So, going back to the hanging example, the prisoner would have been right to conclude that the Judge’s statement had to be wrong to a certain degree: in that if the hanging were to be on Friday it wouldn’t be a surprise. So far, the prisoner is correct.
Now, the prisoner erred in continuing on with that line of thought by assuming that the Judge HAD to be right about the surprise portion, and concluding that he therefore HAD to be wrong about the hanging portion. The prisoner had no reason to conclude that the Judge was right in one particular part of his statement and wrong in another particular part. The prisoner could just as easily (and erroneously) concluded that he would be hanged on a given day and it not be a surprise.
Given that there was a definite contradiction in a portion of what the judge had said, the prisoner shouldn’t have given any reliance to any of the parts of his statement???
Right? Or, better yet, don’t say anything! I have this figured out!
I guess I’ll take a shot at explaining it. It’s not a paradox in my mind.
You can eliminate Friday because it’s the last day and it wouldn’t be a surprise. That’s as far as logic can take you.
The prisoner wants to eliminate Friday and then rephrase the judge’s statement as “the knock will come between Monday and Thursday and it will be a surprise.” But the judge didn’t remove Friday from the equation–the prisoner did. As long as Friday is in the equation, Thursday is an option.
If that doesn’t explain it, I’ll try an analogy. I’m thinking of a whole number between 1 and 5, and it’s not the highest number. Can you guess what it is?
You know the number isn’t 5, so the number is between 1 and 4. Now (using the prisoner’s logic) it can’t be 4 because that is now the highest number. So it must be between 1 and 3, but it can’t be 3. In fact, it can’t be 1 or 2. A paradox!
Now, if you can’t see how ridiculous the logic is when I put it that way, I can’t help you.
Those don’t seem logically analogous at all to me. If the judge had said “You will be hanged next week, but not on Friday” (which is analogous to your situation), the prisoner could not follow the same flow of logic to conclude he won’t be hanged at all. The logical conclusion is predicated on the idea that the prisoner will be hanged and it will be a surprise. The surprise element is what leads the prisoner down the path of faulty logic. The problem, in my opinion, is the prisoner concludes he will not be hanged at all, which is not an option. The initial statement says: 1) You will be hanged. 2) It will be a surprise. The prisoner leads himself to the conclusion he will not be hanged, which invalidates the first part of the statement.
That, right there, is the error: once he concludes that the hanging will not occur at all, he can be surprised when it does happen. Possibly we’d have a problem if he’d concluded that the surprise will not occur – if, for example, he really did make it to a minute before noon on Friday and announced that the expected hanging was unsurprisingly on its way in sixty seconds. If it’d been like that, we could talk for hours about how the judge said stuff that wasn’t true.
But it didn’t go down like that; he concluded that the hanging wouldn’t occur at all, and so got surprised.
Yeah, jtgain, that’s pretty much my resolution of the paradox. To my mind, we have to ground one premise or the other. If we ground the premise that the hanging (or exam) will occur, the prisoner is stuck. But, he says, I’ve just shown I can’t be surprised on any day. Oh, really, then tell us on which day you will be executed? Or, if you prefer, tell us on which day you will know on that day you will be executed? The problem, as you and others have observed, is that the only day on which he can infer this is Friday. Otherwise, his reasoning proves too much and, thus, proves nothing.
Notice that this is a very different resolution from the epistemological one mentioned in the Wiki article linked in the OP and the Unexpected Egg chapter linked by DarrenS in Post #55. That’s an aesthetic preference. I prefer to approach the paradox as a logic problem.
The same paradox, irrelevant distractions removed:
Teacher: Precisely ten minutes from now, I am going to give you a surprise quiz.
Student (thinking to self): Hm… But if I know, as you just told me, that I will receive a quiz ten minutes from now, then it will be no surprise. Thus, I reject your claim that you will give me a surprise quiz ten minutes from now; that would be impossible for you to do. Lucky break for me! I’m gonna take a nap…
Ten minutes later…
Teacher: Surprise! It’s quiz time.
Student: D’oh!
Let me clarify that cryptic comment a little. Essentially, the teacher is saying “There will be a quiz in ten minutes, but Student doesn’t believe it”. Precisely a Moore’s paradoxical statement, capable of being true but not really capable of being believed by Student themself (cf. the canonical “It’s raining, but I don’t believe it”).
The unexpected hanging paradox is nothing more than this, dressed up with the guise of a 5 day delay-chain to distract one from realizing it. Which isn’t to say there’s nothing interesting to say about the unexpected hanging paradox; it’s just that (IMHO) whatever there is interesting to say about it is best addressed to discussions of Moore’s paradox-type investigations directly.
ETA: You know, I hadn’t actually read the Wikipedia article linked in the OP until just now, but I see that my observations have already been made there under the heading of “The epistemological school”. Oh well…
Shodan, in the bellboy and the dollar story, there is no paradox. It’s bad math. You are fooled into thinking that one thing has to do with another.
No. He will not incur a delay penalty because time will never run out. Before time can run out, half of the time must run out. And then again, half of the remaining time must run out, and so on.
You can’t eliminate the fact that there are 5 possibilities. Forget about whether the prisoner performs any reasoning about the situation, there are still multiple days that can be chosen and it would be difficult to predict, therefore anyone would be surprised by the actual day.
An accurate example is if the teacher said you will get a test in either 1, 2, 3, 4 or 5 minutes. Are you trying to tell me you can predict which minute the teacher will give the test?
I’m trying to tell you that it doesn’t matter if we go with “The teacher will give a test in 1, 2, 3, 4, or 5 minutes, or (in contradiction to their claim) not at all” or just “The teacher will give a test in 1 minute or (in contradiction to their claim) not at all”. The same (erroneous) thought process deployed by the student in the former case can be just as well deployed in the latter case, regardless of how many time slots there are [just as I illustrated above].
[If you must, just imagine it as what goes through the student’s head inbetween the penultimate and ultimate time slot when they haven’t already been given the quiz].
Yes, but if you remove the conclusion that the hanging/test will not occur and replace it with “Hmm, I will not be surprised when the hanging/test arrives” then the single vs multiple is substantially different. And I think that the revised conclusion of “will not be surprised” still exposes a very interesting situation with the multiple day/minute option.
The prisoners logic seems correct on the surface up until the final conclusion. If you replace that conclusion with “no surprise” then it still seems correct, yet the judge is able to choose a day that the prisoner is unable to predict and is thus surprised on that particular day. Friday can be logically eliminated, but it doesn’t not seem like Monday or Tuesday can be eliminated (despite the chain of reasoning). Not sure if I would be surprised on Wed or Thurs, probably is somewhat determined by the definition of “surprise”.
I’ll admit I haven’t read all the posts in the thread so my apologies if I’m repeating points already made in other responses to the OP.
I think the key breakdown in the logic is the point when the prisoner decides he can apply the same chain of logic to every day that he applied to Friday. In my opinion, this is a faulty (or at least unproven) premise.
The hanging is not possible on Friday because on Friday morning there is only one possible day left for the hanging to occur on. But that condition does not exist on all five days of the week so the same conclusion cannot be applied to all five days.
If the prisoner wakes up on Friday morning then he knows there is only one possible day remaining in that week for him to be hanged on. But when he woke up on Monday morning there were five possible days remaining in the week for the hanging and he had no way of eliminating four of the possibilities. The Friday logic doesn’t apply to Monday.
If you think Friday can be ruled out, you are not getting it. Because imagine Thursday has passed, and the prisoner satisfies himself that he can’t be hanged on Friday, due to whatever logic. Then, he will be surprised when he is actually hanged, no?
The point is, he can’t deduce anything from the judge’s statement, because, from his point of view, it is one of those contradictory self-referential statements like “this statement is false”. From other people’s points of view, the statement is not self-referential because it refers to the prisoner’s knowledge, not theirs. So for them, the statement can be true and not contradictory.
There is the general notion that there are statements which cannot be proven true within a given system but are true nonetheless. Gödel’s Incompleteness Theorem is a famous example.
The prisoner fails to take this into account. So his assumption that the judge’s statement cannot be proven true within the prisoner’s system and therefore false falls short. Once you understand that the judge isn’t limited to the same set of knowledge and manner of thinking of the prisoner, the prisoner’s argument falls apart. The judge is merely using a different (if twisted) system of logic.
I think Indistinguishable explains the problem exactly.
“I will give you a quiz in 10 minutes, and it will be a surprise quiz.”
The faulty reasoning is to conclude from this that there won’t be a quiz. But the two statements are contradictory. Therefore, one or both of the statements can’t be true. Therefore you can’t reason that because a quiz in 10 minutes won’t be a surprise quiz, that there won’t be a quiz.
You could argue that since you expect there to not be a quiz, therefore when there really is a quiz it will be a surprise, and therefore the statement isn’t contradictory. But why would you reason that there can’t be a quiz in 10 minutes (that the first statement is untrue but the second is true), rather than that there will be a quiz in ten minutes but it won’t be a surprise (that the first statement is true but the second is untrue)?