be set? before the week begins? on the day of the execution? after the prisoner announces his prediction for the day?
c) may the prisoner predict multiple dates as his execution date or just one?
For the sake of example, I’ll pick what I consider reasonable answers to these questions. But you can pick any combination and still get a non-paradox.
a) The prisoner will declare each morning whether it will be today or not.
b) The executioner picks the date before the week begins.
c) The prisoner gets to pick one date.
So, in this case, each morning, the prisoner gets up and tells the executioner whether he believes he’ll die today. If he guesses right, he goes free. If he guesses wrong, he hangs the appointed day. The prisoner has a one in five chance he’ll pick right and go free; even the Friday thing doesn’t matter, because he has to pick the right day in advance. I.e. if he guesses “friday”, and it’s actually Wednesday, he’ll hang on Wednesday.
Now, let’s assume for #c, we allow the prisoner to pick multiple days. Well, for that case, he’ll never hang because each day he predicts today is his day, and when his day actually comes, he’ll predict correctly and walk.
If all the questions are answered, there is no paradox.
Bill H. kinda beat me to this, but the OP really mistated the problem somewhat, which is causing most of the confusion on this thread. The problem should go something like this:
The warden tells the prisoner that he will be executed one day next week (Monday through Friday, if you will). The warden says that the execution day will be chosen in advance and will not be changed. The warden then makes the prisoner the following deal: Each morning the warden will ask the prisoner if today is the chosen execution day.
If the prisoner says yes and he’s right, the warden will let him go free.
If the prisoner says yes and he’s wrong, then he’ll be executed on the spot.
If the prisoner says no and he’s right (i.e. today is not the chosen day) then the game continues.
If the prisoner says no and he’s wrong (i.e. today is the chosen day), then obviously he gets executed.
The salient points are 1) the day is stated to be chosen in advance and will not be changed; 2) the prisoner gets asked once per day if today is the day; 3) The prisoner may only answer “yes” once: if he ever says “yes” and he’s wrong, he’s dead.
So the reasoning of the prisoner should now be clearer: Obviously, if he makes it to Friday and he’s asked if today’s the day, he answers “yes” and goes free. So he rules out Friday as a possible execution day because it’s logically impossible for him to actually be executed on that day; given the constraints of the puzzle, he can’t be executed on Friday because if he were to make it to Friday, all he’d need to do is say “yes” to the question and he’d go free. And from that he reasons backwards that it can’t be Thursday, can’t be Wednesday, etc.
The way to resolve the apparent paradox and discover the flaw in the prisoner’s reasoning is to look at the puzzle from the warden’s point of view. If the warden is intelligent and has thought of the same reasoning that the prisoner has, he will conclude that the prisoner believes that he cannot be executed on any day. Therefore when the prisoner is asked on Monday if today is the day, he will say no. So if the warden chooses Monday as the day, the prisoner will die. The warden can make the same argument for any day except Friday.
So what is the flaw in the prisoner’s reasoning? It’s the “if” part: Friday gets ruled out as the execution day only if the prisoner lives to Friday. Ruling out the other days then follows from that premise. But, if one of the other days turns out to be the execution day, then that first premise is wrong so the whole chain of reasoning falls apart.
Finally (just to say it another way), note that there are two conditions necessary for the prisoner to live to Friday: 1) the prisoner must say “no” on all previous days, and 2) all previous days must not be the actual chosen day. If number 2 is false (i.e. one of the previous days is the chosen day), and the prisoner says “no”, he dies.