The surprise exam paradox

[Ermes Marana]

What it all comes down to is that it is possible for me to make a true statement that you are nevertheless incapable of believing.

The professor is telling the truth, and there is no contradiction for the professor to believe his own statement. But it is impossible for the students to believe him, because the statement references their state of mind, and believing it would lead to a contradiction for them. On Thursday night they cannot believe both that the exam is tomorrow and that they will be surprised. Therefore they must reject the professor's statement, even though the professor continues to know it is true. Thus the Friday exam is a surprise.

An easier example is "it is raining outside, but you don't believe that." Obviously this statement can be made truthfully, but it is impossible for the person you say it to to believe you.

[Garrator]

The reasoning fails because the teacher "takes adavantage" of it, so to say. Because they eliminated every day, any day the exam would be unexpected. Therefore they can't eliminate any day. Once they eliminate a day, that allows the exam to be given to them on that day without them expecting it. I still don't get why, but this is how far I've gotten.

[ocdscale]

To jump to my conclusion: It is impossible to 'surprise' students with a test in a logical sense, but that does not hold true in the real world.
The students' argument is operating in the logical world.
But the paradox's conclusion: "However, the next week, on Wednesday, they have the exam. And all are surprised." is a real world phenomenon, separate from logical truth.
The cleverness of this paradox comes from how well it makes the student's argument seem so reasonable (which I will argue against).

Let's imagine a conversation between the students and the Professor, Thursday morning before class:
Students: Professor, you haven't given the surprise test yet. That leaves only today (Thursday) and Friday available. We know it cannot take place Friday...
Professor: Why can't I give it to you on Friday?
Students: Obviously if we don't get the test today then Friday is the only day left available to you, it will no longer be a surprise.
Professor: I see... but what if I surprise you with the test today?
Students: That's what we were explaining before you rudely interrupted. As we've shown, the test cannot occur on Friday, therefore it must occur today, it is no longer a surprise.
Professor: You got me, congratulations.

Does this exchange fairly capture the student's argument? If not, why not?

Going backwards in time like this, it seems the student's argument is completely logical and completely reasonable. But what happens if we start from the very beginning?

Monday morning, the students approach the Professor and outline their entire argument why the test cannot occur on Friday, Thursday, Wednesday or Tuesdaif it is to remain a surprise. They conclude that if the test is to remain a surprise, it must occur on Monday. However, because the students know this, any test given on Monday will be no surprise at all.
Fair is fair, the Professor agrees and does not give the surprise test on Monday.
Tuesday morning, the students approach the Professor and reiterate their entire argument.
Wednesday morning, same thing.

What is happening? The students are basically arguing that the test cannot be a surprise because they will be "expecting" it every morning. If they are wrong about their expectations, they' will just expect it the next day. Ergo it will not be a surprise.
It is no different than if the students simply said to the Professor: "It is impossible for you to give us a surprise test next week. Every morning we will expect the test to occur on that day. If the test does not come to pass, we will expect it the next day."

These hypothetical students are impossible to surprise with a test. But of course the real world doesn't work that way (which the conclusion of the paradox suggests).

[mrbaggins]

Students: Professor, you haven't given the surprise test yet. That leaves only today (Thursday) and Friday available. We know it cannot take place Friday...
Professor: Why can't I give it to you on Friday?
Students: Obviously if we don't get the test today then Friday is the only day left available to you, it will no longer be a surprise.
Professor: I see... but what if I surprise you with the test today?
Students: That's what we were explaining before you rudely interrupted. As we've shown, the test cannot occur on Friday, therefore it must occur today, it is no longer a surprise.
Professor: You got me, congratulations.

Or how about the slightly altered version, which is what actually happens and (again) shows how the students logic is flawed.

Students: Professor, you haven't given the surprise test yet. That leaves only today (Thursday) and Friday available. We know it cannot take place Friday...
Professor: Why can't I give it to you on Friday?
Students: Obviously if we don't get the test today then Friday is the only day left available to you, it will no longer be a surprise.
Professor: I see... but what if I surprise you with the test today?
Students: That's what we were explaining before you rudely interrupted. As we've shown, the test cannot occur on Friday, therefore it must occur today, it is no longer a surprise.
Professor: But you know what?
Students: What?
Professor: SURPRISE! <Pulls out tests>
Students: You can't do that!
Professor: Were you expecting to do the test today?
Students: No...

[dawolf]

Easier way to look at this.

The students go through all their reasoning over the weekend. On Monday morning, the teacher asks them all to write down which day the exam will happen on. They would have to guess a day at this point, they wouldn't actually know, regardless of the logic they used. That shows clearly that the logic is flawed.

[Karandr]

Assuming that the professor is more clever, crafty and devious than his students, let's say he predicted they would come to the "It can't be any day" conclusion. When he made the statement, he knew the students would conclude that the exam could not happen at all. Thus, he just picked a day. The exam happened, which was a surprise to the students in its own right, as they were not expecting a exam at all.


Prof.: You'll have a test next week, but I'm not telling you what day. Also, you will not expect the exam.
Students: (using the logic in the puzzle) The test cannot happen on any day, thus, we will not have a test. Hah!
Prof.: Hah yourself. I knew you would conclude there would be no test. Thus, the test on any day will come as a surprise.


I'm not sure if I've proven anything here or just re-stated someone else's argument.

[BurningLed]

Well, if the thought process goes, step-by-step, ass uch:

1. If we make it to Thursday with no exam, then we would know it is on Friday.
2. It would not be a surprise then,
3. We can assume it is not on Friday
4. Therefore it has to be Monday-through Thursday
5. If we make it to Wednesday with no exam, we would know it is on Thursday, because it cannot be Friday.

Repeat this until Monday, and we still have an equal probability of it being any day. Because the students suspect it could be any day, they are still at square one; and will remain there because there is no logical way to determine this. But this constitues the 5 pages of discussion reviously.

Or: Professor lied, and gives the exam the Monday of the week afterwards. Nobody suspected, right?

[Yakk]

The Professor says "there will be a surprise exam on Monday".

The Students all say "well, he told us there is an exam on Monday -- so we expect it. Thus we aren't going to be surprised. Thus there cannot be an exam on Monday!"

The Professor has an exam on Monday. The students go "oh noes, you out smarted us".

/thread.

No really. End thread.

Damnit. How do I unsubscribe to a thread?

[rm2010]

I got this exact question in a job interview!

[koobz2142]

Like said before, the idea that you can rule out more than one day relies on the satisfaction of the condition: "has been four days w/o an exam"

if it hasnt, you cant prove friday has the test, or thursday, etc.

Ive heard this another way where it is an experienced substitute teacher
teacher says "will have suprise test"
kid says "cant be friday, cuz we'll know"
sub says "allright, WILL NOT BE FRIDAY"
teacher has already ruled out friday, now restricted to monday-thurs.

AND SO ON.

Assuming the teacher is smart, the only day it cannot be on is friday

[math]

Spoiler:

By reasoning it out and eliminating Wednesday, they all assume that it will not be Wednesday. Which makes picking it a surprise.

[bobleboffon3]

Does this exchange fairly capture the student's argument? If not, why not?

Going backwards in time like this, it seems the student's argument is completely logical and completely reasonable. But what happens if we start from the very beginning?

First of all, I'll say that I don't really like the wording of this version.
The whole "surprise" thing is too hard to define.
I'd use "You will not know when the exam will be" rather than "You'll be surprised by the exam".

Now, let's replay your dialog/the problem using the presumed "knowledge" about the exam date.

The student says : It can't be friday, because if thursday we don't get the exam we will know it's gonna be friday.

This is right ( in fact, not even 100% right. But let's keep going ). Everything else is wrong.
The studend says : Now that we know it can't be friday, if we don't get the exam by wednesday night, we will know it's thursday.
This is where the studend is wrong.

The student says "I know the exam is today - thursday" because it can't be friday.
Let's say the exam doesn't come.
Friday the student says "I know the exam is today because there's no more days left".
The teacher gives the exam, and the student says he can't give the exam, because they knew it would come this day.

This is correct for friday, but let's look back at thursday.
The student said "I know it'll be today - thursday", and then "I know it'll be today - friday".
His first declaration wasn't a fact. It was a guess.
He said "I know" but he was wrong. It wasn't a 100% odds. He never knew for sure. He has no way to know, and the fact that the teacher can just not give it on thursday, is a proof.

So if the teacher decide to give it on thursday, and the student says he knew it, he's wrong. He guessed it, and was right. That's not the same as knowing it.

It's easy to figure out the student logical flaw if you go to one of many extreme, as example, the infinite.
Day 1 : I know it's gonna be today! But it's not.
Day 2 : I know it's gonna be today! But it's not.
...
Day N : I know it's gonna be today! Teacher gives the exam, student says he knew it... but he didn't. He incorrectly guessed N-1 times, and correctly guessed once. That's not knowing something.

Another way to look at the problem :
Let's say there's no exam on monday, tuesday, and wednesday.

Thursday comes, and the student know it can't be on friday, so he declares : I know the exam is gonna be today.
The teacher then says : Ok, are you willing to bet your life on it? If the exam is today, I give you a 100% mark.
If the exam is not today, I shoot you to death.
The student will never accept this bet. Because he doesn't "know". He strongly believes, at best.

Without risking his life, it's easy for him to say he "know" on monday, then "know" again on tuesday after he miss on monday, and then know again on wednesday after he miss on tuesday.
If there was a consequence to wrong guesses, he wouldn't try to guess. Because that's all it is, just guesses. He doesn't know anything.

[Turtlewing]

the students never establish a valid base case.

they say that if the test has not occurred on day 5 than it must occur on day 5, and therefore cannot occur on day 5.

thats:
no (test on day 1, test on day2, test on day3, test on day 4) -> test on day 5
test on day 5 -> no test on day 5

That's a contradiction.

[phlip]

That's a contradiction.

Exactly.

[jestingrabbit]

Thursday comes, and the student know it can't be on friday, so he declares : I know the exam is gonna be today.
The teacher then says : Ok, are you willing to bet your life on it? If the exam is today, I give you a 100% mark.
If the exam is not today, I shoot you to death.
The student will never accept this bet. Because he doesn't "know". He strongly believes, at best.

Okay, but what if we were to turn this on its head a bit. What if when the examiner said "you are going to have an exam next week. I'm not telling you which day, but I am telling you that it will be unexpected (i.e., the day of the exam you won't be sure whether the exam is that day or not)" and a student were to pipe up with "Ok, are you willing to bet your life on it? If I declare on the day that you have it planned, that that is the day its on, then I get to kill you, and otherwise I get 0% on the exam" the examiner would never take that bet either.

The stakes for the two people in the wager are gigantically different in both cases, to begin with. But, more than that, the examiner doesn't really "know" that their pronouncement is true.

[argenbar]

To me the issue with this paradox is that it relies on time dependant knowledge.

In order for the test to occur Friday the following must be true:
- Thursday must have happened
- The test did not occur Thursday

If Friday is the only option left available, it is no longer a surprise - therefore, if by Thursday the test has not occurred, it cannot occur (unless the lecturer lied).

If we try to prove that the exam cannot happen Thursday, the following must be true on WEDNESDAY
- Wednesday must have happened
- The test did not occur Wednesday
AND the case for no exam Friday must also be true.
- Thursday must have happened
- The test did not occur Thursday

So for the logic to hold up, you need to have knowledge of what will have occurred on Thursday before it happens.

To summarise;
- If the exam has not happened by Thursday one can reasonably deduce that the exam will not happen on Friday.
- The logic cannot be applied earlier in the week as you need information of future events.

[RonWessels]

To me the issue with this paradox is that it relies on time dependant knowledge.

How about the simpler version of the same paradox posted by Yakk earlier in this thread:

The Professor says "there will be a surprise exam on Monday".

The Students all say "well, he told us there is an exam on Monday -- so we expect it. Thus we aren't going to be surprised. Thus there cannot be an exam on Monday!"

The Professor has an exam on Monday. The students go "oh noes, you out smarted us".

To me, this "paradox" is equivalent to the "I am telling a lie" statement, just disguised enough to not be obvious.

[Yakk]

Actually, I posted it in the very last post of this thread. You can see that I ended the thread with my post. I even used a slash.

[Ermes Marana]

To me, this "paradox" is equivalent to the "I am telling a lie" statement, just disguised enough to not be obvious.

I disagree, because the "surprise exam" statement is a clear, simple, true statement that does not refer to itself. Very different than "I am telling a lie."

It is equivalent to "it is raining outside, but you don't know that." The "trick" is just that it is possible to make a clear, simple, true statement that logically cannot be believed by the person you say it to because it refers to their state of mind.

[The Chosen One]

If the students decide the test cannot occur on a day because they know it will, then the professor will do the test on ANY day of the week (Yes, including Friday) and it will still be unexpected to the students BECAUSE they expected it.
Some students may argue that the test can't occur because they expected it, but that reaction in itself is an indicator that they didn't expect it.
Also, the professor is Evil and will force the students to submit. (Their exams, that is. Jeez.)

[bobleboffon3]

Thursday comes, and the student know it can't be on friday, so he declares : I know the exam is gonna be today.
The teacher then says : Ok, are you willing to bet your life on it? If the exam is today, I give you a 100% mark.
If the exam is not today, I shoot you to death.
The student will never accept this bet. Because he doesn't "know". He strongly believes, at best.

Okay, but what if we were to turn this on its head a bit. What if when the examiner said "you are going to have an exam next week. I'm not telling you which day, but I am telling you that it will be unexpected (i.e., the day of the exam you won't be sure whether the exam is that day or not)" and a student were to pipe up with "Ok, are you willing to bet your life on it? If I declare on the day that you have it planned, that that is the day its on, then I get to kill you, and otherwise I get 0% on the exam" the examiner would never take that bet either.

The stakes for the two people in the wager are gigantically different in both cases, to begin with. But, more than that, the examiner doesn't really "know" that their pronouncement is true.

Ok, maybe risking his life against a 100% mark ins't a fair bet.
Let's say, in this case, that the teacher is willing to bet 100$ with him on these conditions :

Everyday at 8:00 AM the teacher will ask if the exam is today. If the student answer "Yes" and the exam is actually this day, he wins $100. Else, the student owes the teacher $100. Let's say for the purpose of this example that the student doesn't mind gambling when he knows he has the edge.

Would the student accept this bet? Obviously not.
IF the student says "No" for 4 days, and the exam don't happen, THEN he will say, on friday, that the exam is today. And win $100.
But the teacher isn't dumb, he will not put the exam on friday.

At any of the other 4 days, the teacher can put the exam. If the student say the exam will happen on this day? The teacher says "no" and put it on one of the remaining days, and collect $100.
If the student doesn't say the exam will happen on this day, the teacher tells the students the exam is on this day, and again the student lose $100.

This shows exactly that the student doesn't know. He guess.
The 5 days confuse us, but again, giving more time makes us see how it actually is.

If there was 10000 days to give the exam.
The student would have the same reasonning... Can't be on the 10000th day because I'll know it, so it can't be on the 9999th day because I'll know it so it can't be on the 9998th day because I'll know it, and so on...
But he will not know for any of those days, except the last one, and (arguably) the 9999th.

He will be able to say "I know it's gonna be today" ten thousand times, but it'll be a lie/a guess all but one time.

[Turtlewing]

That's a contradiction.

Exactly.

I'm confused. Your link is to an article on a proof by contradiction. which implies that you're claiming that the students are trying to use a proof by contradiction to establish a base case for the inductive proof they use to iteratively eliminate the remaining days of the week.

However, that's not how their argument is actually structured.

their argument is:
let next week be the set of days {day 1, day 2, day 3, day 4, day 5} which occur in order.
let there be a test on one day next week
if it can be proven that the test must occur on a given day, that day will not be the day of the test. // This line is where the proof falls apart

no (test on day 1 or test on day 2 or test on day 3 or test on day 4)-> test on day 5
test on day 5 -> no test on day 5
no test on day 5 and no(test on day 1 or test on day 2 or test on day 3) -> test on day 4
test on day 4-> no test on day 4
no (test on day 4 or test on day 5) and no(test on day 1 or test on day 2) -> test on day 3
test on day 3 -> no test on day 3
no (test on day 3 or test on day 4 or test on day 5) and no test on day 1 -> test on day 2
test on day 2 -> no test on day 2
no (test on day 2 or test on day 3 or test on day 4 or test on day 5) -> test on day 1
test on day 1 -> no test on day one

Therefore since all days in next week have been eliminated as days on which a test can occur the test cannot occur next week.

The problem is that they're repeatedly using a contradiction as if it's a valid line in an inductive proof. "test on day 5 -> no test on day 5" being a contradiction can only be used to disprove things who's truth would imply it. Not to prove things that require it in order to be true.

After working through this I think I see the confusion:

The students are trying to use a proof by contradiction that there cannot be a test next week. However they're actually mis-using an inductive proof structure and passing it off as a proof by contradiction.

The form of a proof by contradiction is:
To prove P
not P-> Q
Q->not Q
therefore P

what the students are doing is:
to prove P
not P -> Q
Q-> not Q
not P and not Q -> W
W-> not W
not P and not Q and not W -> X
...

You get the idea.

Put more simply:
once you get to a contradiction it's "proof over" you've either got a valid proof by contradiction or you have a failed proof. You can't just say "bare with me a moment" and keep going as if you haven't stated a contradiction.

[phlip]

Put simply... the reasoning starts with "assume the test will be on Friday" and ends with "we will be able to predict the day of the test"... the latter is in contradiction with our given rules, so we have proven the opposite of the original assumption. The conclusion is not "If the test is on Friday then the test is not on Friday", but simply "The test will not be on Friday". Standard proof-by-contradiction.

Also, "P -> not P" is not a contradiction. "P and not P" is. "P -> not P" is logically equivalent to simply "not P".

[Ddanndt]

Well in my opinion the students' reasoning is erroneous:
STEP 1 :

\text{if there is an exam on Friday }\Rightarrow \text{*there was no exam on any day before}\Rightarrow \text{exam can be predicted to be on Friday} \Rightarrow

\text{no exam on Friday }

which is correct

Now using the professor's statements and the result derived from step 1 , the students find in a similar way :

STEP 2 :

\text{if there is an exam on Thursday }\Rightarrow \text{there was no exam on any day before}\Rightarrow \text{exam can be predicted to be on Thursday} \Rightarrow

\text{no exam on Thursday }

which is this time incorrect.

This is because when the students assume that the exam is on Thursday, the implication in STEP 1,

\text{if there is an exam on Friday}\Rightarrow \text{there was no exam on any day before}

becomes false( T imply F) and the whole thing collapses. Well dunno if my reasoning is too simplistic ...

[OllieGarkey]

So where is the reasoning incorrect ?

Either 1) The Evil Professor and the students have different definitions or
2) they reasoned that the prof was honest.

[Ddanndt]

Here's my second attempt to the problem:

Spoiler:

The students should just imagine the week as the Venn diagram below:



If there is no exam from Mon to Thurs this implies that elements Mon to Thurs are in B. Fri can't be in B because the students know there must be an exam next week. Therefore they now assume that Fri must be in A but because that event can be predicted, they know that Fri can't be in A. Therefore they should get:




Now the students know that the exam is next week so the above config is incorrect...
Therefore they have to move an element from B to A (why don't they consider all configurations possible etc moving only Mon, Tue or Wed from B to A and these are unpredictable because they have some chance of happening?)

So the students decide that Thurs isn't in B and move it to A. Of course this event is predictable therefore Thurs isn't in A. And they continue their reasoning until they get:



But using the students' logic weekdays are either Exam Days or Non-Exam days . So their conclusion should have been somewhere along the lines of "next week doesn't exist". Of course next week will come so they can conclude that their teacher sucks at logic...

[Turtlewing]

Also, "P -> not P" is not a contradiction. "P and not P" is. "P -> not P" is logically equivalent to simply "not P".

OK fair enough, but:
P -> Not P
P

reduces to:
P and not P
(because P is true, but P being true implies that P is false).

The contradiction is subtle because it's divided into two steps (establish that P is false, the attest that P is true), but it's still there. Their argument that any given day can't be a test day, is based on the proof that that day must be the test day. This is (P -> not P and P) which is (not P and P).

[Darrell88]

I think that the subtlety here is predicting that an exam will take place and predicting that an exam won't take place.

Let's assume on Thursday evening the students predict that the exam will take place on Friday and stop they their reasoning there. On Friday morning they tell their professor "Today is the exam day". Of course it's impossible for the professor to give the exam because the students have predicted it.

Whereas if the students apply their full logic on Thursday evening and they predict that the exam can't occur on Friday. On Friday morning they tell their teacher "We have predicted that the exam is not today". This time the professor can still give the exam because the students haven't predicted the exam.

The flaw in their logic is that they didn't realise that if the exam occured on a day that they had ruled out by their previous reasoning, this would mean the exam is unpredictable and that therefore it can occur on any day they rule out. So in fact the more they know, the more they don't...

[Iceman]

The flaw is ruling out the exam in its entirety.

As soon as they reasoned that there would be no exam, then any day becomes possible.

If you think its possible there is No exam at all, then you can never rule out Friday in the first place.

[Kayomaro]

I've only skimmed the five or so pages this thread has, and what I'm about to say will probably make the shit hit the fan but... When it's Wednesday, and they've "ruled out" Friday, that shouldn't rule out Thursday. As soon as you add that extra day at the end (being Friday, now that you're trying to rule out Thursday), everything changes. Say I have numbers one through nine, and I tell you that I will count up from one, to a number of my choice and that at some point I will shout "CHEESE!", stop counting and you will not know until it happens. I start at one and count to five and yell. Then someone says "but you couldn't have started counting at all! If you had counted to eight I would have known that you would have yelled at nine. And since I would have known at eight, then I would have known at seven. Etc." I reply "No. Nine wouldn't have been a surprise if and only if I had made it to eight. Which I did not. Just because you rule out the final number doesn't mean that you can count it being ruled out for your next step."

I compare it to kinetics in physics. It's usually a bad idea to use a value that you found from one formula, say time from V1 V2 and A, to find distance. If you make any mistakes, then they will be compounded, which is exactly what the students did here.

And yeah I've got to agree that another flaw is ruling out the exam. As soon as it "can't" happen it's a surprise when it does.

[skullturf]

Certainly, induction in general is a valid method of proof. Induction can sometimes lead to counterintuitive results (cf. the blue eyes on an island problem), but with problems like that one or the surprise exam one, if you state your premises carefully, you can use induction to prove *something*. (Depending on interpretation, there might be room for quibbling on what that "something" is.)

This has already been mentioned in this thread, but isn't one relatively common interpretation of the surprise exam paradox is that it's a proof by contradiction that a certain set of premises is inconsistent?

I mean, clearly one can make a smaller set of premises that's obviously inconsistent. If there is only one day in the problem, and our premises include "You know for sure that there will be an exam tomorrow" and "You know for sure that you do not know what day the exam will be."

[RonWessels]

Any attempt to look for flaws in the induction steps are fruitless. As has been pointed out earlier in this thread, you get the same paradox if the prof announces that "there will be an exam tomorrow and you will be surprised by it". The very act of logically eliminating the exam from happening causes the exam to be possible.

[phlip]

This has already been mentioned in this thread, but isn't one relatively common interpretation of the surprise exam paradox is that it's a proof by contradiction that a certain set of premises is inconsistent?

That was my favoured interpretation for a while, but I realised it doesn't fit... if the premises were inconsistent, it would be impossible for them to all be true at once (by definition), and yet at the end of the story, there was an exam, during the specified time period, and the class was surprised.

[Qaanol]

Here is my take on this:

On any given morning, the student can prove by induction that the exam will not be on any future day.

Case 1: The logical system specified by the professor’s statements is consistent.
In this case, we have proved by contradiction that the exam will be today.

Case 2: The logical system specified by the professor’s statements is inconsistent.
In this case, every statement can be proved true. In particular, the statement “the exam will be today” can be proved true.

Those are the only possible cases. So this means, on every morning, the students can prove that the exam will be that day, regardless of whether the professor’s statements are consistent.

Thus, the professor is incorrect in saying “On the morning of the exam, you will not be able to prove the exam will take place that day.” No matter what day the exam takes place, on that morning the students can definitely prove the exam will take place that day. Thus, a false statement is one of the axioms of the logical system. This proves the system is inconsistent.

tl;dr — Every morning, we can prove the exam will be today. If we’re right, the professor lied. If we’re wrong, we proved a false statement from what the professor said. Thus the professor definitely lied, so every possible statement can be proved from his remarks, and one such statement is the true day of the exam.

[Ermes Marana]

I mean, clearly one can make a smaller set of premises that's obviously inconsistent. If there is only one day in the problem, and our premises include "You know for sure that there will be an exam tomorrow" and "You know for sure that you do not know what day the exam will be."

You've hit on the crux of the matter.

"You know for sure that there will be an exam tomorrow" and "You know for sure that you do not know what day the exam will be" are inconsistent, certainly.

But that isn't what the professor said!

"There will be an exam tomorrow" and "you do not know what day the exam will be" are NOT inconsistent. They are, in fact, true.

The professor never says that the students will be able to believe both his statements. He knows they can't, even though he is telling the truth.

That little extra "you know" is the difference. If the professor had said that, he would have been wrong.

[skullturf]

I mean, clearly one can make a smaller set of premises that's obviously inconsistent. If there is only one day in the problem, and our premises include "You know for sure that there will be an exam tomorrow" and "You know for sure that you do not know what day the exam will be."

You've hit on the crux of the matter.

"You know for sure that there will be an exam tomorrow" and "You know for sure that you do not know what day the exam will be" are inconsistent, certainly.

But that isn't what the professor said!

"There will be an exam tomorrow" and "you do not know what day the exam will be" are NOT inconsistent. They are, in fact, true.

The professor never says that the students will be able to believe both his statements. He knows they can't, even though he is telling the truth.

That little extra "you know" is the difference. If the professor had said that, he would have been wrong.

Very good point. Let's consider the one-day version a little more.

Say the professor tells the student (note: there may as well be only one student)

"There will be an exam tomorrow", and
"You don't know what day the exam will be".

Can those both be true? Certainly: if, for example, either the student is somewhat non-rational, or if the student doesn't know the professor to be truthful.

But it seems to me that if the student is a good logician, and if the student knows the professor to be truthful, then the student deduces that there will in fact be an exam tomorrow. So it looks like the student then does indeed know what day the exam will be. (Ooh, one thing that just occurred to me: if the student knows there will be an exam tomorrow, is that the same thing as the student knowing that the student knows there will be an exam tomorrow? Does this type of thing play a role?)

So it seems to me that in the one-day case, if the student is rational, the student can conclude that the professor's remarks can't all be true.

[EricH]

Say the professor tells the student (note: there may as well be only one student)

"There will be an exam tomorrow", and
"You don't know what day the exam will be".

Can those both be true? Certainly: if, for example, either the student is somewhat non-rational, or if the student doesn't know the professor to be truthful.

But it seems to me that if the student is a good logician, and if the student knows the professor to be truthful, then the student deduces that there will in fact be an exam tomorrow. So it looks like the student then does indeed know what day the exam will be. (Ooh, one thing that just occurred to me: if the student knows there will be an exam tomorrow, is that the same thing as the student knowing that the student knows there will be an exam tomorrow? Does this type of thing play a role?)

So it seems to me that in the one-day case, if the student is rational, the student can conclude that the professor's remarks can't all be true.

And reaching that conclusion creates doubt and uncertainty; now that the student can't trust the truth of the professor's statements, the second one can be true.
Just to make the paradox more compact, if the professor announces "there will be a surprise exam tomorrow," that statement can only be true if it's not provable.
Didn't Godel already show that there must exist statements that are true, but cannot be proven to be true?

[billiams]

I think the reasoning has some interesting qualities which are time sensitive, language sensitive, but not paradoxical in the way it seems at first. To define terms, consider what it means to be surprised. There is clearly the issue of when something is a surprise. Let's say the professor offered the test on Friday week prior, it would be a surprise until the end of class on Thursday. But let's tighten up surprise -- there would be 24 hours where you knew for certain there would be a test on Thursday, assuming at least one day of the week will definitely have a test. The professor says he will stipulate that that is not "really" a surprise, even though from the point of view one has on Monday, it surely would be. It would only be after class on Wed, that you would be sure the only option would be Thursday, not to experience this 24 hour window of knowledge going from Thursday to the Friday of the test. But on Monday, you would know it could be Monday, Tues or Wed, without the 24 hour window, going forward. On Tues, you know it could be that day or Wed. On Wed, you would know it had to be that day, or there would be a potential 24 hour period of not being surprised. If there is a potential, disallowed by the professor of never being aware, even for a moment of the test before it occurs, then you know the next day is a no go if the test doesn't hit you that day, and we are back in the pickle. Tuesday must be the day, but we have the window. Uh oh. If the Professor lightens up and says on Monday you will not be able to guess when the test is, if the test hasn't been given by Thursday you will have a one day heads up, then there is no predicting the test, until and unless the end of class on Thursday, but not on Monday or earlier( there is a Monty Hall issue, but that's another story). Otherwise, the Professor is selling a bill of goods ( or he's buying into wily George's redefinition of surprise unwittingly), and can't promise the test will be given on one of 5 days, and will never allow for a twenty four hour window.

So the Professor relents, and sees that looking forward until Thursday he will be able to really surprise people, but not on Thursday for Friday -- that like picking a marble out of jars, one marble, five jars, pick four empty jars you know that the fifth will have it. The paradox comes from a nutty stipulation of "surprise", and the fallacy ignoratio elenchi , or redefinition thatGeorge introduces. It's logically impossible to avoid the 24 window without generating a contradiction, and if that standard is utilized, it generates a nutty result. Just as if I said the same thing about the jars: "You will be surprised. You will never know which jar has the marble." So you pick till 4, and realize it must be five = fail. Translate: can't be 5, so 4 is the highest, number. So you get to three jars and know it must be 4 = fail. Translate: can't be 4 so three is the highest number. And so on. What is occurring is just a product of a defective definition of "surprise" lurking that can't be met given the other conditions. ( The contradiction argument is not a "proof," but a disproof: A--> P. not P ---> Contra---->invalid ---->Plug in A, A is invalid). Change the def to what we normally mean as third dimensional creatures when we say surprise, meaning at the beginning of the sequence not foreclosing the possibility of this changing going forward. And we are back to being surprised, at least till Thursday. But on Monday, since we know it's not foreclosed by the potential 24 hour window, it would still be a surprise to us if the test were to occur on Friday.

[Danny Uncanny7]

The paradox is a lot easier to see if you narrow it down to just one day.

The professor says, I will give you an exam tomorrow, and it will be on an unexpected day. The students realize that this is impossible, so if it is to be unexpected, it can't be tomorrow. Then they are happy and think the evil professor made a promise he couldn't hold, and don't prepare for the test. However, the next day, they have the exam. And all are surprised.

[Yakk]

If the next poster in this thread has not actually read the thread, will I be surprised?