The "Surprise Exam" paradox
thebigquestions.com
thebigquestions.com
In this case "surprise" seems suspect since it is an emotional and physical response - things which are likely to be difficult to predict using formal systems.
Then of course, there is the fact that the surprise is announced before hand rather than exclaimed - "Surprise!" - at the actual event as is the case with a surprise party...and of course surprise parties often do not surprise the person for which they are planned - indeed, sometimes forewarning is a necessary part of the surprise party plan.
Just as a surprise party is rated a success based upon the general standards for parties even when the person for whom it is thrown is not actually surprised (though acting surprised is often helpful), a surprise exam is successful based upon the relevant academic criteria. One simply does not fail a surprise exam because they were expecting it.
What the professor announced was essentially, "On one day next week, there will be an exam. I am not telling you which day that will be." Whatever paradox we imagine finding is dependent on us deeming the purpose of the exam to be to cause surprise among the students, rather than getting them to study or measuring their knowledge or any of the legitimate purposes of the context.
Of course this does not make as interesting a story.
But it is hardly news that a flawed algorithm may produce flawed results.
The problem is that we are trying to treat ordinary language as if it was psuedocode.
For i = 5 to 1
i = i - 1
if i = 1
then "No Exam"
else next iOnce that becomes a premise, the efforts of the students are of the sort who should be logically classed with:
persons in a state of insanity, whose brains are so disordered and clouded by dark bilious vapors as to cause them pertinaciously to assert that they are monarchs when they are in the greatest poverty; or clothed [in gold] and purple when destitute of any covering; or that their head is made of clay, their body of glass, or that they are gourds...
[Descartes]
One possible interpretation:
"There will be exam this week" means
(1) X is a random variable from the set {1,2,3,4,5}, with unknown distribution.
(2) P( X=x | P(X=x | X>x-1) = 1 ) = 0
Since X <= 5 we know that P(X=5 | X>5-1) = 1. So we know that P(X=5) = 0.
From this we know that P(X=4 | X>3) = 1. So P(X=4) = 0.
From this we know that P(X=3 | X>2) = 1. So P(X=3) = 0.
From this we know that P(X=2 | X>1) = 1. So P(X=2) = 0.
From this we know that P(X=1 | X>0) = 1. So P(X=1) = 0.
Contradiction.
Further implications are based on students assuming there won't be an exam, which makes the exam surprising in the end, and removes contradiction, or assuming that (2) is false, which makes all days possible, which removes contradiction.
For me it's dubious reasoning - if we had contradiction once, it can't be removed by forgetting it for a while. At least that's my intuition.
If I've made errors, please, correct me.
If x is, as you say "a random variable from the set {1,2,3,4,5}, with unknown distribution" then it is true that you cannot, given the methodology of going thru days one by one, be 100% guaranteed to not be able to deduce the day of the test that you must have it that day.
This is because for whatever space of possibilities (in this case 1,2,3,4,5) on the last day of the space, you can deduce (given the methodology) that it must be that day.
But the teacher is not really giving you a gurantee that x is a random variable from the set {1,2,3,4,5}.
the teacher is only giving you a guarantee that x is a member of the set 1,2,3,4,5. You have no idea what x is, nor how the teacher chose it.
So, given the guarantees Version A 1) x is a random variable from the set (1,2,3,4,5) 2) prior to the first day (commencement) you are given a 100% guarantee that proceeding thru opening days one by one until day x, you will not be able to deduce on day x, before opening that day, that the exam will be given that day.
in this case you are right: there can be no solution.
but version b is different: 1) teacher guarantees x is 1 or 2 or 3 or 4 or 5. 2) prior to the first day (commencement) teacher gives 100% guarantee that proceeding thru opening days one by one until day x, you will not be able to deduce on day x, before opening that day, that the exam will be given that day.
in this case you don't really know what set(s) x is a member of.
The teacher could fulfill his obligation for (2) by hard-coding x to be Monday.
The student could reason that "If by Friday we don't have an exam, the exam must be Friday", thinking that the space is 1,2,3,4,5. But it's not: there is no possibility the exam is on Friday, due to hard-coding the exam to be Monday, and therefore Friday is not the 'last day' the exam could be given. It's not any of the days the exam could be given. Nor is Thursday. Nor is Wednesday.
Therefore, the student is simply wrong in supposing (guessing) that this is the space in which the exam is on one of the days of.
Why? Even if X is always 1, it is a random variable from {1,2,3,4,5}, with distribution {1.0, 0.0, 0.0, 0.0, 0.0}, which I don't know so it is unknown for me.
For me teachers method of choice is not important - he can throw the dices, he can set it to 1. For me it's still random variable, because I don't know the result.
If the real space is {1,2}, because teacher will throw a coin, I can still think about it as variable on {1,2,3,4,5}, because it's all the same, when the probabilities are {0.5, 0.5, 0, 0, 0}.
#teacher picks day
DayOfTest = "Monday";
#now go through the week.
for today in (Monday, Tuesday, Wednesday, Thursday, Friday):
print "It's ", today, " today."
if student_is_sure_test_today(today) == true:
if today == DayOfTest:
print "PARADOX!! TEACHER GUARANTEED SURPRISE but student knew test today.."
else:
print "STUDENT ERROR!! Was 'sure' test today, but there is no test today."
else:
if today == DayOfTest:
print "TEST today. Student surprised. No paradox."
exit
else:
print "Day ", today, " over without a test."
But if you run the pseudocode as-is above a million times, the student would NEVER get to say, on a Friday, that they are sure it's that day. You would not print a single PARADOX!!!! error message. Because you don't print a single paradox, we don't have to remove that day (reducing the possible spaces from five to four) to avoid it.
Do you see? If the pseudocode has the teacher giving the day as random monday through friday, then he can only give an 80% guarantee of not tripping a paradox.
but if the teacher can have a secret algorithm (giving the test on Monday) then the student's very first line of reasoning fails: they do not get to trip any paradox error messages or, in order to remove these error messages, remove any possible days from the space.
If someone told you before the roll, that this dice always rolls 6 - you'll know that, and then you won't be surprised.
The dice is the same, your guess is the same, what changed is your knowledge.
In our example teacher provided us with knowledge that is contradictory. You can deduce anything you want from 0=1, so the whole exercise is futile.
There's not much room to interpret here.
We could program this as a betting game. on every day from monday to friday, the student may place a bet of $1000 at even odds, that the test is on that day.
"Sure" means the bet has a 100% chance of being paid off. At even odds, the $1000 has an expected value of $2000.
It doesn't really matter 'why' they placed that bet. If the expected value of the $1000 bet is $2000 then the details don't really matter. The student wins, and there is a paradox.
The definition of the paradox thus shifts from something vague about 'sure' to something concrete: a bet that has 100% chance of being paid off, made with this intention.
Now, you are saying that student argues thusly. If the teacher is making a guarantee that I will be given no sure-bet (bet at 100% chance to win that the test is on that day), then that is a guarantee that I will not be in a position to place that bet on Friday.
This is true.
What is false is the step: "THEREFORE, if on Friday..." (i.e. if I get to Friday)
That is not a warranted beginning of a sentence.
If a test day of Monday is hard-coded into the algorithm, then saying "THEREFORE, if on Friday.." (if I get to Friday) is as spurious as saying "THEREFORE, if Friday afternoon I still haven't been given a test..."
This is a subtle point. One sounds OK, one sounds like it contradicts what we already know: 1) If I haven't been given a test by Friday (and it's friday morning now, the test can still be given)
2) If I haven't been given a test by Friday afternoon (and it's friday afternoon, the test can't be given today anymore...)
It sounds to you like #2 contradicts your guarantee to be given a test on a weekday next week. But in fact, it doesn't CONTRADICT anything, because it's an "if" about something that has 0.0 chance of happening.
You can legitimately say : "If I haven't been given a test by Friday afternoon" (i.e. no test at all week), but you can't DEDUCE anything from that, since it has 0.0 chance of happening. It doesn't matter what the second half of the sentence is.
Now.
Perhaps the first statement (1) is similar. Perhaps it also has 0.0 chance of happening.
Therefore, perhaps it is no better to say: If I haven't been given a test by Friday morning... than If I haven't been given a test by Friday afternoon...
they both have 0.0 chances of happening.
A die that will always fall on 5. Reasoning "If it falls on six" is no better than reasoning "If it falls on 7" UNLESS you have been given some guarantee that there is a nonzero chance it falls on a six!!!
The whole "paradox" hinges on people thinking that by saying "the test will be given on a monday, tuesday, wednesday, thursday, or friday" means that none of these days has the same probability as "the test is not given at all." But in fact, one or more of these days could have that exact same probability: 0.0.
When it's 0.0 due to the stochastic process involved, "reasoning" with it is no better than "reasoning with" the case that the test isn't given at all.
You can see all this very easily with the pseudocode.
If the students are not given any guarantee that there is a non-zero chance the test is given on Friday or on Thursday, it is a mistake to reason anything at all about Friday or Thursday.
Here is the logical fallacy.
You assume that if there is probability 1 that the day is 1,2,3,4, or 5, then it follows that for each of 1,2,3,4,5 there is probability >0.0. That's a mistake. It's a fallacy. You haven't really been told that.
We can invalidate all assertions of a paradox very easily with two parallel universes.
Suppose the teacher is a mad physicist. He splits (forks) his universe in two, preceding as follows.
In one parallel universe the teacher always gives the test on Monday.
In another parallel universe the teacher always gives the test on Tuesday.
In both cases, the students are told: I guarantee you will be surprised by the test. I guarantee that it will be given next week, and on a Monday, Tuesday, Wednesday, Thursday, or Friday.
If the students DON'T KNOW about his mad experiment... how can they possibly be SURE of being tested (and being correct in this deduction) on either Monday or on Tuesday? None of these students could possibly be sure on any of these days.
They are simply mistaken when they begin to argue: "If we make it to Friday, the test must be Friday". This is no better than arguing "If we are not tested at all, then the test must be Friday". Both of those are equally valid arguments. They're both wrong in this scenario.
The scenario where they're NOT both wrong, where "if we make it to Friday, the test must be Friday" is OK (not violating the guarantee to have a test during the week) whereas "if we are not tested at all, then the test must be Friday" is not okay, is if the students know for a fact htat there isn't a 0.0 chance of having a test on a friday.
you simply can't deduce from something that is false. THe student is (in the case of the scenario just outlined, where the test day is monday or tuesday) arguing from false premises when they say "If we haven't had a test by Friday..."
That's an argument from a false premise (in this scenario) and is invalid for that reason.
The question is: are the students given enough information to know that the premise is NOT false ("if we make it to Friday")? If the teacher guarantees them a stochastic process with a nonzero chance of a test on Friday, then they have enough information to know that the premise is NOT false.
since htis is not stipulated, however, the teacher can meet his commitments.
The way to formalize this is to say so. "This is a stochastic process, there is a non-zero chance that you will go through four days without a test (have it on the fifth day".
Obviously, you cannot make this guarantee, while guaranteeing surprise in 100% of cases.
So basically, the only intuitive problem this puzzle poses is that it sounds like the teacher is giving you some kind of guarantee that there is a case where you open four empty envelopes before opening the fifth with the heart in it.
In fact there is no such guarantee.
So, really, my cases A and B can be summarized as follows.
A) In case the teacher makes any guarantee that there is a non-zero chance of a test on Friday and that this chance is assured by a stochastic process, he CANNOT 100% guarantee surprise.
This was my original, knee-jerk interpretation. Of course, the teacher does not ACTUALLY make any such guarantee!!! By saying "test-day = Monday" he has gone back on his word that there is a non-zero chance the test is on Friday: in fact there is a zero chance of that.
Version B, in which he does not guaranteee a non-zero chance on any day of the week, leaves the teacher able to fulfill both his obligations (surprise and test on a weekday) without a paradox.
Really, this is INSANELY clear and simple when you view it clearly. We think he's giving a guarantee that there's a chance the test will be on Friday and a stochastic process determines this. But he's giving no such guarantee.
"Why? Even if X is always 1, it is a random variable from {1,2,3,4,5}, with distribution {1.0, 0.0, 0.0, 0.0, 0.0}"
But
IF, insofar as
"x is always monday",
'random variable from monday,tuesday,wednesday,thursday,friday' with distribution{1.0,0,0,0,0}
"x is always monday",
'random variable from monday,tuesday,wednesday,thursday,friday,not given at all' with distribution{1.0,0,0,0,0,0}
Intuitively, it sounds like we are given some guarantee that Friday is more likely than No Test Given At All.
In fact, we are given no such guarantee.
Insofar as we have such a guarantee, the teacher cannot guarantee surprise.
It's quite simple, really.
But we are given guarantee that no day will have probability 1.0, if I know it wasn't any day before.
From the guarantee we know it won't be Friday. From this we know it won't be Thursday, and so on. From the fact that teacher told us that there will be an exam this week, we know any other options we could add (like "not given at all", or "next monday") have probability of 0.
From this all we see that teacher claims were contradictory.
"that no day will have probability 1.0, if I know it wasn't any day before".
The key is that "if I know it wasn't any day before" CAN'T be a false premise. If the teacher secretly fixes the day at Monday, then the first line of reasoning becomes FALSE.
The student is WRONG to argue "If I make it to Friday..." because that's not possible any more than "If I make it to Saturday..." or "If I make it to next week..." All of these are false premises.
Arguments from a false premise don't lead to any conclusions.
"If your mother is named John, then..." is a mistaken argument, if there is a 0.0 chance your mother is named John (for some reason). It doesn't matter what the other half of the sentence is, if that premise has a 0.0 chance of being true, you can't really deduce anything from it (if it has 0.0 chance of being true).
The thing that makes this thing hard is you don't KNOW how the teacher chooses the day.
If the teacher always picks Monday, but nobody in the class knows it, they don't know that Friday has a 0.0 chance of happening, they begin to reason "If we make it to Friday morning without a test..." and this argument is simply FALSE. It's about something not possible. It is like saying "If 1 = 2..."
The student makes a mistake (if the teacher always gives a test Monday) by saying "If we make it to Friday without a test..." just as the student makes a mistake if they finish the sentence "If 1 = 2..."
I mean, I can continue making arguments: "If 1 = 2, then 2 = 4". Is that a "true" statement? Not really. You can't really deduce anything about 2 being equal to 4 or not if 1 can't equal 2.
You can't really deduce anything about other days of the week, if it's not possible for the teacher to pick Friday.
That is why it matters how the teacher chooses the days.
"If 1 = 2, then 2 = 4" technically this is a true statement. Any implication starting with false premise is true. But you already know that, and you meant that I cannot use the "then" clause of this implication, because it can be true or false.
Main subject:
My reasoning resulted with contradiction, which I blamed on assumption (1) or (2), but it can be that there was hidden assumption (3) "there is a chance that Friday will be choosen".
When we assume (1) (2) and (3) we have contradiction, when we assume (1) and (2), but not (3), we can't use (2) to prove there's no possible day to choose for teacher, so that case is the solution.
OK - count me persuaded. Thanks for disscussion.
Well, yes it is like saying "if 1=2, then 2=4", but that doesn't mean that this line of thinking is an invalid one. It simply means that by showing the result to be wrong, the test at the start is also wrong. The statement isn't attempting to prove that 2=4 by assuming that 1=2. It's saying that we can show that 1 quite clearly doesn't equal 2 because the logic consequence of that would be wrong.
In the same way "If we were able to make it to Friday without a test, then we'll know that the surprise test is Friday" isn't an invalid line of thinking. It's simply showing that it can't be possible for the surprise test to be on Friday.
A philosophy professor tells the class there will be a surprise exam this week. He and the class discuss the exam, and they ‘prove’ that there can be no surprise exam. When everyone has accepted the proof, he announces that there will be an exam on the spot! The exam has only one question: Prove that the exam is indeed a surprise, invalidating the proof that there can be no surprise exam.
This formulation suggests that the exam is a surprise if the students convince themselves that there can be no surprise exam, which is an entertaining trail to follow.
Smullyan also tells an anecdote from his childhood. He and his brother liked to ‘fool’ each other on April Fool’s, and one year his brother announced that the coming April Fool’s Day, he would fool Raymond as he had never been fooled before. Well, the day came and Raymond steeled himself for the prank. He waited, expecting a prank or lie with everything they did and every cinversation they had, but no prank, no lies, no fooling.
As he lay in bed that night, Raymond realized that his brother had, in fact, fooled him. Or had he? How could not fooling someone constitute fooling him? According to Raymond, puzzling over the nature of ‘fooling’ was part of his lifelong journey into philosophy and logic.
Noah looked up. “No, my dear, you are amazed. It is we who are surprised."
This means that the professor cannot guarantee a surprise exam, so his original claim is false. But note that the students' strategy requires them to always assume that the exam will be on the earliest possible date and to plan their study time accordingly, which is probably the behavior that the professor was trying to encourage anyway. It just so happens that he has to lie about the surprise exam in order to do so.
""" The meta-paradox consists of two seemingly incompatible facts. The first is that the surprise exam paradox seems easy to resolve. Those seeing it for the first time typically have the instinctive reaction that the flaw in the students’ reasoning is obvious. Furthermore, most readers who have tried to think it through have had little difficulty resolving it to their own satisfaction.
The second (astonishing) fact is that to date nearly a hundred papers on the paradox have been published, and still no consensus on its correct resolution has been reached. The paradox has even been called a “significant problem” for philosophy [30, chapter 7, section VII]. How can this be? Can such a ridiculous argument really be a major unsolved mystery? If not, why does paper after paper begin by brusquely dismissing all previous work and claiming that it alone presents the long-awaited simple solution that lays the paradox to rest once and for all? """
on Thursday you know that the probability that the exam will be on Friday is 1
on Wednesday you know that the probability that the exam will occur Thursday is .5
etc. etc.
on Sunday you know that the probability that the exam will occur on Monday is .2
you could say the pmf is: .2, .5 .8 .17 .5 and the cdf is .2 .25 .33 .5 1 Which from memory looks like a Beta Distribution.
PS: I suspect this comes from the limits of logic in the real world. If I do the same thing with a really long computer program it might though some internal errors flip that 2 into a 1 even as you simply copy it from one memory location to another.
So as of the moment of notice there is a random choice, and as that day approaches, it becomes less of an ongoing "surprise".
But that's not the case.
The teacher claims that "students won’t know exactly which day until the exams are handed out", whereas by the end of Thursday, they would be 100% certain that the exam was going to be on Friday. Therefore it is impossible for a surprise exam to be on the Friday, so that can be ruled out as a possibility.
Which means that by the end of Wednesday, the only possible day for a surprise exam becomes Thursday, and the same logic which ruled out Friday then applies to Thursday (if it has to be Thursday, it can't be a surprise, so it can't be Thursday), and all the way back through the rest of the week.
The first is the ambiguous definition of surprise. This lead to easy explanation of the paradox. The teacher might as well hand out an exam on a red piece or paper and claim that that is a surprise.
The second is that the teacher invalidly presumes to be able to predict the mental state of his students. This can be stated simply: over the weekend, all his students die unexpectedly. The following week, none of his student are surprised since the set of surprised students is empty.
We can work around those two flaw by ignoring the no-student case and assuming that "surprised" is replaced by a more exact formulation: "on the day the exam occurs, nobody would be able to say ``I knew the exam was today''".
In this formulation, I contend that nothing can be concluded.
The reason for this is that induction can only be applied on facts. The future is not a fact, so to apply induction on a sequence of future events is an error. The only conclusion that one can reach is the contrafactual: if we were Friday then the exam cannot occur on Friday. This is because the reasoning does not rely on any inductive argument but rely only on things that would be facts inside the contrafactual: if we were on Friday now, it would be false that the exam cannot be held on that day and that nobody could say that the exam is to be held that day.
A contrafactual of this nature cannot be built for Thursday, since in that contrafactual ("if we were Thursday...") one would have to reason by induction about teh future. Of course, in the Thursday-contrafactual, one could state the Friday-contrafactual, but you can't build inudction on contrafactual (i.e. non-fact).
He doesn't actually use the word "surprise" anywhere in his statement, and spells out pretty clearly what the nature of the surprise would be - "students won’t know exactly which day until the exams are handed out. Nothing, other than coming in on a particular day, not knowing for certain that the exam will be that day, but getting the exam on that day nevertheless, would do - e.g.an exam on the Friday would not qualify, as you would already know that this was the day of the exam. And nor would red paper.
>The following week, none of his student are surprised since the set of surprised students is empty.
Again, he doesn't claim that anyone would be surprised. He said that they "won’t know exactly which day until the exams are handed out." I suppose you could argue that if they are dead, then they won't even know on that day either, but that's a pretty tenuous point and anyway technically he doesn't explicitly claim that they would know at that point, just that they wouldn't beforehand.
>"on the day the exam occurs, nobody would be able to say ``I knew the exam was today''".
Which is pretty much what he did say.
>A contrafactual of this nature cannot be built for Thursday
Why not? The argument against it being on Friday (and I think we're both agreed that it's impossible for it to be on Friday and still line up with his statement) is that if there's only one possible choice for what day it's on, then the pupils - or at least any that are still alive - would know that the exam was on that day. And if it can't be on Friday, then when it gets to first thing Thursday how many choices are left for possible days for the exam to line up with his statement? One - so we're back to it being a certainty as to which day it would be.
My main point was that future events can't be held as facts.
I used the death of all students as an example. I'm not sure why this would be a tenuous point: it's my central thesis.
Re-reading the paradox as worded, I conclude that the teacher is right: the students won't know exactly which day because they cannot know the future as a fact. They might die, the teacher might die, the sun may explode. I don't consider these as tenuous arguments. They are concrete counter-example that any reasoning involving future events is flawed.
Trying to explain away this paradox by talking about people possibly being dead by next week is about as useful as taking "This statement is false" and explaining it away by saying that it could be in a language where "false" means true.
(Your analogy with pretending that the word false doesn't mean false is entirely off base. I don't see the correspondence. But no matter.)
I think I've found a formulation that won't offend your sense: due to time's arrow and causality (which both go forward in time), one cannot make an induction that goes backward in time. This would cause a circular dependency in the inductive proof.
In this case, the steps of the inductive proof go from Friday, down to Thursday, Wednesday, etc. Yet at the same time, causality goes from Wednesday to Thursday to Friday. One cannot simply dismiss causality.
IOW, we have the inductive process N, N+1, N+2, where N is used to proove N+1, etc and where N = Friday, N+1 Thursday, all the while N state depends on N+1 because due to real-world causality, what happens on Friday depends on Thursday.
You seem to wish to deny causality.
As for causality, I don't really see how that's relevant either. Assuming you've accepted whatever caveats we feel like adding, then nothing (apart from the test having happened) that happens on Monday, Tuesday, Wednesday or Thursday is going to change the fact that if you've got to Friday without the test having happened you'll have certainty that the test will be on Friday, and therefore won't be a surprise.
In other words, the only things required for the paradox to be a paradox are the idea of a test happening on a day that the pupils can't predict and a deadline for when that test must happen by.
(And the reason my analogy about the word false is relevant is that it's pretty clear what the bounds of both paradoxes are, and you "solve" either of them by throwing in things - whether that's events external to the point of the paradox, or word definitions external to the point - that are outside those bounds, but in either case it doesn't tell you anything useful about the paradox itself).
Clever logical conundrum though. Certainly forces you to think deeply on words like "Surprise."
Reminds me of quantum physics. The two quantumly linked properties are the truthfulness of the professor’s statement and the time remaining until the last possible time for the test. When the professor makes the statement at least 5 days before the Friday, it’s true. By the Friday, the statement has become false. Around the Tuesday, the statement is half-true and half-false, an uncertain state.
> …to date nearly a hundred papers on the paradox have been published, and still no consensus on its correct resolution has been reached.
I’d think far more than a hundred papers have been published on the paradoxes of quantum physics. Can I suggest the surprise test paradox and quantum physics have exactly the same underlying principles behind them.
They reason, if there is going to be an exam, and it hasn't happened by Thursday, then it will happen Friday, and we would believe it would happen on Friday and it cannot be a surprise.
They should reason like this: if there is going to be an exam and we believe there will be an exam, and it has not happened on Thursday &c then it cannot be a surprise.
But, they stopped believing the exam was happening. Doh.
EDIT: formatting
The exam can be characterized as "surprising" until the end of Thursday's class. Getting the probability at the end of each day for the exam to be made from now on, we have (M=Monday, T=Tuesday etc):
End of Sunday : P(exam=M,T,W,T,F|S)= 1/5
End of Monday : P(exam=T,W,T,F|M)= 1/4
End of Tuesday : P(exam=W,T,F|T)= 1/3
End of Wednesday : P(exam=T,F|w)= 1/2
End of Thursday : P(exam=F|T) = 1
So, after the end of Thursday class, the probability of the exam is 1, so the students cannot be surprised any more as they are sure that they will be examined Friday. This means, that in every day except Friday the students can be surprised by the exam.
Anybody to point me where the above interpretation is wrong?
1: "You will be surprised to learn the truth value of this is statement, or this statement is false."
Case A
2: "I will not be surprised. I know that the statement is true."
1: "Then the statement is false, and you have been surprised!"
Case B
2: "I will not be surprised. I know that the statement is false."
1: "Then the statement is true, and you have been surprised!"
Case C
1: "I am surprised!"
2: "The statement is true."
"D" being the day at which the exam takes place, "not knowing D on Monday" is not the same proposition as "not knowing D on Thursday". If you change the proposition tested at each inductive step, your induction is invalid.
Similarly, "this sentence is false" is neither true nor false, but just nonsense. "The set of all sets that do not contain themselves" is not a paradox or a contradiction, it simply describes an object that doesn't exist. Likewise, the professor's statement isn't known to be true or false at the time it's made, and either outcome is possible.
You might be interested in verificationism http://plato.stanford.edu/entries/logical-empiricism/#EmpVer... (apologies if you know about it already).
P( exam is today | P(exam is today | exam was not earlier) = 1) = 0
"""One of Kurt Godel’s great insights was that you can go a lot deeper by considering a slightly different sentence: “This sentence is not provable”. If that statement is false, then it’s provable. But surely no false statement should be provable! So maybe the statement is true. In that case, it’s true but not provable, which says something about the limits of logic. It says that not every true statement can be proved.""" [1]
fn 1: http://www.thebigquestions.com/2011/12/13/a-tale-of-three-pa...
As a teacher:
1) you CAN guarantee for your students that there will be a pop quiz Monday through Friday of next week.
2) You CANNOT ALSO guarantee for them that they will be surprised that day. (i.e. that they won't be sure, on that day, that it's that day).
This is because if it hasn't happened by the last day it can happen, it will happen the last day it can happen.
This obviously makes sense. There's no paradox. There's nothing difficult.
You simply cannot GUARANTEE them surprise, since out of whatever possible space of days it can happen, on the last day it won't be a surprise.
Whether that space is 1, 2, 3, 4, or 5 days, on the last of that day it is not a surprise.
For any space of days in which the exam can happen, the last day would not be a surprise;
therefore, you cannot guarantee them both that there will be a space of days in which it can happen, and also GUARANTEE them that it will be a surprise. (in every eventuality. You can make them an 80% guarantee that it's a surprise, while making them a 100% guarantee that it's a quiz between monday and friday of next week. if you can increase the number of days you can increase your guarantee further and further. if I guarantee you a quiz on one of the next 100 days, I can also 99% guarantee you that you won't be sure you're having it that day -- i.e. will be 'surprised' -- on the day that I give it.)
prior probability. pick a day monday to friday. quiz time comes. monday to thursday they're surprised; if it's on friday, they're not surprised.
why you would think you could also surprise them after giving them a chance to "open" every day up to then is beyond me.
THis is like giving you a guarantee: here are five envelopes, one has a heart in it. I guarantee you will be surprised when you open the one with a heart in it, even if you've opened four of the five envelopes already and found them empty.
um....no... you can't guarantee that. it makes perfect sense, and there's no paradox or worth wasting any breath over.
An 80% guarantee is fine, but 100% guarantee that you won't be sure, is incompatible with having a space of envelopes and going through it one by one.
why would anyone think otherwise?
the way in which I discovered this resolution is by coding up a perl script to 'monte carlo' different scenarios. I realized at the location that I made the teacher actually have to choose which day the exam will be, there is a space of days.
(they have to choose - or end up choosing - as they break their first guarantee if they don't have it on any day of the week or more than one day of the week, or whatever. it has to be one and only one day of the week, however they end up getting there.)
it doesn't matter if the teacher is choosing one of five days, 10 days, 100 days, whatever. The methodology is that the students get to go through the days one by one.
If you are choosing one of five days, there is a 20% probability of choosing the last of them; therefore given this methodology you can only 80% guarantee them surprise.
if you are choosing one of ten days, there is a 10% probability of choosing the last of them; therefore given the methodology you can only 90% guarantee them surprise
if choosing one of 100 days, 1% probability you chose the last of them; you can 99% guarantee them surprise.
Bottom line, which you discover if you code it up to run in simulations: at some point the teacher MUST actually choose a day to meet their first guarantee. (If they choose none - or end up giving it on none - then they've broken their first guarantee. Any algorithm that doesn't end up choosing a day 100% of the time is wrong).
At the point of choosing a day, it doesn't matter if you make the Teacher choose monday through thursday, monday through wednesday, monday or tuesday, or hard-code it to choose Monday. Whatever the teacher chooses, if the students have access to the teacher's algorithm (which describes the space of possible days), then the last day of the space they would not have surprise. The teacher can thus give a guarantee equal to the number of days OTHER than the last day, over the number of days in the teacher's space. In the usual sense, this is 80% guarantee of surprise.
of course, another question is if the teacher gets to follow an algorithm the students must only guess at. (They don't know what his algorithm is).
In this case, for your simulation you can just hard-code the teacher always giving the test on Monday. Since the students don't know this is his algorithm, they will be "surprised" (they could have thought that he was picking from one day monday thru friday, they had no way to be sure that he was hard-coded to pick Monday), thus fulfilling both criteria of not being sure of what day the test is, and being given a test monday to friday.
sonofabitch. new solution: The teacher can meet his obligations by being hard-coded to give the test on Monday, but not telling his students that this is the algorithm. Since they could be thinking he's choosing from a space of monday thru friday, they could think that the last day is friday: it's really Monday.
this is an interesting aspect I hadn't considered (a secret algorithm).
In this case I would say my response is more nuanced: 1) To whatever extent the students get to know of the teacher's algorithm for picking days, they are that much less able to be given a guarantee of 100% surprise on the day of the exam.
Therefore, if the teacher is completely secret about his algorithm (i.e. he is NOT 'picking randomly monday thru friday') he can surprise the student. Any information he gives his students about his algorithm takes away from the extent to which he can guarantee their surprise. EVEN IF the algorithm includes randomness.
In other words: if the students know the teacher is picking a random day monday thru friday, then he cannot offer a guarantee of surprising them.
if the students have no idea what algorithm the teacher uses to pick a day (and, therefore, have no way to guess at the space), he can offer a guarantee of surprising them.
Thus if students know nothing at all about the teacher, and he knows this fact, he can meet his obligation by giving the test from a space of one day (Monday) without any randomness or space of possibilities.
* Guys, I made a logical mistake here.
I said "this is because if it hasn't happened by the last day it can happen"... But this supposes the students know what day it can happen. They might not.
For example, the Teacher might give the test on Monday every year. Assuming the students have no contact with last year's students (or didn't bother to ask) they would not know that the space of possible days is Monday.
Therefore, the teacher can make a guarantee that the test is on Monday through Friday, while saying that the students will be surprised.
The students, thinking that the test may be Monday through Friday, would assume that the last day the test 'can happen' is Friday. But that's not a possibility at all. So this assumption would be a mistake.
In fact, if the teacher is completely hiding his algorithm, then the students can be surprised.
They don't know if the space of possible days is "only Monday", "only Tuesday", or "Monday, Tuesday, or Thursday", or whatever.
Each of these spaces fulfills the requirement of some day monday thru friday next week: however, if the teacher keeps the space of possibilities quite secret, then the students cannot make their FIRST step of assumption, which is the last day that the thing "could" happen.
So, I have two answers: 1) If the students know 100% of the teacher's ALGORITHM (how the teacher chooses days) -- even if it is random -- then the teacher cannot 100% guarantee them that they will be surprised. Namely, on the last day of the space the teacher chooses from, they would not be surprised.
if, however, the teachers know NOTHING about the algorithm the teacher uses to determine the day, the teacher is indeed able to make his guarantee.
In other words, if I am the teacher and I am trying to make this guarantee, I can do so, provided that instead of choosing a day randomly from monday through friday, I hard-code the day Monday, while keeping this algorithm secret from the students.
TL;DR: if the teacher tells them only something ABOUT the day (that it is next week, and a monday or a tuesday or a wednesday or a thursday or a friday) then there is no paradox and he can guarantee that they will not know for sure of the day it is happening). If, however, they also know HOW the teacher chose the day (even if the algorithm includes randomness! i.e. 'a random day monday to friday') than the teacher may be unable to make his guarantee.
Give me mathematics any day. Philosophy? You and the lawyers can keep it.