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There a few traps in the question, leading to different refutation or explanation. The first is the ambiguous definition of surprise. This lead to easy explana

There a few traps in the question, leading to different refutation or explanation. The first is the ambiguous definition of surprise. This lead to easy explanation of the paradox. The teacher might as well hand out an exam on a red piece or paper and claim that that is a surprise. The second is that the teacher invalidly presumes to be able to predict the mental state of his students. This can be stated simply: over the weekend, all his students die unexpectedly. The following week, none of his student are surprised since the set of surprised students is empty. We can work around those two flaw by ignoring the no-student case and assuming that "surprised" is replaced by a more exact formulation: "on the day the exam occurs, nobody would be able to say ``I knew the exam was today''". In this formulation, I contend that nothing can be concluded. The reason for this is that induction can only be applied on facts. The future is not a fact, so to apply induction on a sequence of future events is an error. The only conclusion that one can reach is the contrafactual: if we were Friday then the exam cannot occur on Friday. This is because the reasoning does not rely on any inductive argument but rely only on things that would be facts inside the contrafactual: if we were on Friday now, it would be false that the exam cannot be held on that day and that nobody could say that the exam is to be held that day. A contrafactual of this nature cannot be built for Thursday, since in that contrafactual ("if we were Thursday...") one would have to reason by induction about teh future. Of course, in the Thursday-contrafactual, one could state the Friday-contrafactual, but you can't build inudction on contrafactual (i.e. non-fact).

prof_hobart 15y ago
>The first is the ambiguous definition of surprise. He doesn't actually use the word "surprise" anywhere in his statement, and spells out pretty clearly what the nature of the surprise would be - "students won’t know exactly which day until the exams are handed out. Nothing, other than coming in on a particular day, not knowing for certain that the exam will be that day, but getting the exam on that day nevertheless, would do - e.g.an exam on the Friday would not qualify, as you would already know that this was the day of the exam. And nor would red paper. >The following week, none of his student are surprised since the set of surprised students is empty. Again, he doesn't claim that anyone would be surprised. He said that they "won’t know exactly which day until the exams are handed out." I suppose you could argue that if they are dead, then they won't even know on that day either, but that's a pretty tenuous point and anyway technically he doesn't explicitly claim that they would know at that point, just that they wouldn't beforehand. >"on the day the exam occurs, nobody would be able to say ``I knew the exam was today''". Which is pretty much what he did say. >A contrafactual of this nature cannot be built for Thursday Why not? The argument against it being on Friday (and I think we're both agreed that it's impossible for it to be on Friday and still line up with his statement) is that if there's only one possible choice for what day it's on, then the pupils - or at least any that are still alive - would know that the exam was on that day. And if it can't be on Friday, then when it gets to first thing Thursday how many choices are left for possible days for the exam to line up with his statement? One - so we're back to it being a certainty as to which day it would be.
- pierrebai 15y ago
  True, the way it was worded not involve surprises. Instead, the ambiguity is shifted to the meaning of what consitute knowing, which is pretty much equivalent and causes the same confusion. My aim was to try to put aside such semantic argumentation. My main point was that future events can't be held as facts. I used the death of all students as an example. I'm not sure why this would be a tenuous point: it's my central thesis. Re-reading the paradox as worded, I conclude that the teacher is right: the students won't know exactly which day because they cannot know the future as a fact. They might die, the teacher might die, the sun may explode. I don't consider these as tenuous arguments. They are concrete counter-example that any reasoning involving future events is flawed.
  - prof_hobart 15y ago
    It's tenuous because it's almost entirely tangental to the point of the paradox. You could caveat his statement with "assuming that we are all still alive, and in a position to take an exam on any given day next week..." or something like that, and it doesn't change the actual point of the paradox one bit. Trying to explain away this paradox by talking about people possibly being dead by next week is about as useful as taking "This statement is false" and explaining it away by saying that it could be in a language where "false" means true.
    - pierrebai 15y ago
      Then stop focusing on "being alive" bit. I was giving this solely as an example of why making statements about the future and basing arguments about some future state cannot work. Yet you take it as some trick to avoid having to explain the paradox. (Your analogy with pretending that the word false doesn't mean false is entirely off base. I don't see the correspondence. But no matter.) I think I've found a formulation that won't offend your sense: due to time's arrow and causality (which both go forward in time), one cannot make an induction that goes backward in time. This would cause a circular dependency in the inductive proof. In this case, the steps of the inductive proof go from Friday, down to Thursday, Wednesday, etc. Yet at the same time, causality goes from Wednesday to Thursday to Friday. One cannot simply dismiss causality. IOW, we have the inductive process N, N+1, N+2, where N is used to proove N+1, etc and where N = Friday, N+1 Thursday, all the while N state depends on N+1 because due to real-world causality, what happens on Friday depends on Thursday. You seem to wish to deny causality.

prof_hobart 15y ago

>The first is the ambiguous definition of surprise. He doesn't actually use the word "surprise" anywhere in his statement, and spells out pretty clearly what the nature of the surprise would be - "students won’t know exactly which day until the exams are handed out. Nothing, other than coming in on a particular day, not knowing for certain that the exam will be that day, but getting the exam on that day nevertheless, would do - e.g.an exam on the Friday would not qualify, as you would already know that this was the day of the exam. And nor would red paper. >The following week, none of his student are surprised since the set of surprised students is empty. Again, he doesn't claim that anyone would be surprised. He said that they "won’t know exactly which day until the exams are handed out." I suppose you could argue that if they are dead, then they won't even know on that day either, but that's a pretty tenuous point and anyway technically he doesn't explicitly claim that they would know at that point, just that they wouldn't beforehand. >"on the day the exam occurs, nobody would be able to say ``I knew the exam was today''". Which is pretty much what he did say. >A contrafactual of this nature cannot be built for Thursday Why not? The argument against it being on Friday (and I think we're both agreed that it's impossible for it to be on Friday and still line up with his statement) is that if there's only one possible choice for what day it's on, then the pupils - or at least any that are still alive - would know that the exam was on that day. And if it can't be on Friday, then when it gets to first thing Thursday how many choices are left for possible days for the exam to line up with his statement? One - so we're back to it being a certainty as to which day it would be.

pierrebai 15y ago
True, the way it was worded not involve surprises. Instead, the ambiguity is shifted to the meaning of what consitute knowing, which is pretty much equivalent and causes the same confusion. My aim was to try to put aside such semantic argumentation. My main point was that future events can't be held as facts. I used the death of all students as an example. I'm not sure why this would be a tenuous point: it's my central thesis. Re-reading the paradox as worded, I conclude that the teacher is right: the students won't know exactly which day because they cannot know the future as a fact. They might die, the teacher might die, the sun may explode. I don't consider these as tenuous arguments. They are concrete counter-example that any reasoning involving future events is flawed.
- prof_hobart 15y ago
  It's tenuous because it's almost entirely tangental to the point of the paradox. You could caveat his statement with "assuming that we are all still alive, and in a position to take an exam on any given day next week..." or something like that, and it doesn't change the actual point of the paradox one bit. Trying to explain away this paradox by talking about people possibly being dead by next week is about as useful as taking "This statement is false" and explaining it away by saying that it could be in a language where "false" means true.
  - pierrebai 15y ago
    Then stop focusing on "being alive" bit. I was giving this solely as an example of why making statements about the future and basing arguments about some future state cannot work. Yet you take it as some trick to avoid having to explain the paradox. (Your analogy with pretending that the word false doesn't mean false is entirely off base. I don't see the correspondence. But no matter.) I think I've found a formulation that won't offend your sense: due to time's arrow and causality (which both go forward in time), one cannot make an induction that goes backward in time. This would cause a circular dependency in the inductive proof. In this case, the steps of the inductive proof go from Friday, down to Thursday, Wednesday, etc. Yet at the same time, causality goes from Wednesday to Thursday to Friday. One cannot simply dismiss causality. IOW, we have the inductive process N, N+1, N+2, where N is used to proove N+1, etc and where N = Friday, N+1 Thursday, all the while N state depends on N+1 because due to real-world causality, what happens on Friday depends on Thursday. You seem to wish to deny causality.