Surprise Examination Paradox

Great Debates
1

The Surprise Examination Paradox

A teacher tells his pupils on Friday afternoon that on some day of the next school week (Monday through Friday) a surprise exam will be given. The teacher says that the exam will definitely be given, and it will definitely be a surprise to the pupils. One smart pupil reasons that the exam cannot be given at all; he reasons in the following way: if by the end of school on Thursday the exam has not yet been given, the pupil reasons that the exam will have to be given on Friday, since it is the last day left in the week; but if the pupil can predict the exam will be on Friday, then it will not be a surprise. So the exam cannot be scheduled for Friday. Now suppose that the exam has not been given by the end of the school day on Wednesday, the pupil will reason that it has to be given on Thursday, as it is the only day left in the week that the exam can be given on (Friday has already been ruled out); but if Thursday is the only possible day left, a Thursday exam will not be a surprise. So a surprise examination cannot be given on Thursday. The pupil proceeds to use the same logic to eliminate Wednesday, Tuesday, and Monday. He reaches the conclusion that the surprise exam cannot be given at all.

On Tuesday morning, however, the teacher walks into the classroom and gives the exam. It is a surprise to all of the pupils.

The paradox arises because the reasoning of the pupil seems valid, yet the conclusion reached – that the surprise exam cannot be given – is wrong.

My proposed solution to the paradox is as follows:

Take the following rules that are present in the paradox:

1- The exam MUST be given;
2- The exam can only be given if the pupils cannot know when it will be given (i.e. the exam must be a surprise).

Now, let us examine whether the exam can be given on Friday (the last day of the week). For if it can be given on Friday, we have “nipped the problem in the bud.”

The pupil, I maintain, cannot be certain on Thursday that the exam will take place on Friday. For 1 and 2 above contradict one another – that is, 1 tells the student that the exam will be on Friday, and 2 tells the student that it cannot be on Friday. The pupil therefore MUST doubt the truth of either 1 or 2, or assume that one must be wrong. He cannot believe both. So the student cannot be sure whether the exam will or will not be given. So if the teacher gives the exam on Friday morning, the exam will indeed be a surprise because the student could not know the exam would take place.

Now, this solution to the paradox is consistent with both 1 and 2 being true. Both one and 2 could be true, as long as the pupil could not KNOW their necessity for certain. The exam will NECESSARILY be given on Friday (if it is the end of the Thursday school day and it has not been given yet) but the student cannot know if this necessity holds for sure.

(Analogously, there is a difference between the necessity of the law of gravitation, say, and our knowledge of the necessity of gravitation as a law of nature).

So the solution to the paradox is that the exam can be given on any day of the week, including Friday.

This solution bears resemblance to the solutions proposed by Doris Olin and somewhat by Sorensen.

I would like your feedback on this solution. Thanks.

2

Is this an exam? :confused:

3

We had this discussion either here or in GQ a while ago, only it was the “surprise execution” version.

The answer that most makes sense to me is:

The paradox relies on the idea that on friday morning, they’d know the test was coming today if it hadn’t come yet. Then, knowing that it couldn’t be given on friday, on thursday morning, they’d know it’d be that day. Then of course, if they can’t do it on thursday, they’d expect it on wednesday.

Or not.

That’s the break in logic there.

On thursday, they know that it couldn’t be done on friday, so it’d have to be done today. BUT, on wednesday, it could happen on thursday OR friday, so they couldn’t conconclusively say it’d happen on thursday, and from there, they whole ‘paradox’ falls apart.

Basically, from thursday, there is only one choice left, so it must be THE choice. But from wednesday, there are 2 choices left, and you don’t know which it will be.

Of course, there isn’t a conclusive solution to this ‘officially’, as far as I know, but that explanation makes the most sense to me.

4

I read a similar thing, but that was the “Surprise Tiger” version.

5

I think there are many different ways of looking at this paradox, and none of them is terrbily simple. For me, it was best explained in terms of changing your mind, and just what you mean by “knowing” something. So if you “know” that an event is going to happen on Monday and not Tuesday, but then it does not happen on Monday, can you change your mind, and “know” that it’s going to happen on Tuesday?

6

Mr. Shabasson, I think you’re right on the money.

7

I will post my answer to this paradox between now and Friday, but you’ll never guess which day.

8

Aahala: Nice Meta-paradox.

SenorBeef: Yes, the paradox is often stated in terms of a surprise execution. In your proposed solution you state:

“On Thursday, they know that it couldn’t be done on Friday, so it’d have to be done today. BUT, on Wednesday, it could happen on Thursday OR Friday, so they couldn’t conclusively say it’d happen on Thursday, and from there, they whole ‘paradox’ falls apart. Basically, from Thursday, there is only one choice left, so it must be THE choice. But from Wednesday, there are 2 choices left, and you don’t know which it will be."

I’m not sure I agree with your proposed solution. Here’s why: Your solution is essentially that on Wednesday eve, we have two choices: Thursday and Friday. So we cannot know which of the two is the day, so we cannot predict with certainty on which day the exam will be scheduled. The problem with this is: if a Friday exam is impossible, we only have one possibility on Wednesday eve – Thursday. There is only ONE possible day from the perspective of Wednesday eve. So the pupil must conclude on Wednesday eve that the exam must be on Thursday, which of course allows us to fall right back in to the trap of the paradox (for then the Thursday exam won’t be a surprise, and therefore cannot be given on Thursday).

I believe that a Friday exam must be possible, or else the paradox is created. The solution must allow for the possibility of a Friday exam.

Achernar: I would like to hear more about this possible solution to the paradox. My solution does have to do with what counts as “knowing” the exam will be given on a particular day. Yet your solution seems a bit different. Perhaps you could expand on it.

Lucwarm: Have you thought of a similar solution yourself?

9

:o yeah :smiley:

10

the pupil can only predict the exam on friday if it hasn’t been given by the end of the day thursday. it’s not yet the end of the day thursday, so to cross of the exam being on friday is pre-mature. I think that’s the nature of the paradox. To say on monday morning “it can’t be on friday” is not good logic. The exam not being on days monday thru thursday is information the student doesn’t yet have. To make a deduction based on info he doesn’t yet have is not a proper deduction. The paradox stems from this improper deduction.

11

Look at it from the perspective of Wednesday morning, before the possible exam: It can happen either today, or thursday - friday being ruled out for the reasons stated in the paradox. And so one can’t be sure it’s wednesday or not - it can validly be thursday from that perspective.

Reverse to tuesday, and at that point, you can’t conclusively rule wednesday out, because it can still happen on thursday.

There’s no definite solution, but that’s my take. I’m probably explaining it badly - it’s just that removing friday applies that idea that there’s only one choice on friday and therefore can’t be friday, as there must be more than one choice/option for it to be a surprise. And using that “one choice” logic, we can go back through each day sequentially, intuitively, but it ignores the fact that we have more than one choice from the “perspective” of wednesday or earlier.

Yeah, I’m explaining it badly.

The thread I alluded to earlier is here, and has a much more in depth view. I scanned the article, but couldn’t find the poster/explanation I favored… I don’t want to reread the whole thing right now, but you’ll see it when you come across it.

12

Er, I didn’t mean to say “allude”, slip of the brain. Replace it with “mentioned”.

13

Imagine a stack of 7 cards, 6 red and 1 black. You are told to turn over the cards one by one, and you are told that you will be surprised by the black card. Now, if it comes down to 1 card left you would clearly not be surprised when it is black. If it gets down to 2 cards, you would expect the black card next because you would you could not be surprised by the last card. This logic continues, so that at every step you expect the black card. At this point you think to yourself: “The person who told me I would be surprised is wrong, because there is no way I will be surprised. I expect the black card at every step.” And in fact, when the third card turns out to be the black card, your are not surprised, because you expected it.

On further contemplation, you realize that being told that the black card would surprise you does not give you any information about the placement of the black card. Due to the fact that you can expect the black card at every step, there is no place the black card could be that would be preferable to any other. And since no placement is preferable to any other, you realize that you are justified in expecting the black card at every step.

In the case of the students, it was only their own stupidity that caused them to be surprised by the exam. They fell for the teacher’s trick. If you take the teacher’s statement about you being surprised as truth, then you can never expect the exam. There is no point in time when you can say “I expect to be surprised by an exam today.” Consider a student S who expects an exam on Tuesday. On Tuesday morning, when confronted with the statements A) There will be an exam this week and B) It will be a surprise, S will realize that A goes along with his current expectation, but B does not. If S is stupid he will conclude that he should reject his expectation, because it contradicts B. Thus S will be surprised by a Tuesday exam. But if S is smart he will realize that this same dilemma will ovvur every day, and that if he rejects his expectation every time then he will contradict A. Thus S will realize that he had a choice between believeing there will be no exam this week, or in believing that there will be an exam but it won’t be a surprise. The students chose to believe that there would be no exam, but they were just as justified in deciding that there would be a non-surprise exam. If they had done so, then they would not have been surprised.

14

The best modern treatment of this, in my opinion, is from Maria Fasli of the University of Essex. She examines the paradox from the angle of both Self Reference and Common Knowledge. Traditional treatments have considered Self Reference only, some in an epistemological framework and some in a statistical or probabalistic framework. (For the latter, see Hall’s How to Set a Suprise Exam.)

Fasli’s approach allows her to develop a doxastic logic system from first order logic that proves both that (1) the announcement by the teacher is Common Knowledge and (2) the surprise itself is Common Knowledge. This allows there to be a surprise even if the teacher gives the exam on Friday. (Consider temporality: the surprise need not be on the same day as the exam.)

Similar approaches have allowed for solving the Wise Men and Muddy Children paradoxes. I’m going to simplify her approach here for the sake of brevity, so please forgive the informal symbology and obvious leaps across propositions.

Definitions

The following definitions are added to first order logic to develop language L:

Self Reference: Belief by an agent in a proposition about the beliefs of other agents in the agent’s belief — i.e., [B(a) [symbol]Þ[/symbol] [B(b) [symbol]Þ[/symbol] B(a)].

Common Knowledge: Knowledge of [symbol]f[/symbol] by a group when everybody in the group knows [symbol]f[/symbol], and everybody in the group knows that everybody in the group knows [symbol]f[/symbol], etc., for sufficient K — i.e., [K(G([symbol]f[/symbol]) [symbol]Þ[/symbol] K(G(K(G([symbol]f[/symbol]))))…].

Terms and propositions: Constants, variables, functions and quoted sentences such that [symbol]f[/symbol] is a proposition and “[symbol]f[/symbol]” is a term. [symbol]f[/symbol] can be understood by context to mean “[symbol]f[/symbol]” — e.g., [symbol]f[/symbol], “[symbol]f[/symbol]”

Predicates: Knowledge, Everybody Knows, Common Knowledge, and Equality, i.e., K, EK, CK, and = respectively

Wiff: Well formed formula — If t(1) and t(2) are terms, then K(t1), EK(t1), CK(t1), and t(1) = t(2) are wiffs.

Predicated Wiff: Wiff formed from a predicate — If Pn is an n-ary predicate, and t[sub]1[/sub], t[sub]2[/sub] … t[sub]n[/sub] are terms, then Pn(t[sub]1[/sub], t[sub]2[/sub] … t[sub]n[/sub]) is a wiff.

Derived Wiff: Wiff derived from one or more wiffs — If [symbol]f[/symbol] and [symbol]y[/symbol] are wiffs, then so are [symbol]Øf[/symbol], [symbol]f Ú y[/symbol], [symbol]f Û y[/symbol], etc.

Termed Wiff: If x is a variable and [symbol]f[/symbol] is a wiff, then there exists a wiff for every x — i.e., [symbol]"[/symbol]x[symbol]f Ù $[/symbol]x[symbol]f[/symbol].

Proof: If [symbol]f[/symbol] is true in L, then [symbol]f[/symbol] is provable in L — i.e., L [symbol]^ f[/symbol].

Hypothesis

“The exam will definitely be given, and it will definitely be a surprise to the pupils” is Common Knowledge.

Symbologically, this is:

[symbol]f[/symbol]: [symbol]f[/symbol][sub]1[/sub][symbol]Ù[/symbol] (K([symbol]f[/symbol]) [symbol]Þ[/symbol] K([symbol]Ø y[/symbol]))

Where

[symbol]f[/symbol] = The teacher’s entire statement

[symbol]f[/symbol][sub]1[/sub] = “The exam will definitely be given”

[symbol]y[/symbol] = “It will definitely be a surprise”

Now we can begin our proof

Doxastic Axioms

The following axioms are standard axioms of Doxastic Logic. We add them to L:

  1. [symbol]K(f Þ y) Þ (K(f) Þ K(y))[/symbol]

  2. [symbol]K(y) Þ ØK(Øy)[/symbol]

  3. [symbol]"cK(f) Ø K("cf)[/symbol]

  4. [symbol]É (^f Þ K(f))[/symbol]

For clarity, Axiom 4 means "it is necessary that if [symbol]f[/symbol] is provable (in L), then there is knowledge of [symbol]f[/symbol] (in L).

Theorems

The following theorems can now be derived for EK (Everybody Knows) and CK (Common Knowledge):

  1. EK[sub]G/sub [symbol]Û[/symbol] [sub]def[/sub] [symbol]Ù[/symbol] [sub]i[/sub][symbol]e[/symbol][sub]G[/sub] K[sub]i/sub

  2. CK[sub]G/sub [symbol]Û[/symbol] EK[sub]k/G/sub [symbol]"[/symbol]k

  3. CK[sub]G/sub [symbol]Û[/symbol] [sub]def[/sub] EK[sub]G[/sub]([symbol]f Ù [/symbol]CK([symbol]f[/symbol]))

  4. ([symbol]f Þ[/symbol] EK[sub]G[/sub]([symbol]f Ù y[/symbol])) [symbol]Þ[/symbol] ([symbol]f[/symbol] CK[sub]G/sub)

As you can see, we are precipitously close to an inconsistency if we add standard necessity axioms from S4, but we can avoid that by adding some nonstandard axioms.

Lemmas

  1. K(K([symbol]f[/symbol]) [symbol]Þ f[/symbol]) [symbol]Þ[/symbol] K(K([symbol]f[/symbol]) [symbol] Ú [/symbol] K([symbol]Ø f[/symbol]))

I know that Lemma 1 seems self-evident, but bear with me.

  1. K[K([symbol]f[/symbol]) [symbol]Þ[/symbol] K(K([symbol]f[/symbol]))] [symbol]Þ[/symbol] (K([symbol]f[/symbol]) [symbol] Ú [/symbol] K([symbol]Øf[/symbol]))

Now you see why Lemma 1 was necessary.

  1. [K([symbol]Øf[/symbol]) [symbol]Þ[/symbol] K(K([symbol]Øf[/symbol]))] [symbol]Þ[/symbol] (K([symbol]f[/symbol]) [symbol]Ú[/symbol] K([symbol]Øf[/symbol]))

I realize that the Lemmas were tedious, but they serve to help us develop the following axioms for Knowledge in L:

Knowledge Axioms

  1. K([symbol]f Þ y[/symbol]) [symbol]Þ[/symbol] (K([symbol]f[/symbol]) [symbol]Þ[/symbol] K([symbol]y[/symbol]))

  2. K([symbol]f[/symbol]) [symbol]Þ f[/symbol]

  3. [symbol]"[/symbol]xK([symbol]f[/symbol]) [symbol]Þ[/symbol] K([symbol]"[/symbol]x[symbol]f[/symbol])

Now we can add a restricted necessity rule without introducing an inconsistency:

Restricted Necessity Rule

RNR = (L [symbol]^ f[/symbol]) [symbol]Û[/symbol] [symbol]f[/symbol]

At last, we can now add the S4 Axiom:

S4 Axiom

K([symbol]f[/symbol]) [symbol]Þ[/symbol] K(K([symbol]f[/symbol]))

What we’ve come to in what might seem to be a round about way is a FOCK logic that is identical to KCKS4. This is all we need to prove our hypothesis:

Deductions

  1. EK[sub]G/sub — Everybody in the group knows the announcement, from the Teacher’s Statement

  2. EK[sub]G[/sub]([symbol]f[/symbol][sub]1[/sub] [symbol]Ù[/symbol] (K([symbol]f[/symbol]) [symbol]Þ Ø[/symbol] K([symbol]f[/symbol]))) — From Deduction 1 and the Definition of [symbol]f[/symbol]

  3. EK[sub]G/sub [symbol]Ù[/symbol] EK[sub]G[/sub] (K([symbol]f[/symbol]) [symbol]Þ[/symbol] ([symbol]Ø[/symbol]K([symbol]y[/symbol])) — From Deduction 2 and the Theorems

(Note that Fasli has a misprint in her step corresponding to Dedution 3.)

  1. EK[sub]G[/sub](K([symbol]f[/symbol]) [symbol]Þ[/symbol] ([symbol]Ø[/symbol]K([symbol]y[/symbol]))) — From Deduction 3, Modus Ponens

  2. EK[sub]G/sub [symbol]Þ[/symbol] EK[sub]G/sub — From Deduction 4 and the Theorems

Conclusion

CK[sub]G/sub [symbol]Þ[/symbol] CK[sub]G/sub — From Deduction 4, the Knowledge Axioms, and the S4 Axiom

The announcement is Common Knowledge

QED

15

Dammit. Both the announcement and the surprise are Common Knowledge.

16

Perhaps the surprise is that the exam is not in the field of study that the professer teaches?

Math teacher telling you to exspect an exam, then getting one in English Lit. That would be a surprise.

17

Interesting ideas, Libertarian, but doesn’t the fact that the surprise is common knowledge actually guarantee that the students will not be surprised? In other words, if it is common knowledge that the students will not know the exact day of the exam, the students would know that they should expect the exam every day, and thus it would be impossible to surprise them.

In fact this is pretty much the same as my view of the paradox. Since the announcement does not give enough information to determine the date of the exam, the students are justified in expecting the exam every day.

Your explanation of the dilemma treats surprise as a lack of knowledge. But this does not seem right, because clearly there is no possible way the students could have knowledge of the date of the exam. Thus your explanation seems to change the paradox to make it easier to deal with. I have always thought surprise meant that the students would not expect the exam, and have treated expectation as different than specific knowledge.

18

I’m not sure what you’re saying, Nightime, so if I misrepresent you, it’s unintentional. The Common Knowledge applies specifically to two salient facts: (1) the test will be given, and (2) it will be a surprise. If we consider the teacher to be truthful, then those two facts are known to every member of the group.

Even if they expect the test, they will be surprised when they’re not given it. Restating the paradox is what traditional treatments have done. What makes this treatment unique is that it does NOT restate the paradox. It just makes you look at it in a different way, namely this: the giving of the test relieves the surprise.

19

my head hurts…:wink:

20

KALT. You state:

“the pupil can only predict the exam on friday if it hasn’t been given by the end of the day thursday. it’s not yet the end of the day thursday, so to cross off the exam being on friday is pre-mature. I think that’s the nature of the paradox. To say on monday morning “it can’t be on friday” is not good
logic. The exam not being on days monday thru thursday is information the student doesn’t yet have. To make a deduction based on info he doesn’t yet have is not a proper deduction. The paradox stems from this improper deduction.”

The pupil can ask the question on Monday morning, or any time, even before the week has begun, whether the exam can be given of Friday. Look at it from another perspective - that of the teacher: Suppose you are the teacher who has to schedule the day of the exam. You have to make it a surprise (let’s
say your life depended on it). Let’s say you have to make the final and irrevocable exam schedule on Sunday afternoon and hand it in to the principal of the school. Can you schedule the exam for Friday? If no, then Friday cannot be the exam day. So you can see that it makes no difference when the students are thinking about the exam - whether they are contemplating the question on Thursday afternoon, Wednesday afternoon, or
whenever. The teacher has to comply with the rules of the paradox as well. If an exam will not be a surprise on Friday, the teacher cannot schedule it for that day, as he MUST make the exam a surprise.

SENORBEEF. You state:

“Look at it from the perspective of Wednesday morning, before the possible exam: It can happen either today, or thursday - friday being ruled out for the reasons stated in the paradox. And so one can’t be sure it’s wednesday or not - it can validly be thursday from that perspective. Reverse to tuesday, and at that point, you can’t conclusively rule wednesday out, because it can still happen on thursday. [Your solution] ignores the fact
that we have more than one choice from the “perspective” of wednesday or earlier.”

According to the logic of the paradox, on Wednesday morning (before school) you do NOT have three choices. You have ruled out Friday as impossible; by the same logic, you must rule out Thursday, since Thursday would be the last possible day once you rule out Friday. So you have only one choice -
Wednesday. So a Wednesday exam cannot be a surprise, and therefore cannot be given.

NIGHTTIME: Your example involving the 6 red cards and the 1 black card is analogous to the surprise examination paradox ONLY if the person who arranges the cards is doing it in a non-random way. For in the surprise examination paradox, it is a rule that the teacher MUST ensure that the exam will be a surprise. Which means that he must eliminate a Friday exam (unless you agree with my solution that says that even a Friday exam will be
a surprise). So the card paradox will be the exact same paradox. Now, if you are the card arranger, and you MUST make sure the person will be surprised by the placement of the black card, are you going to put the black card last? If you think that the black card cannot be last, then your statement that “there is no place the black card could be that would be preferable to any other” is false. The whole paradox begins again.

LIBERTARIAN: I don’t understand this approach you are mentioning. Perhaps you could couch it in simpler terms. What does it mean to say that there is “common knowledge”?