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Surprise Exams are Conditionally Possible Alex Baia
1. The Paradox of the Surprise Exam A teacher announces that there will be a surprise exam next week. The class whiz, giving the following elimination argument, protests that such an exam is impossible:1 (Elimination Argument) The class meets once every Monday, Wednesday, and Friday. Assuming that the exam occurs on one of these days, it cannot occur on Friday. For if it doesn’t occur by the end of Wednesday, we can know in advance that it will occur on Friday. Therefore, the exam cannot occur on Friday and be a surprise. But with Friday ruled out, the exam cannot occur on Wednesday. For if it doesn’t occur by the end of Monday, we can know in advance that it will occur on Wednesday. Therefore, the exam cannot occur on Wednesday and be a surprise. But with Wednesday and Friday ruled out, we can know in advance that it will occur on Monday. Therefore, it cannot be a surprise. Therefore, no surprise exam can occur next week. What should we make of The Elimination Argument? Prima facie, it seems convincing. Yet, surely, it is possible to give a surprise exam. Such exams actually occur, and students are surprised by them—in the sense of failing to know in advance when they will occur—even when they are announced in advance. We have a paradox: The Elimination Argument seems sound, yet its conclusion seems false.2 1
The elimination argument is adapted from Sorensen (2009). As far as I can tell, the paradox first appeared in O’Connor (1948). Weiss (1952) seems to have first adapted the paradox into its now common form. 2
Sainsbury (1995) defines a paradox as “an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises.”
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One dismissive reaction to the paradox is that the teacher should have simply given the exam but not announced it in advance, thereby guaranteeing that it comes as a surprise. But, like many dismissive reactions to paradoxes, this one misses the point. The interesting question here is whether a surprise exam is possible given its announcement in advance. Intuitively, it is. Yet we have an attractive piece of reasoning —The Elimination Argument—that says that it is not. What counts as a surprise exam? Say that a surprise exam occurs next week just in case two conditions obtain: (Exam) An exam occurs next week. (Surprise) The students do not know, before the exam occurs, on which day it occurs.3 We should distinguish between these conditions in order to be be cognizant of three possibilities with respect to each day on which the class meets: (i) No exam occurs that day; (ii) An exam occurs that day, and it is not a surprise; (iii) An exam occurs that day, and it is a surprise.
2. The Conditional Possibility Solution The surprise exam seems paradoxical insofar as the conclusion of The Elimination Argument conflicts with a belief we have about surprise exams. Call this belief The
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We should use a knowledge criterion of surprise rather than a merely doxastic one. For imagine that a student, Billy, makes a wild guess that the exam will occur on Wednesday and believes his guess correct. Now say that, by chance, the exam occurs on Wednesday. Billy will not be surprised by the exam. That it would be wrong to treat Billy’s irrational and accidentally-correct belief as counting against the exam’s status as a surprise shows that we need a stronger criterion than mere belief.
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Exam Belief; it amounts to something like our pre-theoretical belief about the possibility of surprise exams. What exactly is the content of this belief? Here is a good candidate: (Possibility) A surprise exam—of the sort described in §1—is possible. Notice that the conclusion of The Elimination Argument is just the negation of Possibility. The conclusion says that no surprise exam can occur next week, that such an exam is impossible. Therefore, either The Elimination Argument is unsound (and its conclusion false) or else Possibility is false. Which is it? Surely the conclusion of The Elimination Argument is false and Possibility is true. Let the teacher pick the exam’s date at random. Say that ‘P(Mon: p1, Wed: p2, Fri: p3)’ represents the probabilities that the exam occurs on each day.4 Now imagine that the teacher uses a device that randomly picks an exam day according to the distribution P (Mon: 1/3, Wed: 1/3, Fri: 1/3). In this scenario, could a surprise exam occur? Obviously it could. Since the exam has an equal chance of occurring on any day, no day can be ruled out in advance. If the exam occurs on either Monday or Wednesday, it will be a surprise, since the students will not be able to know, come the start of those days, that it won’t occur on Friday instead. True, if the exam occurs on Friday, it cannot be a surprise. But the impossibility of its being a surprise on Friday is not in conflict with the possibility of its being a surprise on Monday or Wednesday. Suppose that the teacher abides the possibility of the exam’s occurring on Friday and hence the possibility of its not being a surprise. If so, the teacher creates a situation where a surprise exam is possible (since it might occur on Monday or Wednesday),
4 Assume,
of course, that the exam is given exactly once and that p1 + p2 + p3 = 1.
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although this possibility depends on there also being a possibility of the exam’s not being a surprise (since it might occur on Friday). In short, the possibility of the exam’s success is conditional on the possibility of its failure:5 (Conditional Possibility) The possibility of the exam’s occurring and being a surprise is conditional on the possibility of its occurring and not being a surprise.6 There is a potential problem for the randomization response and for Conditional Possibility. Suppose that the randomization device picks Friday. If so, the exam will not be a surprise. According to Sorensen, it follows that the teacher must disallow Friday in the randomized selection.7 But once we accept this, we invariably follow the familiar line of The Elimination Argument, concluding that the teacher must disallow Wednesday and Monday as well. Let us be cautious here. Why think that the teacher must disallow Friday in the selection process? Presumably, to avoid the threat of an unsurprising exam. By the teacher’s lights, the worry is that by allowing Friday, she’ll allow the possibility of an unsurprising exam. By the philosopher’s lights, the worry is that a solution to the paradox where the exam has some possibility of not being a surprise is no solution at all. By such lights, Possibility is arguably too weak a candidate for The Exam Belief. What’s paradoxical about The Elimination Argument, in other words, is allegedly not just that a
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It’s not obvious that the selection needs to be randomized in order to secure the possibility of a surprise exam. Rather, the selection procedure simply needs to not exclude the last day (in this case, Friday) from consideration. 6
The intended logical form of P’s possibility being conditional on Q’s possibility is this: (♢P) ⊃ (♢Q).
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Sorensen (2009).
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surprise exam seems possible, but rather, more strongly, that the teacher ought to be able to guarantee that one occurs: (Guarantee) The teacher, after announcing that a surprise exam will occur next week, can guarantee that such an exam will occur next week. If Guarantee is the content of The Exam Belief, this spells trouble for Conditional Possibility, for Guarantee suggests its falsity. If the teacher can flat-out guarantee that a surprise exam will occur, then the occurrence of a surprise exam surely cannot be conditional on its possible non-occurrence. In response to this objection, first, why identify The Exam Belief with Guarantee? An obvious reason for treating The Elimination Argument as paradoxical is that surprise exams have actually occurred; students have actually taken exams whose dates they could not have predicted. But the empirical observation that surprise exams have actually occurred is great evidence for Possibility and no evidence for Guarantee. The observation that some event, E, has happened in the past is great evidence that E is possible but no evidence that we are able to guarantee E’s occurrence. For that reason, Guarantee seems unmotivated; it’s not obvious why pre-theoretical belief about surprise exams should involve Guarantee. The more natural suggestion, I claim, is that we are pre-theoretically committed to Possibility, a commitment that is entirely consistent with Conditional Possibility. Second, we should not accept Guarantee, since it is self-defeating. How is it selfdefeating? The answer is that The Elimination Argument is invalid; as given, it is at best an enthymeme. However, if we assume the truth of Guarantee as an auxiliary premise,
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The Elimination argument becomes valid. Therefore, by assuming the truth of Guarantee, The Elimination Argument’s conclusion follows, namely, that a surprise exam is impossible. Since this conclusion contradicts Guarantee, Guarantee is self-defeating. To understand the nature of this self-defeat more clearly and to see why it occurs, we need to more closely examine The Elimination Argument.
3. Diagnosing The Elimination Argument The randomization response shows that The Elimination Argument is unsound. But where does the argument go wrong? Let us generalize the surprise exam. Say that there are N possible days to give the exam. The Elimination Argument makes a series of assertions, starting with assertions concerning day N and proceeding until it reaches assertions concerning day 1: (α) For all possible exam days: If the exam cannot occur on day N, then it cannot occur on day N-1 and be a surprise. (1) The exam cannot occur on day N and be a surprise. (2) The exam cannot occur on day N. (3) The exam cannot occur on day N-1 and be a surprise. (4) The exam cannot occur on day N-1. etc. The elimination reasoning begins with the (true) observation that the exam cannot occur on the last possible day and be a surprise. The inference is then made that since the exam cannot occur on this day and be a surprise, therefore the exam cannot occur on this day. From this (together with α), it follows that the exam cannot occur on the second to last
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day and be a surprise. Putting the argument this way makes its flaw more transparent: While (1) is true and while the move from (2) to (3) is valid, the move from (1) to (2) is clearly invalid. From the premise that the exam cannot occur on the last day and be a surprise, it does not follow that the exam cannot occur on the last day. The randomization response shows clearly that this is so: The teacher may allows the exam to possibly occur on the last day in order to secure its status as a surprise should it occur on any other day. We can render The Elimination Argument valid by introducing an auxiliary premise, β: (α) For all possible exam days: If the exam cannot occur on day N, then it cannot occur on day N-1 and be a surprise. (β) The exam can occur only when it is a surprise. (1) The exam cannot occur on day N and be a surprise. (2) The exam cannot occur on day N. (3) The exam cannot occur on day N-1 and be a surprise. (4) The exam cannot occur on day N-1. etc. By introducing (β), the move from (1) to (2) is now permissible, and the argument as a whole is valid. But notice something about (β): It is closely connected to Guarantee. In particular, Guarantee entails (β), for if the teacher can (and does) guarantee that a surprise exam will occur, it follows that the exam can only occur when it is a surprise.8 Since adding Guarantee to The Elimination Argument renders it valid, and since The 8
More precisely, Guarantee entails (β) modulo the assumption that the teacher tries her best to give a surprise exam; i.e. if the teacher can guarantee a surprise exam, she will.
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Elimination Argument’s conclusion is that a surprise exam is impossible, Guarantee turns out to be self-defeating. That’s not so worrisome. Guarantee was never a very good candidate for The Exam Belief to begin with. And now that we see how it is selfdefeating, there is no reason to accept it.
4. Assumptions There are many philosophically-uninteresting ways that Guarantee might turn out true and (hence) Conditional Possibility false. The students might be too slow to think through the logic of the surprise exam or too uninterested to care. Or the students might be too forgetful to remember announcements. Or the teacher might cheat, e.g., by using a memory-erasing gun to immediately erase all students’ memories of the announcement. In these cases, a surprise exam is not merely possible, it can be guaranteed, even when announced in advance. But such “counterexamples” to Conditional Possibility are sufficiently uninteresting, I think, to justify our making some assumptions. First, assume the students are maximally rational: They’re able to think through the logic of the surprise exam as well as anyone. Second, they’re motivated, patient, and memory-reliable: They’ll keep the announcement in mind and spend time doing their best to predict the exam date. Third, the teacher is maximally rational: She’s as capable as anyone of picking an exam date that maximizes the possibility of surprise. Fourth, the teacher won’t use cheats. Fifth, it’s common knowledge among teacher and students that these assumptions hold. With these assumptions on board, the question about the surprise exam is this: When such an ideal teacher is pitted against such ideal students, is a surprise exam possible? I
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claim that it is, but only so long as the teacher allows for the possibility of the exam’s occurring on the last day, and, hence, the possibility of its not being a surprise.
5. Guarantee a Surprise Exam by Self-Defeating Belief? One version of the surprise exam paradox differs in character from the pure elimination version. Call this revised version The Self-Defeat Scenario: (Self-Defeat Scenario) A teacher announces that there will be a surprise exam next week. The class whiz, giving The Elimination Argument, convinces the students that such an exam is impossible. On Friday of the next week, the teacher announces that it is time to take the surprise exam. The students, having believed such an exam to be impossible, are surprised. In Self-Defeat Scenario, the students’ acceptance of The Elimination Argument leads them to believe that a surprise exam is impossible, a belief which turns out to be selfdefeating. How is it self-defeating? If the students buy The Elimination Argument and so accept its conclusion that a surprise exam is impossible, these same students cannot believe that a surprise exam will occur and so cannot know that one will occur.9 That the students cannot know that an exam will occur also ensures that they cannot know when one will occur and hence that they will be surprised regardless of when the exam is given. Self-Defeat Scenario thus makes it seem not only possible for the teacher to succeed in giving a surprise exam on any day, it also makes it seem possible for the teacher to guarantee that such an exam will occur. 9
This assumes that one cannot believe the contradiction that a surprise exam is impossible and that one will occur. It also assumes that one cannot believe both that a surprise exam is impossible and, e.g., that one will occur on Wednesday. While it is dangerous to assume that belief in contradictions is generally impossible, these two assumptions look innocuous in the context of the present discussion.
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However, there is a threat to this guarantee, namely, the possibility that the students are smart enough to realize that belief in the impossibility of a surprise exam is self-defeating. If the students recognize the self-defeating character of this belief, they may reject it, opting instead to either believe that surprise exams are possible after all or else to withhold belief about the possibility of surprise exams altogether. This alternative may be alleged as equally unstable however, as it opens the students up, once again, to The Elimination Argument. If so, then belief in the possibility or impossibility of a surprise exam threatens to achieve liar-esque instability. Self-Defeat Scenario (and the threat of belief instability) would be more troubling were they generated by the acceptance of a sound argument. But, as we’ve seen, The Elimination Argument comes in two varieties. The first is invalid. The second is valid, but the auxiliary premise that renders it valid—namely, Guarantee—is both unmotivated and self-defeating. Ergo, any student who accepts either version of The Elimination Argument is (in some sense) irrational. To the extent that the students are persuaded into irrationally accepting The Elimination Argument argument, the teacher may hope to ensure that a surprise exam occurs (again, so long as the students are uncognizant of selfdefeat). But if the students had believed rationally to being with, they would have never bought The Elimination Argument, and the question of self-defeat would have never arisen. Instead, if the students had believed rationally, they would have believed that surprise exams are possible but that the possibility of their success is conditional on the possibility of their failure.
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References O’Connor, D.J. (1948). “Pragmatic Paradoxes.” Mind 57: 358-9. Sainsbury, R.M. (1995). Paradoxes, 2nd ed. Cambridge: Cambridge University Press. Sorensen, Roy. (2009). “Epistemic Paradoxes.” The Stanford Encyclopedia of Philosophy (Spring 2009 Edition), Edward N. Zalta (ed.), URL = . Weiss, Paul. (1952). “The Prediction Paradox.” Mind 61: 265–9.