The surprise exam paradox

[Yakk]

(exam mon-thur) | (exam fri).

if ~(exam mon-thur) then (exam fri)
+ if (exam fri) then ~(exam fri). Contradiction
Thus ~~(exam mon-thur).

Assuming the axiom ~~X -> X, then we have:
~~(exam mon-thur) gives us (exam mon-thur).

...

That isn't the problem.

As noted, the exact same problem happens with:
1) There will be an exam on Friday.
2) You will be surprised by the exam on Friday: there is no way to prove that the exam is on Friday.

By (1), we can prove trivially that (exam friday).
By (2), this implies ~(exam friday).
Thus (exam friday) and ~(exam friday), contradiction.

Same structure. The only difference is that the original post (2) statement is a bit more convoluted in getting to the contradiction.

[silverhammermba]

By (1), we can prove trivially that (exam friday).
By (2), this implies ~(exam friday).
Thus (exam friday) and ~(exam friday), contradiction.

You're oversimplifying the problem. The student's reasoning is not "the exam must be on Friday therefore the exam is not on Friday" which indeed wouldn't make sense. Rather it is more like "we would know that the exam is on Friday, therefore the exam cannot be on Friday" which is entirely different.

The problem isn't quite so simple. They key is that the students must not know what day the exam is on; that's why the case for Friday isn't a contradiction.

[gmalivuk]

The problem isn't quite so simple.

I'm starting to think it is, though I wasn't so sure before.

The one-day case that Yakk gives is included in any multiday variation of the problem. After all, the very first step of the argument in the "next week" version is to assume Yakk's Friday version. For adding the assumption that the exam didn't happen M-F means that, what the professor said boils down to:
There will be an exam Friday, but it will be a surprise, since at the start of class on Friday, you won't be sure the exam will be that day.

The one-day version is simpler, in that it is immediately obvious that the professor's statement is not provable. That is, we can't be sure the professor is telling the truth. For if we were sure he was being truthful, we would be sure of the day of the exam, which contradicts the second part of his statement.

The students' incorrect reasoning is to conclude from this (being unsure about the professor's truth) that the professor is definitely lying (being sure the professor's statement is false).

That is, the students confuse their assumption of provability with an assumption of truthfulness. So instead of concluding (rightly) that the professor's wording makes his statement unprovable, they conclude (wrongly) that the wording has made the statement false.

(As a separate issue, they also make the mistake of assuming it's the part about there being an exam that's false. We need to be very careful when analyzing the negation of conjunctions, since there are in general two ways an AND sentence can be false.)

[Avram]

Hi This topic is interesting, so I thought I'd sign up for an account and give my 2 cents

The first claim is valid. That is, "The test can't be on Friday because if we haven't gotten the test by Thursday we'll know it's on Friday."

However, there is no way to phrase that without that crucial "if". That is if the test has not been given Monday-Thursday then it can't be on Friday.

So are you saying that the Professor could *possibly* give the test more than once? Because it seems just as logical to say "if the test has been given Monday-Thursday then it can't be on Friday.

Which I think is implied, due to the nature of exam situations. This is certainly implied in versions where it is an execution, rather than an exam, the subjects are being threatened with.

It doesn't matter what happens on M/T/W/H... F is still impossible, because it is impossible to have the exam on the last possible day

Unless--as you seem to suggest--the professor is allowed to give us the exam as many times as he wants . . .

Here's what I'm thinking:

#1: "IF we have NOT gotten the test by Thursday, THEN we'll know it IS on Friday"
#2: "IF we HAVE gotten the test by Thursday, THEN we'll know it is NOT on Friday" (assuming only one exam)

#1 implies "the exam will NOT be on Friday" (because we can't have the exam on a day when we KNOW we will have the exam)
#2 ALSO implies "the exam will NOT be on Friday"

So regardless of what happens prior to Friday, we will NOT have the exam on Friday. So the IF statement isn't actually crucial.

=====

An interesting thought... what if the prof says "you MAY have an exam next week" instead of "you WILL have an exam next week"?

Then we can no longer say to ourselves on Thursday, "AHA! It will be on Friday"

[Yakk]

Let A B C D E be the statement "The exam is on X" for the days of the week.

The statement "there will be at least one exam next week" can be encoded as:
~(A|B|C|D)->E.
~(A|B|C)->(D|E)
~(A|B)->(C|D|E)
~(A)->(B|C|D|E)

Let P(X:Y) be "there is a proof for X given Y". Note that if Z is a subset of Y, then P(X:Z) -> P(X:Y)

Then the statement "you will not be able to prove that the exam will occur when it occurs" is:
~P(A:{}) & ~P(B:~A) & ~P(C:~(A|B)) & ~P(D:~(A|B|C)) & ~P(E:~(A|B|C|D))

Does anyone disagree with my formalization?

PS: Fonkey -- damn, I've never seen a spambot who detects discussions about surprise exams, and posts in them, before!

[Drostie]

But, Drostie, we know that the student's argument ends up being wrong in the end. So either logic is inherently paradoxical or the student made a mistake. I am much more in favor of the latter.

I'm not saying that the students are right; I'm saying that your reasoning is, right now, irrational via special pleading, and if you try to correct it, you'll just get the student's case.

gmalivuk has the right idea, here. The kids have two premises:

(1) We know T ∈ {1,2,...,N}.
(2) If we know T ∈ {1,2,...,i}, then we know T ≠ i, so we know T ∈ {1,2,...,i-1}.

Denying either premise solves the paradox. Now, you might deny the second premise, because the premise would, for i=1, imply that we know T ∈ Ø, a contradiction -- consequently, (2) implies "We can't ever know that T ∈ {1,2,...,i}." You might think that this rephrasing of (2) is so counterintuitive that (2) should just be denied.

Or, you might deny (1), which gmalivuk did. And that's fine, too.

In denying (1), the key is to realize that the professor is only contingently correct, not necessarily correct -- that is, he could have been wrong. If the students had expected the test every day, then there is obviously no day the professor could give the test such that it would be unexpected. So the professor might be wrong; there might simply be no unexpected tests on any of those days.

As such, you never actually have a solid justification for your belief that T ∈ {1,2,...,N}. It could be that this is not true. And without that premise, the argument dies swiftly.

[gmalivuk]

An interesting thought... what if the prof says "you MAY have an exam next week" instead of "you WILL have an exam next week"?

In one sense, this actually is what the professor says. That is, the only information we can be sure of from his statement is that there might be an exam next week. (Which we really knew anyway.)

This is because, if we assume that the prof's statement is true (which the original students do explicitly) and that we know the prof's statement is true (which they do implicitly), we run the students' original argument and arrive at a contradiction.

So since knowing the statement is true leads to a contradiction, it must be that we don't actually know if the statement is true. So all we know is that maybe there'll be an exam next week.

[Drostie]

To be fair, we know a little more -- we know that if it is possible for there to be an unexpected exam next week (given the students' expectations), then there will be an unexpected exam next week.

Which kind of makes it even more obvious why the students' reasoning doesn't work.

[yellomellojello]

To me, it works out that the only possiblity is that the professor was lying (and I admit that the "to me" is crucial, since I have no actual experience in logical mathematics or whatever the field would be called, so I can't follow the equation things at all).

Friday can't be the date of the test because the students would know beforehand, so that seems like a definite to me. All that does is simply move up the last possible date, so the same logic would repeat. Silverhammermba wrote, "3. Come Wednesday, the student does not yet know if the exam will be given on Thursday or not and thus cannot rule out Friday as well. " But that can't be true, because the students already know immediately that Friday cannot be the test date, because it wouldn't be a suprise. Since friday can't be the test date, thursday can't be the test date...

The only other possibility would be that the professor is aware of this logic and so knows that the student expects the test not to occur. But... er.... that just seems kind of lame, so I really don't want that to be the solution.

[silverhammermba]

If the students had expected the test every day, then there is obviously no day the professor could give the test such that it would be unexpected.

That's my biggest problem with every phrasing of this question. The definition of "unexpected" is far too vague in every case that I've seen. What we really need is a complete rephrasing of the question in clear terms so that we can really tear apart the logic. What exactly entails being "unexpected"? Should the question instead say "In order for the test to be given, the students must not be able to figure out the day of the test before the next week begins."?

Also, perhaps I'm just stubborn, but I'm convinced that there is a way to find the flaw in the student's logic without assuming that the professor was lying in his original statement. Plus, the problem is far too simple if we change it to "There may be a surprise exam next week."

[Drostie]

The professor wasn't lying -- he was retroactively telling the truth.

But his statement is not necessarily correct -- there exist possible worlds wherein the professor makes this statement but fails to be correct with it.

[adlaiff6]

The professor's logic is inherently flawed. He cannot hope to create the scenario he dictates.

[bbctol]

The professor's logic is inherently flawed. He cannot hope to create the scenario he dictates.

And yet the test comes, and the students are surprised.
Seriously, people, you can't say that the teacher is lying, or that the student logic is correct, because the TEST COMES, and they're SURPRISED.

EDIT: Epiphany!
If the test comes on Monday, even keeping their logic in mind, they would still be surprised, correct? They know it can't come on Friday, because they know it wouldn't be a surprise. So they work backward, up to Tuesday and then (here comes the flaw) they say it can't be MONDAY, too. But they're logic is wrong, because there are no days before Monday. So we can work backwards, like we did before, and say that it CAN happen, and be a surprise, on not just Monday, but Tuesday, Wednesday, and Thursday, but NOT Friday, because there are no days AFTER Friday that it can be on, just as there are no days BEFORE Monday that it can't be on. SO it can be on any day but Friday and be a surprise.

Sorry if this didn't make any sense to anyone, but I'm bad at explaining things. But I DO actually understand it! Finally!

[Drostie]

I don't think I agree with your method, bbctol.

Again, look at the students' two premises:

(1) "We know that the test is on {1,2,...,N}."
(2) "If we know that the test is on any set S = {1,2,...,i}, then we know that it is not on i, so we know that it's on the set S - {i} = {1,2,...,i-1}."

You're saying that the jump from {1,2} to {1} is justified, but the jump from {1} to {} is not. But you don't have any good basis, I feel, to claim that the latter move is unjustified.

Moreover, you seek to say that the students were right to say "it can't be on Friday" but wrong to say "it can't be on Thursday." As I've commented elsewhere in this thread, that explanation seems wholly irrational.

(1) and (2) together imply a contradiction -- the contradiction is that the test is in the empty set (but nothing is in the empty set, so the test can't be in it).

The straightforward question is, which one of (1) and (2) is false? You can argue for either one.

[Vi]

In the original post, the students think the quiz actually won't be on any day of the week, and won't happen at all. Therefore even if it was on Friday, they would be surprised.

I think one of the problems is that "surprise" is not a very quantifiable term. "Surprise" does not equal "logically it could be a different day besides today." Expectation is a human quality and not a logical one.

[bbctol]

@Drostie
Okay, not having studied any useful set math, let me try this:
On every weekday with at least one day after it, the test will come as a surprise.
If the students are not 100% sure that a test will be on a day, and it is on that day, it is a surprise.
On every day with at least one day after, the test could come tomorrow, so they aren't 100% sure.
Every day except Friday has a day after it.
So even thought the test can't be on Friday, Friday counts as "a day", so the test could be a surprise on Thursday.

[Drostie]

You haven't yet factored in the fact that the test can only be given on a day where it would be a surprise. (That is, everything you've just said is a treatment of a very general case.)

So, when you say that, on Thursday, "the test could come tomorrow," what do you really mean? Could it, if all the kids are expecting it?

[silverhammermba]

I think one of the problems is that "surprise" is not a very quantifiable term. "Surprise" does not equal "logically it could be a different day besides today." Expectation is a human quality and not a logical one.

Exactly. One of my first thoughts when seeing this problem was this:

The test must be a surprise.
The students conclude that Friday is of course absolutely impossible.
From this they conclude that no day would work.
Thus if the professor gives the test on any day (Friday included), it will be a surprise since the students have concluded that it is impossible.

I have yet to think up a simple, logical way of stating the problem that accommodates such a relation.

[Macbi]

The teacher makes a prediction:

1. You (the students) will not be able to, on the morning of the test, walk into the room and say "Sir, the test will be today" with 100% certainty.

Now we can set it up like a game:

The students win if:
On the morning of the test they walk in and say "Sir, the test will be today" and don't say that on any previous day.
or if
The teacher prediction is wrong.

The teacher wins in any other case.

Notice how in the prediction the teacher says "on the morning of the test", therefore he must set a test for that prediction to be right. This means that the students know that there will be a test.
If it gets to Friday and the test has not been yet (and the students have not made an incorect guess) then the students have a winning strategy where they say "Sir, the test will be today" either they are right (They win) or they are wrong and the teachers prediction was incorrect since there was not a test (They win). To stop them using this winning strategy the teacher must either put the test before Friday, or force the students to make an incorrect guess. This means on Thursday the students either guess it will be today, but this invalidates their winning strategy for Friday if they are wrong, or they don't guess it will be today, and run the risk of being wrong. Either way they stand a 50% chance of being right which is what they would have got without using all of that logic.
Similar reasoning works for all other days.

[iw]

Let's go Turing machine here:

Let's say we have a probabilistic Turing machine. It works in the following fashion: after reading a square, it either halts or moves to the right, and it halts when it reads a blank. We want to know how many steps it takes before it halts.

Now, let's say your professor takes a tape with 4 characters on it and runs the machine in secret. He announces, "At no point did I know what the machine was going to do next."

All of a sudden, the paradox vanishes. We have no idea how many steps it took.

Now let's say we have a psychic that claims before we run the machine that "at no point will you know what the machine does." Now we start thinking, "gosh, it can't go 4 steps, because we will know what the 5th step is. And if we know it can't go 4 steps..."

Let's look at it as a directed graph with a marker:

Code: Select all

  T  T  T  T  X
 /  /  /  /  /
1--2--3--4--5


The marker is currently at node "1". The graph is directed to the right and upwards. We move the marker until we reach a terminal node. So what we want to do is make a new restriction that says, "at no point will we be able to predict where the marker will go."

How can you put that into terms? We could try, "at no point will there ever be one and only one possible path for us to take" but it's not what we want. After all, at node 4, we think, if we move to node 5, that will force us to break the rule, and therefore our next move must be to "T". Intuitively, what we're thinking is that node 4 must somehow be illegal as well. But then we'd want node 3 to be illegal for the same reason, etc., etc.

Here is what I think the answer is: there is no way to express such a restriction. Therefore, when the professor says, "the day of the test, you will not know whether I am going to give the test or not," he's basically saying something completely meaningless in terms of the scenario, yet somehow intuitive to humans. It's a logical illusion.

[Yakk]

Macbi, there is a problem: if the students said it would be on Thursday, and where wrong, they wouldn't be surprised when the test showed up on Friday.

...

Try throwing out the ability to prove things by contradiction. Do you still end up with an inconsistent system? ... probably, this is a finite set of possibilities.

[noneuklid]

Well... Ignoring a lot of the above and answering the riddle, the answer is No -- all that really mattered was the possibility that the professor was lying.

That kinda screws up the whole riddle right there, but basically it's like this. On any day of the week, including Friday, there is the possibility that there will or will not be a test... because if the professor was lying, there might NOT be a test on Friday, and thus there was no logical way to conclude that there would be.

Now as to the rest of what y'all are saying, I ain't got nothin' on your fancy diagrams, but I suspect you're representing the problem a bit off. The test could be any day of the week except Thursday or Friday and still be suprising, since you still have 4-N chances remaining. As long as 4-N is greater than one, the test will be a suprise.

[Macbi]

My point was that they only get one guess, since other wise they could just guess everyday.

[Yakk]

Noneuklid, what if there is no possibility that the professor is lieing about there being a test next week? None at all.

[noneuklid]

Noneuklid, what if there is no possibility that the professor is lieing about there being a test next week? None at all.

Edit: Hmm. Misread something somewhere.

Then my second argument takes over, regarding 4-n > 1.

[Disruptive Idiot]

Isn't this true? : the professor's second proposition is false. There is a strategy the students can use to correctly anticipate the test on any day of the week. If they simply expect the test every day, eventually they will correctly expect it and prove the professor wrong.

[bbctol]

So, when you say that, on Thursday, "the test could come tomorrow," what do you really mean? Could it, if all the kids are expecting it?

Thursday morning, you know that IF the test is not on Thursday, it must be on Friday. If it isn't on Thursday, it won't be a surprise. Which means it must be Thursday, which means if it's on Thursday, it won't be a surprise. So it's just been proven that it can't be on Thursday, and it can't be on Friday. But for it to be a "surprise", it's on a day when you don't expect it. So because you've proven that it isn't on Thursday, it will be a surprise if it's on that day.
You do have to admit that, no matter what, it would be a surprise on Monday, though.

[eviljebus]

Here's my thoughs:

I belive the problem lies when the students conclude, "There is not a test next week". Now, their logic is quite sound up until this point, but take a step back and look at the whole situation at this point. The students belive there will be no test any day next week, and they do not expect a test on any day of the next week. If they do not expect a test any day, then the test can be on any day of the next week. Fom here only question left, and (as I belive) the only REAL question, is "Will there be a test".

Going back to the original problem, "There will be a test on a day next week when don't expect it". From that condition I can logically conclude "If they don't expect a test next week, there will be a test next week". Now, do they expect a test? The problem specificly says "they do not expect a test", therefor we can conclude that there is a test because they don't expect a test because they don't belive there will be a test.

Now, I realise I didn't actually answer the real question the problem asks, "Is their reasoning incorrect". As for that, I say it doesn't matter if their reasoning is correct or not it is a giving in a problem. If you were given y=x+5, and told x = 5 and y = 10, you wouldn't ask, is X really 5? It has to be in the context of the problem. As I see it, is there is no "pefect" reasoning if you were the students (that leads me to an infinite loop of logic), but I belive I have proven the scenario works only if that is indeed their reasoning&conclusion, and therefor it can't be anything else.

[Avram]

PS: Fonkey -- damn, I've never seen a spambot who detects discussions about surprise exams, and posts in them, before!

What are you saying? That I'm not contributing anything worthwhile to the discussion?

[Yakk]

PS: Fonkey -- damn, I've never seen a spambot who detects discussions about surprise exams, and posts in them, before!

What are you saying? That I'm not contributing anything worthwhile to the discussion?

Nope, that you are a really smart spambot.

See, non-spambots post to the "Introduction thread" first. :)

It is a joke on the rules of the board.

[notzeb]

Spambots can also post in the Introductions thread.

Most of us are a bit dense, though...

[iw]

A question: Have logicians or philosophers discussed this problem before? Is there some kind of "official" word on this?

[Drostie]

Quine did a relatively famous version, though he didn't claim to be the author.

I'll need to look through my notes to find out more on this.

Edit: The Oxford Journal, Mind, was host to Quine's paper on the subject, as well as a bunch of further discussion. If your college has a subscription to that journal, then the article title is "On A So-Called Paradox," 1953.

Quine basically concludes just what I did: That if the students are going to conclude "There is no test," then they must admit that this is a possibility from the start; and such a position voids their inductive rule, since, on Friday morning, they don't know whether there's either a test on Friday or no test at all.

Chapman and Butler, in 1965, offer an even more interesting take in the same journal, called "On Quine's 'So-Called Paradox'". If you look at my above post where I quoted silverhammerba, you'll see that I divided the kids' argument into two premises. Quine sorta denied (1); Chapman and Butler deny (2) by giving a good definition of "unexpected":

Let us first examine what the schoolmaster actually said. He made two statements: (a) that there would be an examination on one of the five afternoons of the following week; and (b) that the examination would be unexpected. We may define 'unexpected' in this context as meaning that by no process of valid logical argument can the boys at any time predict without contradiction the day of the examination....

...The argument eliminating the last day itself contains two arguments:
(i) The examination must be held on the last day because on the morning of the last day it is the only day left.
(ii) Because of (i) the examination is expected on the morning of the last day and therefore, by (b), cannot be held on that day.
Note that both arguments place the arguer hypothetically on the morning of the last day.

What has been proved by these two arguments is that, for a boy arguing on the morning of the last day, it is necessarily the case that the examination both is and is not held on that day. The conclusion that the examination must be held on the last day is just as warranted as the conclusion that it cannot then be held. Therefore the boys cannot predict, by a valid process of logical argument and without laying themselves open to contradiction, that the examination will be held on the last day. Therefore the examination, even if it is held on the last day, will be unexpected in the required sense.

I might just like this version even more than I like my/Quine's version.

[Drostie]

R. Shaw's 1958 "The Paradox of the Unexpected Examination" in Mind is also a good one.

He makes a good case that at least one version of this paradox is reducible to self-referential paradoxes like the liar paradox. Consider these two rules:

(1) There will be a test on one of the days next week.
(2) Students will not be able to predict, based on (1) alone, which day it is on.

Then there is no contradiction, unless the semester is only 1 day long. Because if there are 2 days, the exam can rightfully be set on the first, and you could not have, only using (1), determined which day it was on. You could also add a premise like:

(3) Students will not be able to predict, based on only (1) and (2) together, which day the test is on.

This bars the last day and the day before that, naturally; but it does not bar any other days, for the same reason. Shaw argues that the error only occurs when you change (2) to (2'), which is:

(2') Students will not be able to predict, based on both (1) and (2') together, which day the test is on.

Shaw argues that this is self-referential and hence invalid.

Even more interesting is G.C. Nerlich's article in 1961, where he argues that the above doesn't actually specify which day it is going to be; so the paradox remains intact -- and then offers his own solution.

I'm going to leave it at that. You must get your hands on a JSTOR subscription if you wanna read what everyone else in the journal had to say.

[lazydrumhead]

this is my attempt to make everything incredibly simple:

can't we say that our conclusion about friday follows the same rules as the result of the 'paradox' ?

can we not say, "on friday, the teacher cannot give us the exam, because it is the last day of the week (which would not surprise us, therefore not following what he said about the exam occuring during this week). however, he might use this tactic against us and surprise us by giving it to us on friday if we thought we could not have the exam on friday"

therefore friday could possibly be a surprise
therefore any day could possibly be a surprise

[iw]

this is my attempt to make everything incredibly simple:

can't we say that our conclusion about friday follows the same rules as the result of the 'paradox' ?

can we not say, "on friday, the teacher cannot give us the exam, because it is the last day of the week (which would not surprise us, therefore not following what he said about the exam occuring during this week). however, he might use this tactic against us and surprise us by giving it to us on friday if we thought we could not have the exam on friday"

therefore friday could possibly be a surprise
therefore any day could possibly be a surprise

It's not quite that simple: the student's reasoning that the exam cannot be given on Friday (assuming the professor keeps his word) is completely sound, because it has to be a surprise that morning. It's the resulting induction that makes things sticky.

[lazydrumhead]

no, if he has tricked his class into believing that there would be no test on friday (by making them think he lied about having it during the week due to their logical analysis) then his class would NOT believe they were having the exam on friday, therefore it would be a surprise to them on friday if they recieved the exam.

[Drostie]

It is worth asking, if your professor said, "There will be an unexpected test tomorrow," does the same issue occur? (Quine would say it could. Consider what happens if everyone says, "Well, we're going to expect it tomorrow; so it's impossible" and then fails to expect it. Chapman and Butler would extend this by saying that the students cannot come to the desired conclusion by any logical manner. And Nerlich would argue that the primary paradox, here, is that the students' argument doesn't specify a day -- whereas on the Friday-only version, it can.)

[iw]

It is worth asking, if your professor said, "There will be an unexpected test tomorrow," does the same issue occur? (Quine would say it could. Consider what happens if everyone says, "Well, we're going to expect it tomorrow; so it's impossible" and then fails to expect it. Chapman and Butler would extend this by saying that the students cannot come to the desired conclusion by any logical manner. And Nerlich would argue that the primary paradox, here, is that the students' argument doesn't specify a day -- whereas on the Friday-only version, it can.)

I still think that this gets too hung up on the idea of "expected", because the problem still exists without that concept. Let's say you split it up, and said, "There will be a test tomorrow." Taking that as truth, he then says, "you won't know if I give it tomorrow or not," which is clearly false, and therefore the professor is lying. But if you extend those same restrictions to five days, the paradox appears. So, looking back at the original problem, the students are perfectly justified in ruling out Friday. It's recursing on that conclusion that gives a problem.

I think it's better to look at it as flipping a coin five times and knowing
that heads will come up at least once AND you will not know which flip will be heads right before you flip it.

[Yakk]

Demonstrate why "There is a test tommorrow" and "You cannot know what day the test is on" means the professor is clearly lieing, while the OP doesn't.

Logic:
There is a test tommorrow.
But we cannot know there is a test tommorrow.
So because we know there is test tommorrow, there cannot be a test tommorrow.

Tommorrow comes, there is a test, and we are surprised.

Same shit, different pile.

If you cannot accept the possibility that the logic professor creates a self-referential contradictory system...