Why do you think that makes it impossible in the 5 day scenario?
Did you take my spoiler test? How do you reconcile real world surprsie with the “impossibility” of it?
Do you say this because you think the prisoners logic is valid? If so, I think that is assuming something that hasn’t been proven.
It’s impossible for me to accept the judge’s claim that I’ll be hanged because he said it would be a surprise, but as per the reasoning given in the story, since the judge told me about it, there’s no day on which it could occur on which it could be a surprise–if I accept the judge’s word that I’ll be hanged.
I didn’t say it’s impossible to be surprised. I said it’s impossible to accept the judge’s word that I’ll be hanged. The things the judge said made it impossible for me to both understand what he said and accept what he said. If I understand his claims, I can’t accept them. That doesn’t mean I won’t be hanged or surprised, and it doesn’t even mean I can’t believe on independent grounds that I’ll be hanged and surprised. It just means I can’t accept the judge’s claim that I’ll be hanged and surprised.
I haven’t seen a good explanation as to why the prisoner’s logic is invalid. But I don’t think I was relying on the prisoner’s logic. Here’s my own argument.
The claim I’m arguing for is this. When the judge says “You will be surprised,” he makes it impossible for a hearer to both understand and accept what the judge says when he says “you will be hanged.” For if the hearer understands both of the judge’s statements, then he understands that, according to the judge, he will be both hanged and surprised. Can he accept this? No. For suppose that he did. Then upon accepting the judge’s claim that he will be hanged, he becomes unsurprisable by the hanging. But this is contrary to the other claim we’ve supposed he accepted–the judge’s claim that he’ll be surprised.
That argument assumes that a hearer will know that upon accepting that he will be hanged, he becomes unsurprisable by the hanging. It is not necessarily true that any old hearer would realize this. But it is stipulated in the story as it is usually told that the prisoner knows enough to know this about his own surprisability. And it’s a reasonable assumption for anyone to make–if I’m told about X ahead of time (barring a knock on the head or something–the kind of thing the story clearly means to be out of the picture) then I am not surprisable by X’s occurance.
I too don’t understand why the judge’s comments are contradictory, when everything that he said turned out to be true.
However, maybe I am getting closer to understanding it by imaging the following situation:
An eccentric millionaire walks up to you and your friend, and hands each of you an envelope, and says “One of these envelopes contains a million dollar cheque, but you won’t know if it’s your envelope until you open it”.
Now, I could respond to this and say “Both of those statements can not be true under all conditions. For example, what if I wait for my friend to open his envelope first? Then I’ll know – before I open it - whether or not the cheque is in mine”.
Now, in this instance, I can understand why the millionaire’s statement can’t be true under all conditions… BUT… in this scenario, the millionaire can’t control all variables, ie, who opens the envelope first. In the hanging paradox, the judge appears to be in full control of all variables… so… well… umm… am I getting closer?
How does that make him unsurprisable, he still doesn’t know which day it will be. That’s the surprise, that he can’t predict in advance which day it will occur on.
You seem to be thinking the “surprise” is purely based on whether he will be hanged at all that week or not. You can still know that you will be hanged and not know on which day (again as per my spoiler test).
I mentioned this once above but I’d like to bring it up again and see what people’s thoughts on it are. Now, my purpose in bringing this up is not to claim that this is “equivalent” to the OP’s paradox or any other so far mentioned, whatever that would mean; however, this may still be interesting and help us to cast light on analysis of those other paradoxes anyway, despite not being exactly the same as them. Fair enough?
The Professor loves long, drawn-out, arduous pop quizzes. He can’t get enough of them. But, for him, the fun is all ruined if there’s no surprise; just as much as he loves to spring quizzes on students who haven’t seen them coming, he hates giving quizzes to students who can see them coming.
So the rules of The Professor’s classroom are as follows: every morning, The Professor asks if any students can legitimately prove that they will be given a quiz that day. Any student who cannot is forced to take a six-hour long quiz. Any student who can (with their proof being declared valid by the the very expensive, proudly error-free ProofChecker 3000) is sent to recess instead.
No student has ever managed to make it to recess… Oh, they’re all dead-certain each morning that they’ll be given a quiz that day. But any attempt to demonstrate it to the standards of the ProofChecker 3000 always goes the same way: “Professor, the machine rejected my argument as fallacious.” “But of course it did! Why, did you think it would accept your alleged ‘proof’ that you’d be given a quiz today? But then you’d be sent to recess instead of the quiz, clearly demonstrating that ‘proof’ to be invalid. The machine never accepts invalid reasoning; it was very expensive, you know…”
And thus every student gets the quiz every day, and the playground remains perpetually deserted. The quiz is a sure thing.
Yet, trying to show the ProofChecker 3000 the above argument never works. The question is, why not?
Alas, my own post could use some proofchecking… Obviously, there should only have been one “the” in the bolded bit.
Because if a student could prove that he would be given a quiz that day, then he’d get sent to recess instead of being given that quiz. Therefore no such proof can exist.
Right, sure. Absolutely. What I meant to ask with “Why not?” was this: presumably, the ProofChecker 3000 only rejects fallacious reasoning. Yet we offered an argument above purporting to show that a quiz is a sure thing, an argument which, as noted, is bound to be rejected by the ProofChecker 3000. Is there something fallacious about that argument? What would or could the ProofChecker 3000 point to as its reason for not accepting it?
What argument, exactly?
You wrote that "any attempt to demonstrate it to the standards of the ProofChecker 3000 always goes the same way: “Professor, the machine rejected my argument as fallacious.” “But of course it did! … thus every student gets the quiz every day, and the playground remains perpetually deserted. The quiz is a sure thing. Yet, trying to show the ProofChecker 3000 the above argument never works. The question is, why not?”
You mentioned “the above argument” – but I didn’t actually see it. What reasoning was (a) presented, and (b) rejected as fallacious?
Show me the argument and I’ll show you what the PC3K would or could point to.
As it happens, you yourself relaid the foundation for the argument as well. You pointed out that no student can ever convince the ProofChecker 3000 that he’ll be given a quiz. Therefore, no student will ever make it to recess; i.e., every student will always be given a quiz.
I have apparently just proven, in the above paragraph, that every student is given a quiz every day; in particular, taking myself to be one of the students, I could conclude from this that I will be given a quiz today. And yet the ProofChecker 3000 won’t accept this reasoning. What’s wrong with it?
Your proof can’t be accurate on the same day as recess by definition. No real reasoning about your specific proof is required, the setup of the problem creates a situation that can’t happen.
You can’t be guaranteed of event X at the same time you are guaranteed of not event X.
If it accepts the “all students are always given quizzes” apparent proof, then you won’t be given a quiz – which would mean that not all students are always given quizzes. Such a result would involve claiming both that you will and won’t be given a quiz, and that’s what the PC3K could point to when rejecting the apparent proof.
In short, this is the setup of your problem:
not X if and only if X
Re: The Other Waldo Pepper:
When demonstrating the problem with the offered proof, it would point to something outside the proof itself, rather than a mistaken step within the proof?
Anyway, is your position that I haven’t correctly demonstrated that all students are always given quizzes, then? Or have I? Can we be sure that all students are always given quizzes, and if we can be sure, do our grounds for certainty not constitute a legitimate proof?
Could a non-student, temporarily interrupting class and whose use of the machine has no direct effect on the distribution of the quiz, present the apparent proof that “All students get quizzes every day” and have the ProofChecker 3000 accept it? If not, what would the machine point to as the problem in that case? If so, how could the machine justify not accepting the same chain-of-logic from a student? Does the validity of a chain of reasoning depend on who is presenting it?
Just to clarify for RaftPeople: I know that the ProofChecker 3000 can’t accept an alleged proof of quiz-time without thereby invalidating it by causing recess-time instead. That is, after all, what I designed the situation to involve. But your statement “No real reasoning about your specific proof is required” is the interesting thing. Surely, the idea goes, if a chain of reasoning is not acceptable as a legitimate proof, then there is some particular unsupported step in it which can be pointed out as not logically justified. You deny this?
I do deny that you even need to examine the proof in any way shape or form.
As long as we are told “for not X to be true X must be true”, the proof and the checker and recess are not very relevant.
I believe so.
If it were a student, then it would have a “direct effect on the distribution of the quiz”. That’s what’s doing all the work; it’s like how you might be correct if you said I can’t speak, but I can’t be right if I’m the one saying it. (See also the classic quip about hearsay and the truth of the matter asserted: if you’re a witness who claims the guy cried out “I’m alive,” then it doesn’t matter that he was claiming to be alive; a hearsay objection will be overruled, because your testimony would be evidence of that even if he’d said “I’m dead.”)
I agree with RaftPeople. The point of the paradox isn’t that the prisoner will be surprised by being hanged. The point is that he won’t know on whch day. (BTW, RaftPeople, I liked your little thought experiment on Page 3.)
As for whether the one-day version of the paradox is equivalent to the multiple-day verson, I notice that no one has stepped up to the plate to defend that they’re the same. Ned Hall discusses this issue in the essay I linked in Post #99 (and that’s his name, btw, not Block). His argument, to me persuasive, is to consider a 100 day “week” (see p.16). Obviously, one can surprise the students/prisoner with that much time. So multi-day is different from one-day. The two-day scenario is tough, and Hall addresses this too, but I think the right answer is that there are some number of days (four may be the minimum) over which backwards induction doesn’t work.
I apologize for not making my position clearer in Posts #97 and #100: the one-day version is equivalent to the multiple-day version so long as (a) the student/prisoner concludes that the event rather than the surprise won’t happen, sure as (b) we’re explicitly told that “he retires to his cell confident that the hanging will not occur at all.”
Hall’s argument also involves the following: “To reckon this, the student needs to consider how probable he will take an exam to be, should Friday morning arrive with no exam having yet taken place. Plausibly, the student should, in such a situation, consider it much more likely that the professor is being mischieviously deceptive…” – except, as per the OP, he doesn’t. Likewise, we’re told that “he must rely on his knowledge of the professor’s character, and judge the likelihood that she is lying, or being mischievous. Perhaps these hypotheses should, in light of his background knowledge, receive such a low probability that he counts as justifiably believing that the exam will take place before Friday. If so, he can conclude that come Friday, he will still justifiably believe that an exam is scheduled” – except, again, the OP makes clear that this isn’t someone who still believes the event is scheduled.
Hall frames it correctly at the very end: “He first pretends that the professor has merely announced an exam, and from this premise correctly concludes that if it takes place on Friday, it won’t be a surprise. He then lifts the pretense, and appeals to the (incompatible) premise that she has announced a surprise exam; maintaining the illicitly drawn conclusion that if the exam takes place on Friday, it won’t be a surprise, he concludes that if she has spoken truly then the exam cannot take place on Friday.” For a student who reaches that conclusion, even Friday alone suffices.
What I see a lot of in this thread is someone has a point of view, runs some logic from there. Another person has a point of view, runs their logic. Both end up at different places. Both are “right” from their point of view. If only someone could come up with an amusing little fable highlighting this.
PBear42: Ned Hall, noted. Thought exp: tks.
I don’t think you can say they are the same even in that condition. The 5 day problem retains it’s attribute that you can in the real world surprise someone, which changes your subsequent analysis of the problem and the prisoner’s logic. You can not conclude all of the exact same things about the prisoner’s logic between the 1 and 5 days even if it plays out the way you stated.
They are fundamentally different.