partly_warmer,
I understand your position now, but don’t agree.
It certainly seems from the man’s argument that he can not be suprised. But common sense tells us he can.
When the guards knock on the door on Tuesday morning, the prisoner will not know their intentions.
Your last paragraph is correct, but I feel it rather misses the point. Yes we can avoid the paradox, but can we solve it ? hawthorne seems to think not, and I’m starting to agree.
You need to make the next step: the prisoner has “proved” A is impossible, but if the execution is done on any day it will be a surprise.
See: http://www.wischik.com/lu/philosophy/surprise-exam.html
http://mathworld.wolfram.com/UnexpectedHangingParadox.html
http://www.jaworski.co.uk/m10/10_paradox.html
This one’s not a paradox, just a result which relies on backward induction (which I’m suspicious of). See http://www.economics.soton.ac.uk/courses/ec614/ec614_topic1.pdf slide 3 for the basic PD and slide 21 and 22 for the finitely iterated. Put simply: if cooperation can only be achieved by appeal to the benefits of continuing cooperation, cooperation can’t get going because it must end in the last iteration. Since it must end in the last iteration, there’s no point cooperating in the second last. Etc.
[sub](Hmm, on preview that doesn’t look that simple, but it’s close to bed time and I’ve had a few, so maybe someone else will make more sense.)[/sub]
One can be emotionally surprised at an event they know is coming, such as a death of a loved one and can even be surprised when the time of the event is certain.
But emotions are not the content of paradoxes or logic. They have no truth value, ;they are either consistent or inconsistent with one’s internal system. If you felt the judge meant to surprise the prisoner emotionally, this whole thread is meaningless as to reason. We might as well be discussing which tastes better, apples or oranges. It’s just a matter of taste. Nothing comes of it logically.
Among other things, for a problem to be even considered as a possible paradox, the content must involve matters with truth value. In this case, the judge’s sentence meant the prisoner could not logically determine the time of the sentence, but the prisoner discovered the LOGICAL inconsistency thru induction.
i see a hole in the prisoner’s inductions, as well. he considers all the days of the week in one fell swoop. but he shouldn’t.
look at it this way. okay, he can’t be executed on sunday, because he would expect it. and saturday is off as well, because (excluding sunday) saturday is the last day, and he would expect it.
but he can’t keep excluding days past saturday. for instance, he says now that friday is the last day, so he can’t be executed on friday. but on the actual day of Friday he could still be executed on Friday or Saturday. so no matter what logical loopholes he’s forced himself through to discount the rest of his days, the fact remains that he would be surprised to be handed a blindfold on any day.
jb
Say we turn this into a game. The game has 7 turns. In each turn the prisoner player first writes on a hidden piece of paper whether he thinks he will be executed on the day or not. Afterwards, the executioner player says whether the prisoner will be executed or not. If the executioner says the prisoner will be executed, and the prisoner writes “no execution” then the executioner wins the game. If the executioner says “execution” and the prisoner writes “execution” then the prisoner wins. It is against the rules for the exutioner to not say “execution” once during the 7 turns.
Obviously, in this game, the prisoner need only write “execution” every turn, since there is nothing wrong with being surprised that he is not executed. So the best strategy for the prisoner in this game is to write “execution” every turn, and then the executioner can not possibly be correct in saying that the prisoner will be executed by surprise. So there exists a winning strategy for the prisoner.
So when the prisoner reasons that the executioner will not execute him on Sunday because it would not be a surprise this is fallacious. The prisoner is assuming that a strategy exists for the executioner that gives him a possibility of winning.
So to translate the prisoner’s argument into the game:
The prisoner reasons, if there exists a strategy that allows the executioner a chance of winning, he will use that strategy. That strategy could not be to not say “execution” until Sunday, because that gives him no chance of winning. Therefor the executioner will not say “execution” on Sunday.
So the prisoner’s conclusion is based on a false premise, because there exists no winning strategy for the executioner, and therefor, he cannot possibly use that strategy, and therefor no one can say whether the executioner will not say “execution” until Sunday. He very well might – this would only make him lose (just make his statement that the prisoner would be surprised false).
So any objections? Have I solved it?
Here are some possible objections:
The original problem doesn’t speak of winning or losing, so a better formulation would be that it is illegal for the executioner to say “execution” unless it will be a surprise, i.e. if the prisoner has not written “no execution.” But this is just a game with no legal moves for the executioner, since the executioner cannot know whether the prisoner has written “no execution” so he must always say “no execution.” On the last turn the rules become self-contradictory when he must say both “no execution” and “execution” or break the rules.
The game representation is flawwed, because it doesn’t address the irrationality of the prisoner believing he will be executed every day. This objection is wrong, because you can’t have your cake and eat it too. If you are going to allow temporal reasoning, you have to allow the prisoner to change his mind every morning. If instead you force the prisoner to choose a single day when he believes the execution will occur then it becomes a different game – in effect you are forcing him to write all 7 notes on the first day and only write “execution” once. In this game, the prisoner clearly only has a 1 in 7 chance of winning. But clearly, the line of reasoning that the prisoner takes in the original problem is absurd if this is the game representation.
So, any more objections? If not then you saw the solution to the Executioner’s Paradox on the Straight Dope first.
To my way of thinking, the problem is that the prisoner can’t logically eliminate Saturday and can’t therefore use that logic to eliminate the rest of the days either.
Saturday’s logic doesn’t lead to a conclusion only to a paradox, if you will. It’s stopping part way thru circular referencing. By eliminating Sunday, he concludes he WILL get a blindfold. By KNOWING he will get a blindfold, he concludes he MUST get breakfast because the judge won’t give him the blindfold if he knows it’s coming. So now he KNOWS he’ll get breakfast. That means his first assumption is wrong. His KNOWING he was going to get a blindfold is the reason he now KNOWS he is going to get breakfast. There is no conclusion to be drawn from this. It can go on forever. Since he now KNOWS he’ll get breakfast, it leaves the posibility open again for him to get a blindfold.
The judge made the mistake of letting him eliminate Saturday with opposing conclusions.
I hope I explained that reasonably well.
Jim
[hijack]
Actually, that doesn’t compute, because we’re assuming that if God is omnipotent, he can create an object that is immovable.
[/hijack]
Please continue; I have been intrigued as to the flaw in this paradox (the OP’s one) for many years. 
I took a class on paradoxes my last semester in college, and this one came up. No resolution was forthcoming, so I’m still interested in it.
Aha, this answers it for me, thanks. I’ve been wondering about this one for years, but I didn’t know how to formulate the exact objection you mention here, or why it was wrong. You can’t guarantee that the prisoner will be surprised, because you can’t guarantee that the prisoner won’t change his mind every day… I guess another way of looking at it is, it’s possible for someone not to be surprised by his own death, even if he doesn’t know when it is, if he’s always expecting it.
Incidentally, I tried to simplify this problem to one day, but it didn’t help:
JUDGE: You will be executed tomorrow morning, but only if you’re surprised by it.
PRISONER: Well, I guess that’s fair. (Hee hee…)
JUDGE: What are you laughing at?
PRISONER: Oh, nothing…
(The next morning comes around…)
GUARD: Wake up! It’s time for your breakfast or blindfold!
PRISONER: Aha! Well since I’m expecting the execution today, I’m not surprised by it! Thus you can’t execute me. I’m anticipating the execution, and so therefore, I’m not.
GUARD: The judge knew you’d feel that way. Here’s your blindfold.
PRISONER: Ah, crap…
So why not try to answer the question I asked you:
Are you saying that the judge’s claim - that the prisoner would be surprised - is false?
murphyboy99, I think this might make what partly_warmer is saying clearer:
"A man is to be killed by firing squad. The judge says that his crime was so heinous that, to increase his suffering, he cannot know which day he will die until the morning of the execution.
[The judge says that] the execution will happen next Monday. Monday morning at 09:00 there will be a knock on his cell door; the guard will give him a blindfold. But the judge stipulates that until [the guard] opens the door he will not know this."
The extra days are not necessary for the “paradox” and only serve to distract you from what’s going on.
The man reasons that he cannot be surprised by the execution on Monday, so therefore the execution cannot happen.
Of course, the flaw in his reasoning is when he realizes that he cannot both be executed and surprised, he assumes he won’t be executed rather than assuming he won’t be surprised.
kg m²/s²
I disagree that there is no such thing as a paradox. While it’s true that a statement that is paradoxical can be undone by amending it or rewriting it, a statement or concept in and of itself can be a paradox.
Consider:
“This statement is false.”
As a reference to a previous or subsequent statement it’s need not be a paradox. However, when viewed as self referential it is. If it’s true, then it’s false, but if it’s actually false, then it must be true. See? As another person said, it’s both “A” and “Not A” at the same time.
Gosh what a mess I made. Let me take another crack:
So why not take a crack at the question I asked you: **Are you saying that the judge’s claim - that the prisoner would be surprised - is false? **
Agree.
Let’s suppose that a logician friend says to you “I’m going to knock on your door next Saturday at noon, AND it will be a surprise to you.” If you believe that it’s logically impossible for your friend to carry out his plan, then you cannot be certain that your friend will in fact knock on the door on Saturday at noon. Thus, on the Saturday when the knock comes, it will (arguably) be a surprise!!
Well, they can’t both be true, so it’s either that or the other claim (that the prisoner would be executed).
Me? I don’t like the prisoner’s chances.
PS: I think it would increase my suffering to know the execution date in advance.
Hmm. This is getting fun. I’d like to ponder a couple questions a bit more before answering, but I did think of a way to execute the prisoner, that is, a way to make sure he’s surprised.
On the first day, drug him unconscious. When he wakes up tell him he’s been asleep for days, but don’t tell him how many. Since the prisoner doesn’t know what day it is, any day they come for him will now be a surprise.
(I’m not saying it was a paradox in the first place, though. I see this as a way for the judge to save face for having said something illogical. Or at least something he and the prisoner felt was illogical.)
“This statement is false”
Sentences are nearly always used to make statements outside of the context of the sentence itself. Letting a sentence refer to itself may or may not be valid. In this case the statement is invalid.
Note that the intent of the speaker “This statement is false” is not to convey any information. Therefore no meaning can be taken from this sentence. Valid sentences must make sense. It’s as much as if the speaker said “Truth is false” or “Truth is toy boat tulips”. No meaningful content.
Language is meant to describe static, non-self referential situations. Thus, it’s invalid for me to point to an atom and say “There’s an electron at the top of the shell, now.” because for part of the sentence – as I speak it – the statement is true, and for part of the sentence, it isn’t. There’s no provision in language to constantly reevaluate the validity of something. It’s like the joke: “What time is it?” “Seven-thirty one, and ten seconds… eleven seconds… twelve seconds.” The question clearly stretches language beyond conventional usage. One can’t require mankind’s invention “language” to avoid creating unanswerable questions. Or to avoid creating invalid statements.
The flip-flop nature of sentences dealing with real time situations seems to have something to do with the “This statement is false”, as well. That’s perhaps why there’s a nagging suspicion as one reads the sentence that there is some meaning there. Evaluating words, particularly complex combinations of words is a several step process. First, one identifies the individual words, then one assembles them into a grammatical sentence. So far so good. “This statement is false” appears to have valid words in all the proper slots for nouns and verbs. The next step is to evaluate the truth content of the words. Having got this far, there’s an impulse to think that meaning must be now be forthcoming. But now one is dealing with the sentence independent of how it was parsed. It is not valid to go back and constantly reparse the statement.
Take two sentences:
A) “This statement is false.”
B) “Sentence A is invalid.”
It can be seen at once that sentence B is coherent. The statement may be wrong or right, but it’s a completely valid sentence. The “target” of sentence B is sentence A, and having understood the meaning of sentence B, we are now free to examine the target and decide whether sentence B is true or false. Note that B is a valid sentence precisely because I can point to it and say “Sentence B is true”, and that another person will understand what I mean, and that they can meaningfully agree or disagree.
However, asking whether “This statement is false” alone is true (or even coherent) is effectively treating the sentence in the role of both A and B. Again, once the “B sentence” is parsed correctly, it is not reinterpreted, but making A and B the “same sentence” force it to be reinterpreted. Constantly reinterpreted, one might say. This seems to be an invalid process for our current language.
The prisoner’s analysis ignores that for people time goes in only one direction.
partly_warmer, if everyone here is like me, then we’re more than convinced that inconsistent statements exist, but we’re waiting for your argument of why the judge’s statement is one.
My game theory classes dealt with this one, too, and as far as I know no one’s satisfactorily dealt with it yet. There’s a really good overview of the problem in Martin Gardner’s book The Unexpected Hanging: And other Mathematical Diversions. Unfortunately, I can’t find the chapter itself online, but a good library should have a copy somewhere.
Either “This sentence is false” is valid, or “This sentence is true” is invalid. As far as ordinary language goes, yes, it is outside the boundaries and therefore odd, but most of philosophy seems to be geared towards showing that language fails outside its scope. Big deal.
The sentence makes sense to an English speaker, and the assignment of truth values to sentences other than this one leads me to conclude that we can validly ask whether “This sentence is false” is true or false. The answer, IMHO, is not to restrict ourselves to friendly sentences, but to come up with a more general theory of truth that can handle sentences like that one.
Self-reference is very definitely allowable in ordinary language, btw. This post is evidence as to why that is so. 