Wednesday, August 31, 2005

The Surprise Examination Paradox

I really like this one. Here's how it goes: your teacher tells you (i) she's going to give the class a surprise exam next week, and (ii) you won't be able to work out beforehand on which day it will be. Using this information, you work out that it can't be on Friday (the last day), or else you'd be able to know this as soon as class ended the day before, contrary to the second condition. With Friday excluded from consideration, Thursday is now the last possible day, so we can exclude it by the same reasoning. Similarly for Wednesday, Tuesday, and finally Monday. So you conclude that there cannot be any such exam. This chain of reasoning guarantees that when the teacher finally gives the exam (say, on Wednesday), you're all surprised, just like she said you'd be.

I originally thought that this paradox would only arise when the speaker was known to be fallible. The reason you can't know beforehand what day the exam will be, is because when running through the reasoning, you come to suspect that what the teacher said is false. But what if the teacher was known to be infallibly truthful? Imagine it is God that makes the announcement (and suppose we know that God cannot speak falsehoods). My thought was that the announcement would then simply be false, i.e. the exclusionary argument proves that this statement could never be made by a being known to be infallible.

But I've changed my mind about this. It seems that if God were to make such a statement, we would be thrown into confusion by it, and would be unable to work out on what day the surprise exam would be. So the statement would turn out true, even though we know the speaker is truthful and so cannot (unlike before) suspect that there will be no exam. (But then what's wrong with the exclusionary reasoning? Most puzzling!)

A commenter over at Opinatrety distills the core of the problem:
I have a coin in my hand. But you will never know if I really have a coin in my hand before I open it.

So you do not know if I have a coin in my hand or not. Then, I open it. Yes, there is a coin. My statements are right.

As he notes, the trick lies in saying 'you will never know'. By saying that, you throw the listener into a state of epistemic conclusion, thereby ensuring the truth of that very statement. Even the known-infallible God could do this, by saying to you: "X is true, but you cannot know it". After all, you'd then reason, "If God says I cannot know X, and he's never wrong, then there must be something dodgy about it... I'd better suspend belief" -- so then you don't know X any more! Even though you were told it from an infallible source! Tricky.


  1. I am not sure you can build the logic on itself.

    And the logic implies (in one interpretation) you would expect the most likely day to be the next day every day. As a result you expect it EVERY day and are cheating the definition of "not surprised". then again maybe that is a permissable rebuttal??

    Anyway if that is not permited then - If the teacher was infallable which day would you conclude the test was on? You still have no answer.You may be sure it is not friday and confident it isn't thurdsay (but not sure) but wednessday might surprise you as might tuesday or monday.
    (UNLESS you expect the test EVERY day).

  2. I mean it seems to be this creeping growth of "expectation" (you could use the argument to define surprise as impossible but only by constantly expecting a surprise - which I admit is posible) that to a large extent creates the paradox. Then you can loose sight of this if you use the frame of reference in time to distract from what is happening to the participant to make it counter intuitive.

  3. If you are expecting the exam every day then you have not worked out which specific day it is on. (So mere "expectation" is not enough. You need knowledge.)

    Why are you any less confident about Thursday than Friday? You say we can be sure it's not Friday. Suppose you're right about this. Now imagine what would happen if the test was scheduled for Thursday... you'd get to Wednesday night, realise the test hadn't happened yet, and thus be able to work out that it will be Thursday (since we are sure it won't be Friday). Now, by imagining this we have shown that it can't be on Thursday either, else we'd be able to work it out the night before, contrary to condition (ii). So we can be sure it's not Thursday either.

    Lather, rinse, and repeat.

  4. > Why are you any less confident about Thursday than Friday?

    OK I will take the extreme case.
    Lets say it MUST be thursday or friday ... if your logic dictates that it must be thursday all that needs to happen is for it to be friday and you are "surprised" (except in as far of course that you just expected it on every available day). As soon as you make a conclusion you open up the potential avenue for you to be "surprised".

    However I think as you get less and less days it becomes less of an argument to accuse the participant of constantly expecting the test because for example if it is only friday then it makes little sense to say it being friday is a "surprise".

    However I also admit you may argue that it is imposible to guarnatee a surprise to someone who is in a constant state of expectation. In which case the paradox is solved !

    So my question I guess is how many days (out of five) is it legitimate for you to "expect surprise"?

  5. All that we mean by "surprise" in this context is that you are not able to KNOW beforehand that the test would be this day. (So being in "a constant state of expectation" is irrelevant, because it clearly isn't knowledge of which specific day the exam will be.)

    The paradox here is that the exclusionary reasoning seems perfectly sound, and yet when looked at holistically one has no way of knowing which day the test would be on. So, the challenge is to show what is wrong with the exclusionary proof that the test cannot be on X-day (for whichever X-day the test ends up being given on).

    My solution, explained in the main post, is that the "you cannot know" comment throws the listeners into "epistemic confusion", thereby ensuring that they cannot be confident of their own reasoning or proofs.

  6. OK how about this - a more technical reasoning

    it ia friday - e can say the last day possible i friday - P=1.00 impossible (unless we get really confused)

    it is thursday P(=friday) less than P(=thursday) (otherwise your first step cannot be to exclude friday) P(=friday) is thus somthing less than .5. it is hard to define P(=thursday)/P(=friday) because your logic implies both are close to zero relitive to any alternative.

    This leaves two posibilities -
    1) you can follow a policy of P(=tomorrow)/P(=today) = 0 but that requires each day to be infinitly more likely than the next AND it forces you to expect a surprize every day no matter how many millions of days there are.

    Or you can say P(=tomorrow)/P(=today) is either indeterminate or some rational number in which case the exclusionary principle falls apart.

    I suggest exclsuion is only possible in a relitivistic sense not in an absolute one. call it my theory of logistic relitivity :) (relevant if we see surprise relitivisticaly)

    Usually the exclusionary principle would work because you probably would not be using a logic system that implies everything has a low/zero apparent probability.

    Somthing to do with maths with zeros and infinities I guess.

    > The paradox here is that the exclusionary reasoning seems perfectly sound

    I think I might need to know what approach is likely to be satisfactory to you since I seem to be missing the mark even though I think I am adressing the point. Anyway I hope my former explination is a little better.

    > My solution, explained in the main post, is that the "you cannot know" comment throws the listeners into "epistemic confusion", thereby ensuring that they cannot be confident of their own reasoning or proofs.

    the above is legitimate but it implies the person saying "you cannot know" knows the effect it will have on all those he talks to and htat that effect will be similar in effect (possible for a god I guess).

  7. Friday night, the only way a student can be surprised is if he's concluded the teacher was lying about the quiz, an unlikely event.

    Thurdsday, a student might be surprised if he think's she didn't think the surprise thing through carefully, or was lying to get students to start studying for a real quiz earlier. This is slightly more likely than the Friday scenario.

    And with each day working back, the inference gets weaker.

    If I were a teacher trying to do this, I'd give the test on Monday or Tuesday, assuming the students were thinking this out clearly. If the students thought the way I thought, I'd only have a 50% chance of surprising them, but that's better than nothing.

    BTW, Richard, are you familiar with the game theory version of this paradox (involving repeated playing of the prisioner's dilema)?

  8. Genius, we can (seemingly) work out that the test cannot possibly be on the Friday. So once we've proved that, we know that P(Friday) = zero. We know this even on Monday, because we 'think ahead' and engage in this reasoning about what would happen if the test were on Friday.

    Given that P(Friday) = 0, the rest of your logic falls apart. You're not allowed to divide by zero, so P(Thurs)/P(Fri) is undefined, for one thing.

    Anyway, if you accept the first step of the reasoning -- that P(Fri) = 0 -- then repeated application of the same reasoning then gives you P(Thurs) = 0, and so forth. [This responds to Hallq too.]

    So I think we need to stop the cascade before it begins, and deny that the students can know that it won't be on Friday. If the test were scheduled for Friday, the students wouldn't be in a position to know this even on Thursday night, for the "you cannot know" comment should have thrown them into epistemic confusion.

    Hallq - As I explained to Genius before, it's a mistake to focus on "surprise". That isn't really relevant here. The question is whether the students are in a position to work out what day the test will be.

    The Prisoner's Dilemma version of this is interesting: that's where you have a set number of iterations, right? Their opponent is going to default on the last one, so they might as well try to default on the one before that too, but their opponent knows this so then they reason...

  9. > Given that P(Friday) = 0, the rest of your logic falls apart.

    haha - keep in mind it is you who are (in this instant) arguing in favour of the paradox. I.e. under at least some frameworks your point above must be flawed. Or as an experimental scientist might say - if the universe doesn’t behave according to logic it is logic that is flawed not the universe.

    Anyway as you said

    > You're not allowed to divide by zero, so P(Thurs)/P(Fri) is undefined, for one thing.

    Which was my point - the exclusionary logic (in this form - since you picked this one) creates a paradox (undefined equation) and then tries to build logic out of it. I think this is what your "epistemic confusion" encompasses anyway so we are inclined to end up supporting each others arguments by accident here.

    Anyway specifically the problems are

    1) You can only say P(=Friday) = zero if you define P(=mon-thurs) as 1 - this is a fundamental part of the problem.

    2) On thursday - you can't BOTH say P(=Friday) is 0 and also "if it is Friday the test is on Friday" because you have declared it CAN'T be Friday therefore there is no meaningful "if". Maths is a bit like 1*(test is on mon-thurs)+ 0*(if it is Friday I know it is on Friday). But we can substitute ANYTHING into the zero probability set including things that do or don’t breach the rules of the game particularly since the current inhabitant of that set (which encourages us to define it as 0) clearly does breach the rules of the game anyway as per (1).

    However I guess this just restates the issue you have with the exclusionary principle as opposed to anything else.

    However, I think it might be possible if we mathematically defined "surprised" and all the other variables to your satisfaction at any arbitrary level - which the answer might just drop out. Not considering those variables makes it really hard for us to separate wood from trees rather like trying to convince someone relativity is not a paradox without being able to use actual examples (which just reinforce how paradoxical it is) or maths.

    > Hallq - As I explained to Genius before, it's a mistake to focus on "surprise".

    I am not entirely satisfied by your confusion concept it may well encompass an answer to the problem but not in a way that logically emerges from the preconditions which I though was what we were supposed to be looking for (and the reason why you don’t want us wandering into "what is surprise". It actually looks pretty similar to an attempt to redefine what surprises people.

    Anyway, all the student has to do is to rationally reject your use of exclusionary logic (which may be possible depending on various parameters because in some context it seems to fail and thus not become confused. Anyway I don’t think it makes the problem any more impossible to just say "and by the way epistemic confusion wont be the main reason you can't solve it" (I hedged a bit to make it practical)

    I do have a creative solution for you that skips all of this though it works as follows

    1) if you can guarantee the student will be surprised it implies you know the student WONT (as opposed to can't) "guess" it.
    2) Therefore you need not concern yourself with "cant".
    3) this implies you know what the student will guess (otherwise you can't be sure they wont guess right!)
    4) this implies it could be any day the student doesnt guess - this brings be back to my origional point about how many days (and more precisely which of the five days) they will /can guess...

  10. In some Universities, we teach on Saturdays too, you know!

  11. It does seem clear that there's a contradiction inherent in the two statements a) there will be a test on one of Monday, Tuesday, ... , Friday, and b) at no time prior to the test itself will you know which day the test is on. (Note that this is a slightly stronger statement than your original statement; I've clarified b) a bit so as to exclude the possibility that the teacher meant that "beforehand", i.e. prior to the entire week, the students would not know what day it's on.) If b) is true, then by the your "exclusionary" analysis, a) must be false, and there is no test at all. If there is a test, then there will be a time, namely Thursday night, when the students will indeed know what day it's on. I don't see the need to invent this "epistemic confusion"; it seems to me that your original thought was right, that the two statements are simply contradictory.

  12. Covaithe,

    You are claiming that it is impossible to be surprised - and yet the problem demonstrates that it IS possible for you to be surprised - If you want to reject two key components of the paradox (1- that you can be surprised and 2- that the two things can exist together) we probably need to adress it directly because we are moving further from the origional paradox.

    Anyway - to examine your rejection of the "surprise"
    Specifically – if the test was on Tuesday would you wake up on tuesday expecting it?
    Or conversely - if it was in Friday would you wake up on Mon-Thurs expecting the test?

    here is a realistic theoretical solution where none of the propositions are rejected

    Imagine this: Teacher watches you carefully to see if you show any sign of preparing for the test and if you do then he doesnt put the test on that day.
    Why doesnt the kid "pretend to be preparing in order to fool the teacher?" well A - the teacher is a bit more experienced at these things - that's why they are the teacher and B- it isnt worth any rational students time to worry so much about the "surprise" and so little about the test.
    Which leaves just one way the strategy doesnt work - if the students are in a constant state of expectation.

  13. Covaithe, how can the two statements be contradictory when they both come out true? The teacher gives the test on Wednesday, and we weren't able to know this beforehand. No contradiction.

    What this shows, I think, is that there is a flaw in the exclusionary reasoning. The reasoning is, in a sense, self-defeating. As you note, it leads to the conclusion that there can be no test. But this means that if there is a test, you won't know when it is (because you falsely expect there to be none at all!). So the conclusion of the reasoning undermines the premise that if the test were on Friday, you could know this beforehand. Once you believe its conclusion, this crucial premise is no longer true.

  14. As you note, it leads to the conclusion that there can be no test. But this means that if there is a test...

    If we take b) to be true, then the exclusionary argument leads to the conclusion that there will be no test. At that point, you can say whatever you like about what might happen if there is a test. If there is a test, the students will not know when it is; that's true. If there is a test, all ravens will be red. If there is a test, green monkeys will fly out of my butt. These are all true, vacuously, since there is no test. You have another recent post about this kind of vacuously true conditional.

    I think it's only possible to be surprised in such a situation if we foolishly assume that b) is true. In other words, if a teacher actually said a) and b) to a group of students, the smart ones would realize that they're contradictory, and would assume that b) is false, or at least misleading, because of the exclusionary argument. In other words, if a teacher actually said a) and b) to me, I would assume that he or she didn't mean b) strictly as I stated it, because that would lead to an absurdity. I would assume that he or she meant simply not to tell me which day the test would be given. It could perfectly well be on Friday, even though that would mean that I would know Thursday night when the test was.

    So, again, I think that a perfectly infallible being could not say a) and b) together.

    Now that I've written all that, I'm thinking that maybe you mean to imply that this situation is related to the Epimenides paradox family of propositions, e.g. "This statement is false", which can be neither true nor false without contradiction, and down which path lies Goedel's theorem. So, "a) and b) are true" leads to a contradiction, and "a) and not b) are true" also leads to a contradiction. I haven't really quite thought that through to see if it makes sense, but if so, the conclusion still must be that, given a), b) cannot be either true or false.

  15. "These are all true, vacuously, since there is no test."

    On the contrary, there is a test. It's on Wednesday, and the students can't work this out beforehand. Both (a) and (b) are true, and hence not contradictory. (There's no better disproof of a so-called "impossibility" than showing it to be actual!)

    You're assuming that the exclusionary reasoning is sound. Only then can we guarantee that "there is no test" and thus that the various conditions are vacuous. But the exclusionary reasoning is not sound. This is immediately obvious from the fact that both (a) and (b) are actually true, contrary to the exclusionary conclusion. The difficulty, of course, is showing just where the exclusionary reasoning goes wrong, because it does appear quite compelling. That's where my talk of 'epistemic confusion' comes in.

  16. Hmm, perhaps I am coming from this from the opposite direction, in that I'm trying to describe a second statement (bX), besides (a): "There will be a test this week", such that the exclusionary argument is correct and valid for (bX). In other words, so that (bX) ==> not (a). And then, there are two discussions: 1) what happens when someone claims both (a) and (bX), and 2) what makes exclusion correct for (bX) as opposed to your original statement, or my (b), which I admit may not quite capture what's necessary for (bX). I kind of thought that you have been mostly talking about 1), rather than 2), but I could be wrong.

    I guess what I'm trying to say is that I don't see the point of talking about what it means when someone claims both (a) and (bY), for some (bY) that does not imply not-(a). We can talk about what the difference is between (bY) and (bX), and try to figure out why the exclusionary argument fails for (bY), and that's fine, but that's not what I thought we were talking about. Or, at least, not the only thing we were talking about; I thought we were also trying to talk about what happens when someone asserts (bX). Hence my assumption that the exclusionary argument works.

    I think that when exclusion fails, it fails at the recursion point. Consider (b2): "There will be no test on Friday". Obviously this does not allow us to conclude anything about Thursday. What about (b3): "There will be no test on the last day the test could possibly be on"? Looks like we can still rule out Friday, but what about Thursday? Not so clear, because it's not clear if the exclusion of the last day applies to any set of possible days the test might be on, or only to the initial given set. If the rule were clearly defined in such a way as to allow recursion, then maybe exclusion would be right: (b4): "Let d be a day. If there are no days after d that the test could possibly be on, then the test cannot possibly be on d." Does that work? It seems to; clearly the test isn't on Friday under (b4); and assuming that the test can't be on day N, (b4) tells us directly that it can't be on day N-1 either. This is exactly what mathematical induction requires in order to conclude that there is no day that the test could possibly be on.

    So I think it is possible to have a statement of the problem in such a way that the exclusionary argument is valid and shows that there's no test. It's also pretty difficult to actually construct such a statement. (Or, at least, it took me a long time to come up with one that I'm (currently) convinced works.) Which means it's not totally absurd to talk about what happens when the exclusionary argument is right. But I've really typed way too much for the moment.

  17. *shrug*, I was talking about the argument given, i.e. your statement (b), rather than some different bX or bY that we can only imagine. But if you can find a bX such that the exclusionary reasoning really does work, then sure, I'd be happy to discuss that too.

    I agree that b4 is the sort of principle that underlies the exclusionary argument. But recursion is clearly no problem for it. So that's why I think we need to say the problem lies before that: in fact the students are not able to conclude that the test will not be on the last day -- even for Friday.

    This is clear from the later examples discussed in my main post. If God says "X is true but you cannot know it", then you are not in a position to know X. To learn X by testimony, you must be in a position to rely upon the speaker. But when the speaker says "you cannot know it", and you trust them, then you will be thrown into epistemic confusion. You will be able to rely upon them no longer. So you are no longer in a position to be able to draw any conclusions from what they have told you.

  18. Though I should add that your denotation is confusing. Your (b) is a statement made by the teachers. I assume that your hypothetical (bX) is likewise. But b4 is not a statement made by the teacher, it is a rule of reasoning employed by the students after the teacher tells them (a) and (b). That's quite different.

    Further, b4 would not make an interesting instance of bX. If the teacher asserted b4, she would be lying, since it blatantly contradicts (a). There would be no paradox anymore. So that is not a case worth talking about. In fact, this would be the case for any such assertion of bX. So I retract my earlier claim about being happy to discuss that. That isn't where the paradox is at all.

  19. But wholistically we know exclusionary logic does not provide good reason to prefer any particular day to all others. If that is the objective - from a wholistic point of view it is surprising we are using this logic at all.

  20. But if God says "X is true, but you cannot know that X is true", how is this different from Epimenides, being a Cretan, saying "All Cretans are liars"? The latter statement has been very well analyzed, and leads to Goedel's Incompleteness theorem. What's different about the first?

    I agree that the teacher asserting (a) and (b4) is not such a case, though, it's just a simple falsehood.

    I guess my notation is confusing, because I think you've mistaken what I meant by it. I meant (b4) to be something that the teacher explicitly asserts, *not* the reasoning process the students go through. Yes, (b4) looks a lot like the exclusionary argument itself, but I think that in order to make the exclusionary argument at all, you need a premise like (b4). I don't think (b3) suffices, and I don't think (b) suffices.

  21. Oh, yes, I agree that (b4) is crucial to the exclusionary argument. I just don't think the teacher needs to say it. When the students are reasoning, they take (b4) to be entailed by (b). The resolution of the paradox lies in showing why they are mistaken in making this inference.

    "What's different about the first?"

    The liar sentence is both true and false. Its truth entails its falsity. There's nothing like that going on in God's sentence. It's just plain true.

    The puzzle here is epistemic (concerning what we can know) rather than semantic (concerning what is true).

  22. Ok, after a bit of thought I agree that God's sentence isn't like the liar sentence. Whoops.

    So, if I understand correctly, you're making a distinction between "X is true" and "X is known to be true", yes? It seems to me that God saying "X is true" naturally implies "X is known to be true". Isn't that why we have God saying it, instead of, say, me, in this example? But then, we have a simple contradiction. Let Y be "X is known to be true". Then, God saying "X is true" implies Y, and God saying "X is not known to be true" is identical with not-Y. God saying both implies Y and not-Y.

    Unless you're saying "God says X" does not imply "X is known". But if it doesn't, then why are we talking about God?

  23. More specifically, I'd distinguish between "X is true" and "S is in a position to know X". After all, X is known by God, at least. The question is whether we are in a position to know it. And while it normally is the case that if God tells us X then we can thereby know X (because we know God is infallible, etc.), it's always possible for new evidence to come along and cast doubt on this. That's what happens when God adds the further claim, "But you cannot know X". We could have known it, if he hadn't added this further phrase. But because he did add it, this changes our epistemic position with regard to X. We are no longer in a position to accept God's testimony (we are instead thrown into a state of epistemic confusion).

  24. Truepath comments:

    "As far as the surprise problem I simply don't see the paradox. I mean if by surprise the teacher means "the students will literally be surprised when I give this exam" then we don't have any problem or contradiction. After all it's not contradictory for the students to be surprised on friday, they could just be dumb.

    Alright well another possibility as to what you mean is the formal approach but we already covered that. So what's left? Well you could try to formulate the approach by asking whether it is rational for the students to be surprised. But since you aren't talking about a stipulative formal system anymore but actual rationality (to the extent that makes sense) you can't force in the assumption the teacher is infallible and the paradox again fails to generate.

    Perhaps then you say but what about the possible world in which the teacher is really infallible. Well at worst then the consequence of the surprise quiz paradox is that such a world is not possible and it's no worse than the irresistible object and immovable force puzzle.

    But all this is unnecessery. The burden is on the person presenting the paradox to offer up specifics that do make it troubling and I just don't see them.

    I think my main post (above) does just that.

  25. I still don't see how it's tricky at all. We just need to remove the ambiguity in surprise.

    Of course if we mean surprise as in literally real individuals will be surprised there is absolutely no problem. Some people experience a certain emotion, no biggie. This is one sense in which the statement can be true.

    Suppose instead the statement is that I (GOD) will never give a test on the last day of the week nor on any day on which every later day has already been eliminated on these grounds. This statement clearly implies god will not give the test this week so we just proved a truth speaking being could never make such a claim.

    Alright I presume your response to these two points will be, 'sure but I want to know what happens when GOD tells an ideally rational agent that they will not know when the test will be this week.' But now this just depends on what you mean by ideally rational. In particular will an ideally rational agent infer any conclusion from a contradictory system.

    I think the only reasonable answer to this question is no (but if you answer yes just work out the specifics of when it is allowed and the paradox will disappear the same way. An ideally rationally agent would, if given a contradictory set of axioms, realize they are contradictory and infer nothing from them. Yet, the supposed 'paradox' arises only by having the ideally rational agent accept propositions that together yield contradiction. Such an agent would simply refuse to reach any conclusion from these propositions all together and could thus be 'surprised' by a friday test. (Not that at worst if the argument did go through it would merely prove that god couldn't make the statement).

    Ultimately in your version there are two confusions going on. First is the confusion between surprise/know as a formal notion and as the real world concept we find familiar. Secondly is our temptation to assume that a perfectly rational being is just a regular guy who is a big smarter. Being perfectly rational is going to cause the agent to often behave in ways that violate our expectations for people.

    In particular when I say, "the perfectly rational being realizes it's inconsistent and concludes nothing" you are going to want to yell, "but on thursday night surely he has no doubts anymore." But this rests upon the idea that the perfectly rational being will behave as we do and assign a high credence to shorter deductions. Since on thursday night the deduction the test will be friday is so trivial it tugs at us to say of course the rational being will know about the test tomorrow. However, if we are really going to be serious about the idea of a perfectly rational actor then they absolutely won't give lesser credences to more remote conclusions.

    I'll actually give you an argument that this in principle can't be a paradox in my next comment.

  26. Alright so despite what I think is a dissolution of the problem one might still think there is a sneaky way to define the notion so the paradox returns. I argue this can't be so once we sufficiently preciscify our notions.

    Suppose our agent A has a well defined belief state in response to being in the situation of waking up on day D and not having had a test yet. If we supply a coherent notion (rational/knowledge/whatever) that specifies a coherent constraint on this agent's belief that means there is some function like the above that is consistant with every requirement we want to hold of his beliefs.

    Either the agent's belief about when the quiz will be always satisfies B(D)=D or it does not (i.e. he thinks the quiz will be today every day or he does not). If he thinks the quiz will be today every day then it is impossible for any teacher to give a surprise quiz. If he has a day where B(D) neq D then the surprise quiz can be given. If at any point this story fails you specified an incoherent set of constraints about how the agent's beliefs are constrained to behave.

  27. In fact I think I'm going to go post my full argument at my blog rather than just give half-assed versions here. Both of this issue and the other one (

    Note that I do think this was a very interesting philosophical problem to start with, I just think it was solved way back when.

  28. Why do you keep talking about 'surprise'? I mean, I know it's in the title of the paradox. But the actual substance, as I spell it out above, is simply in terms of what's knowable. Not what can be known or worked out using such-and-such formal methods, but just what can be known or worked out, simpliciter.

    The paradox here, unlike your God case, is that what the teacher says is true. There will be an examination during the week, and you won't be able to work out beforehand which day it is on. This contradicts the apparently sound exclusionary reasoning, so the challenge is to show why the exclusionary reasoning actually fails. (My solution invokes the notion of agents being thrown into a state of epistemic confusion, which prevents their knowing things they otherwise might.)


    I was just using surprise as a synonym for not know. The exact same reasoning applies just the same.

    If you mean an actual teacher talking to actual students and using our natural language concept of know no problem (they are failible, probabilities build up and the students indeed don't know on the day of the students).

    If you want to go any farther beyond this you can't rely on our intuitive notion of knowledge because like most other natural language terms it's vague and imprecisce in exactly these sorts of extreme cases. Thus in order to generate a paradox you need to:

    1) List out some predicate K that your agent supposedly posseses.

    2) Given me reasons to believe it's logically possible for the agent to actually have a predicate like this.

    3) Give me reason to believe with respect to that predicate that the teacher/god's claim is true.


    The problem is that once you depart from our intuitive notion of knowledge the argument that the statement is necessarily true is no longer obvious. In particular if you allow a bizarre notion of knowledge that lets the student know that the quiz will be on monday and know that it will be on tuesday and etc.. then the statement made by the teacher is simply straight out false.

    On the other hand if you demand as one of your axioms of knowledge that it not reach contradictory results then your system can never infer that the student knows the test will even be on friday since use of such a scheme must already be blocked by some part of your definition of knowledge.


    Ultimately though I did agree you *could* make a paradox out of this by making in complicated in a fashion that lets you embed the liar. This seems to be what you are sorta reaching at, well more preciscely you want to demand the existance of a predicate K that functions as a truth predicate for it's own theory. But this isn't a paradox about the surprise examination it's a well known restriction on what sorts of consistant theories you can have (remember all the Kripke stuff about crazy hierarchies with multi-valued logics ....he did this because it's impossible to have a predicate that obeys the constraints you want).


    In short if you want to give me the supposed features your notion of knowledge has that generate a paradox I'd be glad to continue the debate. If you can't I think that's very strong evidence that there is no such paradox, i.e., it's been successfully dissolved.

  30. To be more clear:

    I claim that any notion of knowledge you try and offer will fail to generate the paradox in one of three ways.

    1) The notion of knowledge will allow the reasoner to know contradictory things. Thus the teacher's statement is flat out false since the students know the test will occur on monday and on tuesday and on wednesday and etc..

    2) The notion of knowledge you offer will avoid allowing inconsistant beliefs in a fashion that prevents it from making the inferences needed in the problem, e.g., you will defined knowledge in such a way that it 'forsees' the problem it is about to get into and rejects the reasoning that leads to the knowledge the test will be on some particular day. Once again no problem the teacher says something true and fufills it by giving the test on the day the students in fact did not know the test would occur.

    3) You will define knowledge in a fashion that is internally inconsistent to avoid the first two problems. Once again we fail to have a problem.

  31. There's nothing wrong with the ordinary conception of knowledge, and no reason not to use it here.

    Now, knowledge entails truth, so your option #1 is clearly ruled out.

    My own solution is something more like your #2. But note that the claim is not just that the students "do not in fact know" the day of the test, but that they could not know it (i.e. by means of any chain of good reasoning). This is puzzling, because the exclusionary argument looks like good reasoning, so the question is why it is not in fact so, or at least why it can't yield knowledge in the reasoner. (It sounds like you might be sympathetic to the answer I give here. No?)

  32. Suppose it is now Thursday and your teacher tells you, after the class is over for that day, that (i) she's going to give the class a surprise exam this week, and (ii) you won't be able to work out beforehand on which day it will be. (Assume that the week ends on Sunday and that there are no classes on weekends.)

    Does the original version of the paradox add anything to this simplified version?

  33. Not really, or at least that's the suggestion of the last part of my post, with the discussion of: "I have a coin in my hand. But you will never know if I really have a coin in my hand before I open it."

  34. Then again, perhaps there is the following difference:

    In the simple (only one option remaining) case, even if you can't attain "knowledge" for tricky reasons, you can at least settle on a most likely hypothesis (they have a coin in their hand; the test will be Friday), and almost certainly turn out to be right about it.

    In the original case, though, it seems like you really can't narrow down the 5 options to any better than a chance guess between them (and the sixth option of no test at all).

  35. The teacher makes two relevant statements:

    (1) the exam will be this week; and

    (2) the students won't know when the exam will be.

    We are supposed to assume that the teacher is truthful, so we must assume that (1) and (2) are true. If (1) is true, then given the assumptions I made explicit in my parenthetical remark above, if follows necessarily that the exam will be on Friday. (I’m assuming the simpler version I proposed.) The students can draw elementary inferences and are aware of the relevant assumptions, so they must know this. Yet this is contradicted by proposition (2). It seems to me that there is no paradox here, but simply a description of a problem predicated on a logical impossibility. One might as well imagine a “paradox” involving a truthful being who claims that 2 + 2 = 5.

  36. Pablo - see my responses to Covaithe, upthread. In the original case, both statements from the teacher turn out to be true. There is an exam, and the students really didn't (and couldn't)* know beforehand when it would be.

    This can really happen. And a situation cannot be logically impossible if it is actual.

    * = in your simplified version, it's less clear whether the true conclusion cannot be known. It can at least be reliably guessed at, as per my previous comment. But the latter half of my original post argues that we still couldn't have knowledge in such a case.

  37. Richard, your response to Covaithe presupposes that the two statements come out true. I am contesting that presupposition.

    Suppose, to simplify, that the teacher tells the students that he will take an exam on Friday and that they won't know that he will take an exam on Friday. Would you still deny that, as I claimed, it's impossible to consistently believe that the teacher is truthful?

  38. Yes, they can consistently believe that, and indeed be right in doing so because both claims turn out true. The point is that the students' belief in claim (1) will not (in light of (2)) constitute knowledge.

  39. [I have consolidated the two previous comments and corrected an omission.]

    The point is that the students' belief in claim (1) will not (in light of (2)) constitute knowledge.

    What are your grounds for saying that the students' belief in claim (1) will not constitute knowledge in light of (2)? Would you also be prepared to say that, if a truthful being told you that 2 + 2 = 4, your belief in that proposition would not constitute knowledge? If not, what's the difference between the two cases?

    (Notice, incidentally, that I didn't ask whether the students could consistently believe that the teacher is truthful. The question is whether we, in the real world, can consistently believe that proposition. I claim that we can't because there is no logically possible world in which both (1) and (2) are true. Since there is no such world, there are no possible students considering whether they can consistently believe that their teacher is truthful and correctly reaching the conclusion that she is.)

  40. If an infallible being (or one I consider infallible) tells me that I cannot know that P, then that undercuts my other grounds for believing P, and throws me into a state of epistemic confusion. It raises doubts of a sort that, we may think, are incompatible with knowledge.

    The case where P is "2+2=4" is unclear. It's possible that my antecedent belief in P is so robust that I would sooner doubt the infallibility of the being than my knowledge of P. In that case, the agent would be speaking falsely in asserting that I cannot know P; so the scenario would be impossible as you suggest. But I do not think our knowledge is always so robust and impervious to epistemic confusion. Especially on contingent matters, like the question whether there will be an exam on Friday, I see no reason at all to deny that one's epistemic position may be undercut, and thus the scenario described is possible.

  41. The teacher is truthful. The students know that she is. Necessarily, if a truthful agent asserts P, then P is true. The students know this analytic truth. The teacher has asserted (1), and the students know this too. So the students know that (1). If someone knows that Q, that person cannot be thrown into a state of epistemic confusion regarding Q. Only if the students are in a state of epistemic confusion regarding (1) can (2) be true. The students are not in a state of epistemic confusion regarding (1). So (2) cannot be true. So the scenario is logically impossible.

  42. "If someone knows that Q, that person cannot be thrown into a state of epistemic confusion regarding Q."

    Why think that? Seems false to me. The students may initially know (i.e. have a non-gettiered justified true belief) that the teacher is truthful and hence that her assertion (1) is true, only to have (the warrant for) both these beliefs undermined by her later utterence of (2).

    We know from other cases that knowledge can be 'lost' in virtue of acquiring new defeaters for one's belief. So why not in this case?

  43. As I was assuming, knowledge is indefeasible justified true belief. If you deny that knowledge is indefeasible, fine; just assume the term 'know' and its cognates in my previous comment express the concept of superknowledge, where someone superknows P iff she indefeasibly knows P. What step in my argument, thus restated, would you deny?

  44. That the students superknow anything.

  45. Okay, at this stage we can only continue the debate by straying too far away from the topic of the original post. It was fun while it lasted.


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