Another Surprise Examination

I’d like to see what you think about one (and a half) versions of the surprise examination paradox. I’m not actually sure this case is paradoxical; it’s a variant of one of the variants Williamson considers in Knowledge and Its Limits.

It goes like this:

Mr. Chips is teaching a class that meets Monday through Friday for the whole semester (which is, let’s say, ten weeks long). On the first day of class, he announces, “There will be a test one day during the semester; I’ve scheduled it already, but I won’t tell you when it is yet.”

On the second day of class, he announces, “About that test: It’s not on the last day; and if it is on a certain day, you can’t come to know now that it’s on the next day–unless yesterday you already knew that it wasn’t on the next day. And when I say ‘next day’, I’m counting weekends.”

Mr. Chips is extraordinarily reliable, such that it would be reasonable to take any ordinary assertion of his as conferring knowledge.

The students reason as follows:

Yesterday, we knew only that the exam would be on some weekday.

Today, we know that the exam won’t be on the last Friday, because Mr. Chips told us so.

The exam can’t be on the last Thursday, either. Because if it were on the last Thursday, we wouldn’t have come to know that the exam wasn’t on the next day. But we have come to know that the exam isn’t on the next day (the last Friday); and we didn’t know that yesterday.

Nor can the exam be on the last Wednesday. Because we have come to know (in the previous paragraph) that the exam isn’t on the last Thursday, we didn’t know that yesterday, and according to what Mr. Chips said we couldn’t have come to know that if the exam was on the last Wednesday.

Similary, the exam can’t be on the last Monday or Tuesday.

But the exam might be on the next-to-last Friday. Because we already knew yesterday that the exam wouldn’t be on the next day, since class doesn’t meet Saturday. So what Mr. Chips said doesn’t rule out that the exam isn’t on the last Friday, or any earlier day.

So we know that the exam doesn’t take place in the last week, but it might take place any other time.

My questions: Suppose the exam is in fact scheduled for the next-to-last Wednesday. Do the students come to know that the exam isn’t any day during the last week?

Suppose the exam is in fact scheduled for the last Wednesday. Do the students have the right to be annoyed at Mr. Chips? (They’d have the right to be annoyed at him if he told them a falsehood; but that might not be the only possible grounds for annoyance. I’m deliberately leaving this vague.)


Comments

Another Surprise Examination — 12 Comments

  1. Matt, to both questions, I say no. I want a solution to the paradox that explains the mistake in the reasoning, so that implies that they don’t know it’s not during the last week. If Chips gives the test the last Friday, it would be appropriate to be annoyed–actually, downright hostility would be OK too! But not if he gives it any other day of the week.

  2. I also want a solution to the paradox that explains the mistake in the reasoning–but I just don’t see a mistake in this case! That’s why when the exam in fact isn’t in the last week, I am inclined to say that they know that the exam won’t be during the last week.

    By annoyance I’m really thinking of something like this: Suppose I say to you, “There’s an egg in this box, but you can’t know it.” Now, if you assume that you can gain knowledge from what I say, then you can know that there is an egg in the box; but then what I said was false, so you can’t gain knowledge from what I say; by reductio, you can’t gain knowledge from what I say. So you don’t know that there’s an egg in the box (unless you can know that I’m more likely to lie about the second conjunct than the first).

    But–if there is an egg in the box, I was telling the truth. Still, I think you have the right to be annoyed at me–for ‘blindspotting’ you (that’s Roy Sorenson’s term, and I think that’s even how he uses it).

    If the exam is on the last Monday through Thursday, I think Mr. Chips has blindspotted the students, and they have the right to be annoyed for that reason. Though it might take a little work to explain why I think you have the right to be annoyed when someone blindspots you in this way….

  3. I see, Matt, about blindspotting here. And I agree as well that it is hard to see what mistake there is in there reasoning. But there is one. I swear it. Really, I’m serious…:-)

    And, if there is, then the students weren’t blindspotted. They’re just too logically coy for their own good! Chips should laugh at them for their logical legerdemain, and say, “You deserve what you got!”

  4. I think there’s a mistake, but we can’t know what it is. 😉

    Notice, though, that if the term were only two days long (not counting the days on which the announcement was made), what Mr. Chips says would just boil down to my egg statement.

  5. Yes, that’s right. And if Chips tells you it’s one of the two days and not the last of the two, then you can legitimately be pissed off at him when he gives the exam. That’s the difference between single-and multi-premise closure, right?

  6. Wait–do you mean if that’s all he tells you? If all he tells you is that it’s one of two days, and not the last of the two, then he’s OK giving the exam on the next-to-last day.

    I’m not sure the difference between single- and multi-premise closure is decisive here. (I’m also not entirely sure there is a fundamental difference–as an anonymous referee pointed out to me recently, even single-premise closure relies on the fact that P entails Q as well as on P.) In the two-day Mr. Chips case, the students need to infer from “We know the test isn’t on the last day” and “If the test is on the next-to-last day, we don’t know the test isn’t on the next-to-last day” to “The test isn’t on the next-to-last day”; then they need to reflect on their reasoning to realize that they know the test isn’t on the next-to-last day; then they put these together to realize that it’s incompatible with something else Mr. C said (that there would be a test) and, by reductio, infer that they can’t get knowledge (or perhaps knowledge of knowledge) from what Mr. C says after all. At least the first step (the inference to “the test isn’t on the next-to-last day”) seems to involve multi-premise closure.

  7. oops, enthymeme city here!

    So, what I meant was this. Add to my last description the further information that you gave about the case in your original example. Allow that the students know what Chips tells them, and also allow the conjunction formation needed to get in a position to apply single-premise closure to conclude that the test has to be on the first of the two days. *That’s* what I took to be analogous to the egg case, and then they’d be right to express ire.

    Well, the referee wasn’t me! And applying single premise closure doesn’t require two premises: it requires knowing the conjunction in question, and deducing q from it. I’m puzzled how the remark about single premise closure could be used to question the distinction between single- and multi-premise cases.

  8. The students might have been blindspotted despite the fact that the argument they offer is perfectly sound. So there is not need to show that the argument is bad (I doubt that it is). Here’s how. We have supposed that Chips is reliable and these are true. The argument goes this way, if I’m tracking you.

    1. You know the exam is not on the last day (viz., the last Friday of classes)
    2. You know that if the exam were on day D, then you would not know now that
    the exam is not on day D+1 (unless you knew yesterday that it was not on
    D+1).
    3. Suppose for reductio that the exam is on Thursday.
    4. You do not know now that the exam is not on Friday (3,2, plus kn. is factive)
    *Contradiction* (4,1)
    5. The exam is not on Thursday. RAA

    Nothing wrong with that reasoning. But now suppose it is a fact that the exam is on Wednesday. Suppose further that (1) and (2) are both true. It follows that you do not know now that (5) is true. So you have sound argument that (5) is true and you do not know now that (5) is true. What has happened is that, if there is an exam on one day during the last week, then the students cannot know that their argument is sound even if it is sound. It does not follow that the argument couldn’t be sound (as it certainly appears to be). For if it is sound and I don’t know it, then I don’t know now that (5) is true. In that case the exam could be given on Wednesday.

  9. Here’s the shorter story. If I don’t know that the argment for (5) is sound, then (since that is my only evidence for (5) as the situation is described), then I do not know that (5) is true. If I do not know that (5) is true then the argument that I know the exam is not on Wednesday is in fact unsound. So it is true that the exam *could* be on Wednesday. Indeed, I could know that the exam is on Wednesday.

  10. Matt,
    I’ve got a quick question on the “egg example”. You say that what Chips says in the two-day version of the surprise exam boils down to the egg statement. But the two-day version doesn’t seem to resolve in the way you suggest the egg example resolves. In the egg example you suggest that since Chips is reliable only if unreliable (or, you gain knowledge from him only if you don’t), you don’t know (by his assertion that there is an egg in the box) that there is an egg in the box. I’m less sure that is the conclusion we should reach, but let’s assume it is.
    In the two-day surprise exam we have these premises.
    1. Chips is reliably asserts (2) and (3).
    2. The exam is not on the last day (viz., the last Friday of classes).
    3. If the exam were on day D, then you would not know now that
    the exam is not on day D+1 (unless you knew yesterday that it was not on
    D+1).
    4. The exam is on Thursday. Assumption
    5. You do not now know the exam is on Friday (3,4)
    6. You now know the exam is on Friday. (1,2) Contradiction

    Now suppose the assumption in (4) is in fact true. Does it follow that either (1) is false? That would be the analogy with the egg example where Chips is reliable only if he isn’t. In that case the exam is on Thursday and I can’t know it. But I don’t think it follows that (1) is false. What seems to follow is that either (1) is false or I am unreliable in my derivation of (5) or (6). So Chips is unreliable or I am unreliable in my derivation. If the latter, then the exam might be on Thursday despite Chips’s reliability. And if I am not reliable in my derivation of (5) or (6), then I could know the exam is on Thursday.

  11. Mike, I think the two-day surprise works like this:

    Mr. Chips says: (1) The exam is not on Friday
    (2) If the exam is on Thursday, you can’t know it’s not on Friday.

    If it’s on Friday, Chips lied in (1). (Similarly, if there’s no egg in the box, I lied.) If it’s on Thursday, and the students can know it’s not on Friday, then Chips lied in (2). (If there’s an egg in the box and you can know it, I lied.) So if the students assume that Chips’ statements give them knowledge, one statement must be false, contradicting the assumption. So they can’t get knowledge of Chips’ statements. But Chips’ statements can be true then (if the exam is on Thursday; cf. the case where you decide not to believe me, but there is an egg in the box). The students are blindspotted.

    In your analysis I think it doesn’t follow that 1 is false–but it follows that the students can’t use 1 to gain knowledge of 2 or 3.

    I’m not completely sure about this–but I’m out the door to pack for the FEW! Very cowardly of me I know. Maybe I’ll e-mail you and Jon about this when I return (or put up another post on the paradox so you can point out that I never answered your original objections).

  12. Matt,
    Do you have a paper on this? If you do and don’t mind circulating it, I’d like to see the complete argument. In your version of the two-day surprise above you have these premises (I will add one premise, just to make this assumption explicit).
    1. Chips is very reliable and asserts (2) and (3).
    2. The exam is not on Friday.
    3. If the exam is on Thursday, then you cannot know that it is not on Friday.

    Now the question is whether the students could know that the exam is on Thursday, given (1)-(3). From (1) and (2) you have the students derive that,

    4. We know the exam is not on Friday.

    But that doesn’t follow. After all, there is no assumption in the argument that these students are themselves reliable. If the students are sufficiently bad at drawing inferences they might not reach that conclusion. If I am dyslogic (if you will) I might nearly always conclude from the very reliable assertion P that ~P. The fact that on this occassion I happen to conclude P from the very reliable assertion P, does not entail that I know P. But suppose they do reliably arrive at (4). Now suppose (5) is true.

    5. Chips asserts that the exam is on Thursday.

    The students now draw the conclusion in (6).

    6. We know the exam is on Thursday.

    Can the students now conclude that (1) is false? No, they can’t. From (3), (4) and (6) the students might derive a contradiction. But then it follows that,

    7. Either we unreliably reached a contradiction or premise (1) is false.

    And there’s no need to assume “dyslogia” for students to unreliably reach that contradiction. So the students have no basis for rejecting (1) that Chips is very reliable and they know that Chips said the exam is on Thursday. Looks to me like they can know that the exam is on Thursday.

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