Thursday, June 07, 2007

The Surprise Exam

Whereas Moore’s paradox hardly seems paradoxical at all (no more so than how, although I may not, at some time, be saying anything, I cannot say so truthfully at that time), I’ve found the Surprise Exam to be very puzzling (although experts say that the former is much deeper than the latter). We are to imagine that a good teacher wants her class to revise their lessons, so she decides to set them an exam for the following week, but she does not want them to cram their heads with facts the night before the exam, so she tells them that it will be a surprise exam—they won’t know which of those five days it will be on until she plonks the paper on their desks. One pupil works out that there is therefore no need for him to revise, such an exam being impossible, as follows. The exam cannot be on Friday because if it were then on Thursday evening the class would know that it was the following day. So the exam must be on or before Thursday. But therefore Thursday is similarly ruled out (since Friday has been ruled out the class could know on Wednesday evening that the exam was the following day). And so on, going backwards through the week like that until even Monday is ruled out.
......But of course he was wrong because the exam was on Thursday, and it surprised him most of all because he did not expect it on any day that week! And similarly, anyone else who on Wednesday, for example, worked out that it would definitely be on Thursday because Friday was (as the last possible day) ruled out, would have obtained an inappropriately justified true belief (i.e. would not really have known) that it would be on Thursday, because for such a reason Thursday (as the new last possible day) should also have been ruled out. Now, the earlier in the week the exam is, the less expectation the class could justifiably have, the day before the exam, of it being on the following day, but the reason why the induction failed was that the exam could even have been on Friday (without the teacher being wrong). Suppose it got to Thursday evening, with as yet no exam. Those who thought that therefore it would probably be the next day would probably have overlooked the possibility of the teacher being wrong about there being an exam at all, that week, rather than wrong about it being a surprise—they could only know that it was on Friday if they knew that their teacher was not wrong about it being that week, but they would have less reason to believe what she said about the exam the more they believed that it was likely to be the next day.
......That what the teacher said turns out to be correct even with the exam on Friday, because of it being apparently wrong in some respect, is certainly odd, but hardly contradictory. And so the induction fails, and so the exam could also (less oddly) be on Thursday, or even earlier (even less oddly). It is certainly odd though—e.g. suppose that one pupil trusted the teacher so much that he had complete faith in what the teacher said about the exam (not unjustifiably, because this teacher is indeed sufficiently trustworthy). Prima facie he might know that what she said was correct (as indeed it was, as usual), but then if we also suppose that she had set the exam for Friday then he could, the previous evening, have known that the exam would be on Friday whilst also knowing that he did not know which day it would be on!
......(Added June 8:) Having slept on it (and there's a mystery, that good thinking can get done while we're unconscious!), this seems less puzzling. Since the teacher expressed herself that way (saying that the pupils would not know when the exam would be until the paper was on their desks) she probably had in mind a day early in the week (as otherwise it would have made more sense to say just that she would not tell them which day it was going to be). In the unlikely event of her intentions being frustrated, so that the exam had to be on Friday, the pupils could hardly (given only the above information) be justifiably sure that there would be an exam at all that week. So, the lazy pupil's induction could not get started (the exam might be on Friday, for all that it is probably not intended to be), and the truth of the faithful pupil's belief (that the exam would be the next day) would have been too much a matter of chance to be justified (?)

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