Here’s Smullyan on the problem. He discusses the surprise exam version. I couldn’t find my copy but easily found this on the web. (www.gregology.net/paradoxes/#unexpected). This is supposed to be from Forever Undecided: A Puzzle Guide to Godel.
As per my previous post, just figure the prisoner is a blithering idiot. Possibly one who falls down a lot and drools on himself quite a bit.
No, I’m addressing the prisoner’s reasoning directly: as a blithering idiot, he rules out a Friday hanging when he’s only justified in ruling out a surprise Friday hanging. One can’t go further in faulting his reasoning; if not for making that stupid mistake, he (a) wouldn’t have ruled out a Friday hanging and thus (b) couldn’t have mistakenly concluded that “the hanging will not occur at all.”
I don’t know why you figure we need to go further; you explicitly agree that he can “indeed” be surprised by a Friday hanging if he “somehow” manages to convince himself that Friday is ruled out – and while you’re curious about why his reasoning doesn’t look half-bad, that’s irrelevant to figuring out why it’s wrong.
Still, if you need more, I don’t see that Smullyan’s analysis differs from what’s already been said in this thread: he says we should note that the judge/professor actually said two things, adds that believing the exam will occur would make it impossible for a surprise exam to occur, and adds that mistakenly ruling out the exam instead means the exam can then both occur and be a surprise. If that helps you, then you’re welcome to it – but how does that differ from previous posts?
“Figure the prisoner is a blithering idiot” is a strategy that does not deal with the scenario on its own terms. You might as well be working on the following problem:
A judge tells a prisoner the prisoner is going to be hanged on Friday, and that it will be a surprise. But the prisoner has a strange mental condition that causes him to disbelieve everything said by people wearing wigs. The judge is wearing a wig. So the prisoner believes he won’t be hanged on Friday. Then, on Friday, he’s hanged, and it’s a surprise.
That’s basically the kind of thing you get when you “figure the prisoner is a blithering idiot” and as you can see, there’s simply no puzzle presented there at all.
In the Unexpected Hanging, there is a puzzle. The prisoner has a line of reasoning that is apparently valid. But it leads from apparently true premises to a false conclusion. Where did he go wrong? Was there a step in his reasoning that wasn’t valid after all? Was there a premise he relied on that was false after all? That’s the puzzle. Figuring the prisoner is a blithering idiot doesn’t really help with that puzzle, since it undermines the main premise of the scenario–that the prisoner uses reasoning that looks, plausibly, as though it is valid.
And, just to beat a dead horse, what I think is, it turns out the prisoner uses a false premise. He thinks that if he survives til Thursday, he’ll think he’s going to be hung on Friday. That’s a reasonable thing for him to think–but it turns out, it’s false. Were he to survive til Thursday, having gone through the reasoning he’s going through, he won’t believe he’s going to be hung on Friday. QED. I solved the paradox. Drinks for everyone!
Hm. I dunno. On the formalization I started exploring (of course, not the only possible one), it really is true that if the prisoner survives till Thursday, he will think he’s going to be hung on Friday (because he initially accepts the judge’s words that he will be hung within the week, and only ever adds premises to his beliefs (specifically, by learning ones of the form “I am not hung on Day X”), never removing any). Granted, he will also think other things at that point, perhaps even to the point of contradiction. But, nonetheless, he will still have that belief.
Anyway, quals out of the way, I should get that part 2 up soon…
Where did he go wrong? In ruling out a hanging rather than ruling out a surprise hanging. Was there a step in his reasoning that wasn’t valid? Yes, when he figured there wouldn’t be a hanging rather than figuring it wouldn’t be a surprise.
Now, that, right there, would settle the whole thing; call it Explanation #1. The problem with Explanation #1 is that some folks in this thread prefer to postulate that he’s not a blithering idiot:
. . . and explicitly emphasizing that, no, the prisoner really is a blithering idiot helps with that problem. If someone incorrectly suggests that it must be an interesting problem involving some prisoner who avoids poor thinking while making reasonable conclusions – well, see, that’s when you deploy Explanation #2: no, the prisoner is a blithering idiot, such that Explanation #1 can suffice.
Explanation #1 explains why the reasoning isn’t valid. Explanation #2 explains why the prisoner uses reasoning that isn’t valid. (Explanation #2 isn’t, strictly speaking, necessary – but poster after poster after poster seemed to think the puzzle must involve a prisoner who isn’t a blithering idiot, and Explanation #2 helps deal with that.)
Well, yeah, but it doesn’t need to be that specialized; he already uses a false premise days in advance of that. Forget the fact that he won’t believe he’ll be hung on Friday come Thursday; he already undermined his assumption on Monday, like I’d quoted back in post #95. "It has been pointed out that the students’ reasoning has this peculiar feature: in order to rule out any of the days, they must assume that the quiz will be given and that it will be a surprise. But their alleged conclusion is that it cannot be given or else will not be a surprise, undermining that very assumption."
I don’t know whether “even to the point of contradiction” actually means “nonetheless, he will still have that belief.” We’re flatly told, in the OP’s version (as with Ned Hall’s formulation, and in Smullyan’s phrasing) that “he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.”
He doesn’t just seem to add new beliefs; he seems to drop the old one.
It’s not necessary to read him as assuming that he’ll be hung. Rather, he can be read as supposing he’ll be hung, i.e., for a reductio. Then he believes he finds an argument that, given that supposition, a contradiction can be derived. So, as seems reasonable, he drops those supposition.
In other words, there’s nothing particularly irrational about arriving at a conclusion that undermines its own “assumptions,” if one is doing what’s called an indirect proof or a reductio argument. And it seems that this is the kind of reasoning the prisoner is trying to engage in.
Use line numbers man! Line numbers!
Here:
- Joe will be hung on M or T
- When Joe is hung it will be a surprise.
- Suppose Joe will be hung on T.
- Then early T morning, before he is hung, he will know he’s to be hung on T.
- So he won’t be surprised.
- 5 contradicts 2.
- So we reject our supposition (line 3). Joe won’t be hung on T.
Is there anything wrong yet? If so, what’s the line number where things go wrong? How does it go wrong exactly?
I’ve already explained why this is not an apt response.
Listen, in fact, I agree with you. The prisoner is an idiot. But that’s immaterial. He seems to be an idiot with a valid argument with true premises. Let’s attack the argument and not the man, as they say!
If your formalization has him, on Thursday, thinking he’s going to be hung on Friday, then it may not correctly capture the problem. I say this because as the problem is stated, the prisoner makes it all the way to Wednesday (IIRC) thinking he’s not going to be hung. He thinks this because he accepts the conclusion of his Sunday-Evening reasoning. And I can’t see why he’d make it to Wednesday accepting that reasoning, but suddenly change his mind come Thursday.
Well, on my formalization, if the prisoner is capable of carrying out Sunday evening reasoning leading to the the (incorrect) conclusion that he is not going to be hung at all (from the judge’s premises), then this would furthermore lead the prisoner into contradiction, from those same premises of the judge’s (since he would also be capable of concluding, directly from the first part of the judge’s assertion, that he is going to be hung at some point). Which seems a fine formalization of the paradox to me; if the prisoner is capable of carrying out reasoning that reaches a contradiction, even starting from premises which turn out to be true (as, presumably, they would end up, even allowing for such contradictory reasoning from the prisoner, since my formalization of the judge’s assertion “You will be surprised” is as “If you only prove true things, then you will not prove that you will be hung on…”), then this would certainly be cause for concern. Thus, the question I am investigating becomes, is the prisoner capable of carrying out Sunday evening reasoning leading to the conclusion that he will not be hung at all? And my plan for approaching this is to first boil it down into some concrete system of proof rules the prisoner is assumed to be able to use and then mindlessly plow through them purely formally to see where he can get from those.
If there is some specific aspect in which my formalization (spelt out before) fails to properly capture or engage with the meat of the paradox’s setup, I don’t see it, but of course, I’d be open to having it pointed out to me.
Which is to say, it’s not that he would change his mind on Thursday. He never changes his mind, in my model. He just adds new premises to his beliefs. The worst that can happen is that he reaches an inconsistent corpus of beliefs; and if he ever concludes that he won’t be hung/given the quiz, then he’ll certainly end up finding himself unable to consistently reconcile this with new premises about which days did and didn’t have quizzes/hangings. But he never abandons previous beliefs (after all, being the idealized rational prisoner/student, he’s only ever supposed to form “beliefs” when he already has watertight proofs of them).
I’m confused over what the student’s reasoning is supposed to be. It’s hard to keep track of the changes. Could we pause and have an agreed-upon formulation? Before faulting his reasoning, we should at least agree on what the reasoning is supposed to be, else we may end up talking past each other.
So never mind the flaw in the reasoning for the moment. What’s the reasoning itself supposed to be? Frylock, I know you have formulated it quite a few times now, but could you suggest a definitive version for our consideration?
I suspect that we’ll disagree on what his reasoning is even supposed to be. Shouldn’t we sort this out first?
Line 4 presupposes that Joe isn’t a blithering idiot, just like Line 5. The apparent problem dissolves if we postulate someone foolish enough to conclude he won’t be hung on T rather than concluding that he won’t be surprised – and, as per Line 7, he is exactly that foolish, which demonstrably makes Line 5 and Line 4 false: upon concluding that he won’t be hung on T, he won’t know he’s to be hung on T, and so can be surprised.
Line five just follows trivially from 4, given a plausible definition of “surprise.”
But I think you’re right that the problem is in line 4. There’s nothing in the argument to justify the conclusion that Joe will know he’s going to be hung on T. There’s a hidden premise snuck in there. You’re saying the premise is “Joe is not an idiot.” You’re at least this right: “Joe is not an idiot” would justify line 4.
It looks to me like what you’re saying is pretty similar to what I’m saying in a way. You think there’s a hidden premise that Joe isn’t stupid. I think there’s a hidden premise that (or premises justifying a notion that) if Joe makes it through Monday, then he’ll know he’s to be hung on Tuesday. In a way, my hidden premise is a way of spelling out what it means for Joe not to be “an idiot” in this scenario. A reasonably intelligent person would generally reason that if you have two possibilities, and one’s been eliminated, then the other one is certain to be true.
Where we differ more widely is in our respective bases for calling the hidden premises. You’re saying he is an idiot because he doesn’t see that there is some alternative reasoning that would have led him to conclude he’d be hung but not surprised. One thing wrong with what you’re saying here is the fact that what actually happens is that he is both hung and surprised. His reasoning that he wouldn’t be hung turns out to be faulty. But had he reasoned (as you recommend) that he wouldn’t be surprised, that too would have been faulty. So it’s not clear to me what the basis for your recommendation is. Why should he have reasoned as you recommend, and why is he an idiot for doing otherwise, given that both his reasoning and your recommended reasoning lead to false conclusions?
Meanwhile, I show my hidden premise is false by a direct observation about the scenario. The premise says that once he gets through Monday, he’ll know he’s to be hung on Tuesday. But in fact, if you look to see what actually happens, you can see that the premise is false. He gets through Monday, and continues to believe (as a result of his reasoning) that he won’t be hung at all. I’ve spotted a hidden premise which can be straightforwardly shown false simply by direct appeal to facts given in the scenario. That seems adequate to me. Why not to you?
Believing that he won’t be hung at all doesn’t preclude also believing that he will be hung on Tuesday.
You’re right to point that out. What I should have said is that in the scenario as its generally presented, the prisoner doesn’t only believe he won’t be hung, but also lacks the belief that he will be hung. It’s the latter that makes false any premise (or conclusion drawn from hidden premises) that the prisoner will believe himself to be hung on Friday if he makes it til Thursday (or til Tuesday in the two-day version).
But in the scenario as generally presented, there’s no reason for the prisoner to fail to draw the belief that he will be hung (on the model of the rational prisoner I have adopted; one who draws all logical consequences of the judge’s assertions as his beliefs); if he’s willing to draw beliefs from the premises of the judge’s assertions at all, then he very definitely should draw that belief that he will be hung, as that was, itself, directly one of the judge’s assertions.
Put another way, the scenario as generally presented shows us one line of reasoning the prisoner could follow, but doesn’t present us with anything stopping him from also following another line of reasoning in addition, even to the point of contradiction.
I guess I disagree. I think the scenario presents us with something stopping him from also following another line of reasoning. Namely, that something is the scenario itself. The scenario presents itself as telling us the entirety of the prisoner’s reasoning. That it’s the entirety of the prisoner’s reasoning means there’s not some other line of reasoning that we can hypothesize that he followed.
In other words, I don’t think the scenario shows just a line the prisoner could follow, rather, it shows the line he did follow.
Of course, we have to assume some things about his mental processes that aren’t given in the scenario, simply to be able to understand him as a human being for example. But if the scenario is supposed to allow for the prisoner to have engaged in some separate line of reasoning than the one given, I’d expect that to be made explicit since it’d be pretty significantly relevant to the point of the scenario itself.
I’m still interested to see the next post in your promised series, by the way. 
Ah, ok. I guess our difference is that I’m reading the judge’s “You will be surprised” as “There will not be any reasoning you could follow to…”, rather than merely “You will happen to not follow reasoning to…”? Thus, for me, distinctions between reasoning the prisoner could follow but ended up not following and reasoning the prisoner could and did follow are not particularly relevant, though for other formalizations of the judge’s “You will be surprised”, they may be.
I, as well, am still interested in making the next post, but I keep putting it off (first exams and then Halloween, let me make my excuse). Tonight!
I think this shows the importance of the question of whether or not the prisoner can do anything to spare himself (especially by changing the nature of his surprise).
The paradox seems to stem from something like a post hoc fallacy. Post Hoc misassigns causation (event A precedes event B ergo event A caused event B) while in this case there is a misattrubution of correlation.
The statements of the Judge are only syntactically related and not logically or necessarily related.
He could just as easily have said: You will be executed and fairies wear boots and you gotta believe me.
The idea of being surprised is unrelated to the execution except by the grammar and syntax of the statement. If the prisoner is not surprised-- for whatever reason and by whatever logic or lack of logic-- he will still be executed. (Likewise, unfortunately, for the students and their quiz.)
(One way out would be to commit suicide and leave a note saying: shows what you know!)