Surprise Quiz Paradox

Philosophy Forums Science, Technology, and Math
21

The fact that Friday can still be chosen is the kicker here. This means that even if the professor doesn’t use a randomization but simply decides for himself, with full knowledge of everything, which day the quiz will be on, he is free to pick M-Th and know that it will be a surprise. Because he could STILL have picked Friday, even though it wouldn’t then be a surprise.

That is the heart of the paradox. Friday can be chosen, knowing full well it will violate the surprise condition; but if Friday is NOT chosen, then any other day will fulfil the surprise condition. The students cannot truly know that Friday will not be chosen, because Friday has already been defined as a possibilty and the professor can employ any number of underlying reasons for choosing which day to have the quiz, including randomization. The students do not, according to the problem as defined, have access to the cause or reasons for which day is chosen.

I suppose you could then revive the paradox by modifying it slightly, " the professor then tells the students he will not choose a day where the quiz being on that day would be known before the quiz is announced." In that case Friday is off the table from the start, which alters the condition “there will be a quiz M-F next week” to “there will be a quiz M-Th next week”. So the problem itself is different. That’s the essence of the paradox, it just depends on how you define the selection methodology behind why the day of the quiz was chosen. If there is a randomization factor then it is possible to have a surprise quiz M-Th; if the professor promises not to pick what amounts to Friday, then the problem is changed and we aren’t even talking about the same situation anymore. So to uphold the M-F condition as defined by the professor, it could not be the case that the professor would himself rule out Friday as an option, therefore if Friday is still possible all conditions of the situation are upheld, so long as Friday is not actually selected.

Hence back to what I wrote above, that this situation is possible to occur non-contradictingly if the quiz is M-Th, but impossible to occur non-contradictingly if the quiz is on Friday. If the quiz ends up being on a M-Th, AND Friday was not explicitly ruled out by the professor at the beginning, then all conditions are met and the paradox vanishes. That is the only resolution of the paradox, everything else (altering or infringing upon the M-F condition, or having the quiz actually occur on Friday) is a violation and upholds the paradoxical nature of the situation.

22

I agree with your summary of the problem: On Thursday, there are multiple possible outcomes, and so the outcome that the quiz is on Thursday it is still surprising.

It’s still logically weird. Even though we can logically exclude Friday, because the only way for the test to be given on Friday would be for the premises to be false, that is still a live option for logically evaluating whether a quiz on Thursday is ‘surprising’. Friday can’t be the day the quiz is given, but is treated as a live possibility for the purpose of determining if a quiz on any other day is surprising.

I return to thinking that this is ultimately about the problem of knowledge. Logically, it can’t be Friday, but epistemically we can’t know that it can’t be Friday with enough certainty that we won’t be surprised on Thursday.

You raise an interesting point starting here:

The paradox depends on the professor saying that the quiz will be M-F. If the professor secretly excludes Friday, e.g. re-rolls the die if it lands on either 6 or 5, so that Friday is excluded in practice but the students don’t know that it is, the paradox still works. But if he tells the students he’s doing that, it doesn’t. Similarly, if he were to say “the quiz will be M-Sa”, or “the quiz will be M-F next week or not at all”, then the quiz could be given on Friday and still be a surprise, because the professor has planted a live possibility.

Epistemically, suggesting that the quiz could fall on a day that can’t actually be picked is necessary to create uncertainty in the minds of the students, allowing for the collapse of that uncertainty on the day the quiz is announced.

23

Right, so a savvy student might think to himself (but perhaps avoid telling the professor and other students), “Well the professor cannot pick Friday, but he cannot tell us that he won’t pick Friday, so I know the quiz will be M-Th and I really have no idea which of those days it will be.”

That seems to solve the problem as it was given to me and the rest of the class by my epistemology professor so many years ago. But what if we spice it up?

What if the professor really goes out of his way to define to the students the surprise condition, saying that he promises not to pick a day for the quiz whereby the students would know for certain, before the quiz was announced, that that particular day would be the day of the quiz? Now the students are correct to protest and rule out Friday as an option, and then subsequently point out that the professor is lying when he claims the quiz will be M-F. But the professor calmly explains he never said that, he only said the quiz would be sometime next week.

Now, further puzzled, the students try to reason backwards from Friday not being an option to Thursday also not being an option. Now they know for certain that the quiz will not be on Friday. Can they therefore also rule out Thursday since Thursday has become the new “last option”? It seems like it but I am not convinced. Think about the students’ perspective Wednesday when class finishes: they may think “no quiz so far, and we know it cannot be on Friday, so it must be tomorrow!” However, they should also therefore realize that is cannot be tomorrow for the same reason that they assume it cannot be on Friday. I.e. they know in advance. But do they REALLY know this?

There is a disconnect here between what they students suppose they know ahead of time, and what they might really be able to know or not know. Perhaps the professor will put the quiz on Friday just to mess with them, or just to claim “Surprise! You thought there could not be a surprise quiz on Friday, and you were right about that, so I decided to put the quiz on Friday as a surprise!”

Ultimately the professor can violate the logic of the situation, from the students’ perspective, in a way that still upholds the logic of the situation. This mere possibility must be factored into the students’ logical analysis before hand. And none of this also touches upon the fact that regardless of any of that, any given student is free to guess a particular day and may be right or wrong about that, which for him/her could mean that they broke the surprise condition on their own either by clever deduction or a simply lucky guess.

The gap between what students can logically deduce-predict and what the professor can choose, seems to be the problem here. Because even if the professor violates his own rules from the prior perspective of the students’ reasoning, he can still maintain those same rules in reality when the quiz actually occurs. At the end of the day, the students are free to deduce and logically evaluate and predict any and every possibility they like, forming their best guesses, but none of them really knows which day the quiz will be on.

24

The sun is going to surprise rise anytime in the next 24 hours.

25

I think “know for certain” has some of the same problems as ‘surprise’, as your subsequent analysis shows.

I’ve been playing with “the day of the quiz cannot be logically deduced in advance.” My formal logic is rusty, but in principle it should be objective. For example, if the quiz will be M-F, and it hasn’t been given by Th, we could logically deduce that it must be on F:

Let M be the statement “the quiz is on Monday”, and Tu,W,Th,F be similar statements for the other days.
Given

  1. M \vee Tu \vee W \vee Tg \vee F
  2. \neg M \land \neg Tu \land \neg W \land \neg Th
    F follows by eliminating the disjunctions.

But on Monday, that’s a conditional statement:
\neg M \land \neg Tu \land \neg W \land \neg Th \implies F

The student is trying to use the ‘surprise’ premise to eliminate the conditional and get \neg F: if the quiz is on Friday, we could logically deduce it Thursday, so it’s not on Friday. But I don’t think that’s a proper move in formal logic (though again, it’s been a while since I’ve done formal logic).

I think the best you can get is F \implies \neg F, which produces a contradiction only if the quiz is on Friday.

I’m not sure if any of this is clear, but to me it vindicates what we’ve been saying: the quiz could be on Friday so long as it is not actually on Friday. The inconsistency of it being on Friday can’t be used to deduce that it’s not on any other day.

Reading up on this I landed on the Stanford Encyclopedia of Philosophy page for “The Logic of Conditionals”, which gives me the impression that 1) this paradox is related to deeper problems, and 2) even at its peak my understanding of formal logic was beginner level.

(I note that putting “not logically deducible” in the premise makes the logic self-referential, which is what creates the problems in paradoxes like the Liar’s Paradox. I don’t think that’s a coincidence.)

26

Well if you eliminate it by logic, and rely on that logic, you will be surprised when the logic fails.

So there’s that.

27

I think the quiz can be on Friday, because the students already ruled it out, therefore having the quiz on Friday (which is supposed to be impossible) would actually be a surprise.

Besides, the whole thing of guessing correctly has to be taken into consideration. If a student really thinks the quiz will be on a certain day, for whatever reason, and it turns out he is right (even for some random reason and he was lucky) then the quiz, for him, was not a surprise. So the entire idea of the quiz being a surprise is no sort of absolute and should not be treated as such. There is no PERFECT way to guarantee a surprise quiz but there are PRACTICAL ways to (nearly) guarantee it. Like putting it on Friday after everyone already thinks it cannot possibly be on Friday.

After all what is more likely to be violated, if one must be: the premise “there will be a quiz next week…” or the other premise “the day of the quiz will be a surprise”? Clearly if its Friday morning and no quiz so far, the students are as justified to say there will not in fact be any quiz this week (violation of premise 1) as to say the quiz will not be a surprise (violation of premise 2). And even if some students accept the violation of the second premise and think the quiz will be on Friday, that can still be counted as little more than a lucky guess on their part. After all, they already assumed the second premise can be falsified, so why not also or instead the first premise? If the first premise is falsified then the quiz being on Friday returns to being surprisable.

28

You’re a freaking genius!

29

The Professor states it is a surprise quiz. This is absolutely reasonable. It could happen any day, any time.

The professor then makes it clear that the surprise will be the lowest denominator - the day, as opposed to the hour (as per the above quote).

Other than that l don’t see the issue. Revise Sunday night for Monday, revise Monday night for Tuesday, etc.