Unfortunately I may not be able to participate in this thread much more. I just acquired some tight deadlines.
I’m sorry not to directly respond to some posts addressed to me above. I want to ask this. In the five day case, where do you think the flaw is in the prisoner’s argument that he can’t be hanged on Friday?
Here’s the prisoner’s argument. Is there a flaw in it? And is it a faithful representation of the first part of the classic Unexpected Hanging paradox?:
Premise A. I will be hanged on either Monday, Tuesday, Wednesday, Thursday or Friday.
Premise B. When I find out I am just about to be hanged, I will be surprised.
Premise C. I will not forget any of the premises of this argument at any time in the next week, and will be able to draw simple logical conclusions from them at all times.
Definition: “X will be surprised about Y at time Z” means there is no time interval prior to Y and bounded by Y during the entirety of which X believes Y will happen.
Suppose I am to be hanged on Friday.
Then on Friday morning, I will not have been hanged on Monday, Tuesday, Wednesday or Thursday.
From 2 and premise A and C, it follows that on Friday morning, I will know that I am to be hanged on Friday.
From 3 and the Definition, it follows that I will not be surprised by my hanging.
Line four contradicts premise B.
By supposing I am to be hanged on Friday, I reached a contradiction.
But it doesn’t change my subsequent analysis of the problem or his logic; so long as I’m analyzing the prisoner, I don’t care that real-world folks could get fooled by the 5-day problem but not the 1-day problem. So long as we’re talking about someone who concludes that the event won’t occur (instead of concluding that the surprise won’t occur), he can be fooled.
No, if it plays out the way I stated, then all that’s relevant is that he ‘joyfully returns to his cell confident that the hanging will not occur at all’ upon concluding that it can’t happen on (a) Friday, or (b) any day; so long as it’s a given that he reaches that incorrect conclusion, it doesn’t matter how implausibly foolish it’d be for someone in the real world to reason that way. We’re flatly told in the set-up that he’s reached the wrong conclusion regardless of why he reached the wrong conclusion.
If you tell someone you’ll surprise him by punching him ten seconds from now – and then do so – what matters is whether he mistakenly concludes there won’t be a punch rather than concluding that it won’t be a surprise; what matters is whether it “plays out” that way.
As per Hall, the students in the hypothetical really didn’t expect the exam. As per the OP, the prisoner in the hypothetical really did conclude that the hanging wouldn’t occur at all. Like the man said, don’t fight the hypothetical; call it the magic of fiat or something, but just take it as a given and reason from there.
Maybe we mean different things when we say the “problem” is or is not the same.
You seem to be saying that in the 1 day and the 5 day case, as long as the prisoner used the same error that Ned Hall describes, then the “problem” is the same regardless of any other details about the setup.
For me, a big part of the “problem” is “given the statement from the judge, what can correctly be concluded by the prisoner.” This doesn’t mean the prisoner’s reasoning is not also part of the problem, but it is not the only interesting thing going on.
In the 1 day case, a smarter prisoner can only do marginally better by correctly concluding that he has been given contradictory information and thus can’t determine which statement to believe.
In the 5 day case, a smarter prisoner can do better than marginally better, he can correctly conclude that it is entirely reasonable and within the judge’s power to make both statements come true and he had better begin working on his escape plan.
They turn out to be true only if the prisoner follows a very specific line of reasoning. The judge can’t really control this and if it happens the judge gets lucky, if not the judge is wrong, in the 1 day scenario only.
In the 5 day scenario the judge does not need to rely on the prisoner to follow any specific line of reasoning to be successful.
In some other hypothetical, maybe the judge wouldn’t get lucky. But he did, and that’s why the 5-day and 1-day scenarios are equivalent: they both involve a situation where the prisoner reaches his remarkable conclusion, regardless of what that prisoner could correctly conclude.
We can of course move on to all sorts of other hypotheticals. But as per the situation in the OP, what trips up the prisoner contemplating a span of days is the same factor that would trip him up in a one-day scenario: foolishly concluding that the hanging won’t happen (a) on that day, or (b) at all – such that he can then be surprised like the judge said when he gets the hangman’s visit just like the judge said.
Sure he does. The prisoner could easily follow invalid reasoning in the 5 day scenario and end up unsurprised, just as in the 1 day case. E.g., the prisoner might, by delusional hallucination, come to believe that God has told them that they will have a quiz on such-and-such a day, and thus end up unsurprised. It’s not exactly the same as in the 1-day case, fine, but the judge’s assertions coming out true does still depend on the prisoner being a certain way in the 5-day case.
ETA: Actually, the reply The Other Waldo Pepper gives is much better. Oh well…
If we use a definition of surprise to include any manner of poor thinking then there is no problem ever for any scenario, but that, I think is an uninteresting position.
If we assume that “surprise” means that the prisoner can not, using the data available and valid logic, determine in advance the day the hanging will occur on, then we have an interesting problem and again the 5 day is different from the 1 day.
That’s not my concern. From the OP: “This puzzle truly has me paradoxically confounded. I understand perfectly both sides of the puzzle, and simply can not resolve them, despite the alleged “explanations” contained within the Wikipedia article. I find it hard to believe that this paradox can not be resolved” – it can be resolved, because there’s nothing paradoxical about it; the guy is surprised because he made a mistake, plain and simple. Regardless of whether you find it interesting that he employed demonstrably poor thinking, the answer is that he employed demonstrably poor thinking.
But it’s unclear that the prisoner can be justified in believing the judge in the 1 day case when the judge announces that there will be a quiz, just as it’s unclear that the prisoner can be justified in believing the judge when the judge announces that the quiz will be a surprise (or, at least, the prisoner does not seem capable of being justified in believing both, and thus, whatever justification there can be in any particular instance, it would have to be something beyond a mere “Believe what the judge says” principle). And thus, it’s not necessarily valid logic for the prisoner to conclude that there will be a quiz. Or perhaps it is, on some account. But it’s not cut-and-dry.
Incidentally, though surely this obvious point has been explicitly noted somewhere upthread and has been at least implicit in some posts, it’s worth re-pointing out that this paradox turns on a proposition which asserts a certain property of itself; such self-reference is of course is at the center of many paradoxes (“This sentence is not true”, “RaftPeople will never believe this sentence, even though it’s true”, etc.). Specifically, when the judge says “… and you will be surprised by the quiz”, what he means, as you begin to point out, is “… and you will not be able to logically deduce the date of the quiz”. Except, of course, any nontrivial deductive proof starts from some premises. And so what the judge is saying, spelt out fully, is “… and you will not be able to logically deduce the date of the quiz, using (only) the premises that of each previous day you have not been given the quiz, and this very setup I am telling you right now, and your own awareness of it.”. So, it may just be the aspects of the elephant which interest me, but connecting this problem and its solution it to those other classic paradoxes of self-reference and one’s thoughts on their resolutions seems a good path to pursue.
Which really is determined by our rules regarding the term “surprise” and whether we assume the prisoner has all logical capabilities that are possible, and whether we consider the judge’s statement to be conditionally believed initially, etc. etc.
You may not consider part of the problem to determine what can you conclude given the information the prisoner had, but I think that’s a pretty interesting problem. Heck, Ned Hall though it was 59 pages interesting.
It seems like we need to translate all of this stuff into a formal language to be able to analyze it, but it also seems like it would get murky because the real world isn’t black and white.
The Other Waldo Pepper, as I drove home I realized the following which I had forgotten along the way:
My primary reason for stating that the 1 day is different than the 5 day is because some posters were saying something to the effect “The judge made a contradictory statement and it’s more obvious if you just use 1 day.” Which really is a statement about a problem prior to the prisoner’s reasoning. So, I don’t know which camp you are in, do you only think the 1 day is the same if you include the same reasoning by the prisoner? Or do you also think the 1 day is the same as the 5 day from the outset due to the judge’s inherently conflicting statements?
Here’s a recap on why I think the 1 day is not the same as the 5 day:
***** The 1 Day Scenario *****
The judge makes 2 statements about 2 future events
The prisoner can correctly conclude that the statements can’t be accepted as fact AND have both events come to pass
Why?: because there is only 1 day and as long as the hanging is accepted as fact (which it must be for both statements to be accepted as fact), then the surprise has been eliminated.
***** The 5 Day Scenario *****
The judge makes 2 statements about 2 future events
The prisoner can accept as fact both statements AND have both events come true.
Why?: That’s the interesting and controversial part. We know from real world examples that it works ok.
So, Indistinguishable, Frylock (and maybe The Other Waldo Pepper, depending on your stance on this), or anyone else: Can you show that the 5 day scenario is the same as the 1 day scenario, can you show that point #2 in the 5 day is not accurate?
Well, what is your interpretation of what the judge means by “surprise”?
In both the 1 day case, and the 5 day case, the prisoner, after accepting as fact both statements, can use reasoning to rule out any possible day for the quiz, as well as use (trivial) reasoning to establish that there will be a quiz, thus reasoning themselves into a contradiction, and ergo, ex falso quodlibet, they will be able to reason themselves into any conclusion one puts forth; they will be able to establish all of “There will be a quiz on Monday”, “There won’t be a quiz on Monday”, “There will be a quiz at some point”, “There won’t be any quiz”, “2 + 2 = 7”, and so on. The prisoner who accepts as fact both the declarations of the judge is therefore, in both cases, surprised by any particular date for the quiz, in the sense that they were able to reason themselves into ruling out that particular date, and, in both cases, is unsurprised by any particular date for the quiz, in the sense that they were able to reason themselves into concluding that that must be the particular date of the quiz (and, of course, will be similarly surprised/unsurprised by any other proposition one asks about, for they will, in their explosive contradiction, be able to reason themselves into both it and its negation).
The only way for the prisoner to avoid putting themselves into a situation where valid reasoning from the things they’ve accepted as fact leads to a contradiction, is by not accepting everything the judge says as fact. (This even though it may well, and indeed, on some account of “surprise”, definitely will, be the case that everything judge says really is fact. Similarly to the case with “RaftPeople will never believe this sentence”.)
Okay, The Other Waldo Pepper, now I understand your point. And you’re right that, if we assume the prisoner concludes the hanging can’t take place, the two conditions can be satisfied in both scenarios. But, I submit this doesn’t make them equivalent, because the reason the prisoner’s reasoning is faulty differs. In the one-day scenario, he should realize that he can’t know what to believe. In the multiple-day scenario, he should realize that backwards induction doesn’t work (if there are more than, say, four days available) and that both conditions can be satisfied quite easily. To me, why his reasoning is faulty is the interesting question. That it’s faulty we already knew.
Another way to look at this is to imagine a dialogue between us and the prisoner. He announces his conclusion that he can’t be hanged. We tell him he’s mistaken. The hanging is going to occur. This was the tweak I added by using a lottery in my first post. Hall does it by introducing an “Iron Law” that there must be an exam during the week. Now, what should a reasonable prisoner conclude? In the one-day scenario, he should conclude he will be hanged and it won’t be a surprise. In the multiple-day scenario, he should conclude he will be hanged and on which day can be a surprise. IOW, as RaftPeople says, in the one-day scenario, the conditions conflict, in the multiple-day one they don’t.
I agree with the rest of RaftPeople’s analysis also, but you didn’t find that persuasive, so I’m trying another tack.
It’s difficult to be precise, but generally it means does not have information (either explicitly given or correctly deduced) that the event will happen on that particular day.
Valid reasoning? If so, what is the valid reasoning for the 5 day case?
I will tell you right now that my counter to the valid reasoning is that the real world results prove that there is something wrong with the reasoning. You can’t use the reverse chain of days, or maybe you think you can. If so, can you show it symbolically? I ask because it seems there can be subtle differences between the various ways you could represent ideas like “accept as fact”, etc.