Surprise Examination Paradox

Great Debates
21

I may be using a different definition of “surprise.” I think a lot of the trouble with this is that it is unclear what definitions to use. For example, I was using “surprise” to mean “unexpected” (the students would definitely not expect the exam). Thus from my point of view, the students could expect the exam every day, and to defeat the paradox they would only have to ignore the teacher’s announcement because it didn’t give them any help.

But another definition of “surprise” could be “unable to be deduced from the information.” In this case the students would be surprised no matter what, because there is no way to deduce the date of the exam from the evidence.

So you can see that depending on your definition of surprise, the students will either never be surprised, or be surprised no matter what. But perhaps I am still missing the point. Maybe what I should be trying to to do is determine the flaw in the student’s reasoning.

Well, that seems clear. The student is using a premise A (there will be a surprise exam this week) to prove Not A ( there will not be a surprise exam this week). Thus, the student is arguing A -> Not A and he should have thrown out the argument. That was the flaw.

The student cannot eliminate a Friday exam, because to do so he would violate the premise of the very argument he is using to eliminate Friday. The student is taking the teacher’s words as absolute truth, but on Friday morning it is literally impossible for him to still accept those words as absolute truth. He cannot accept that there will both be an exam and that he will be surprised. Thus there is no way for him to deduce whether there will be an exam, because his only possible information (the teacher’s words) is now in doubt.

This is why I rejected Libertarian’s explanation: it is impossible for the students to know that both the teacher’s statements are true. Even if they are true, the students can never know that. IF it was possible to know both statements were true, then they COULD eliminate Friday. But it isn’t, so they can’t.

Its sort of like if I said “Daniel cannot know this sentence is true.” Everyone could know the sentence was true, expect for Daniel. In the same way, everyone can know that what the teacher said was true. Except for the students.

22

Daniel wrote:

It’s a formal term in Doxastic Logic.

It’s defined in my post, but I’ll give you a less technical answer since you indicated that you’re having trouble following it. Common Knowledge is knowledge that is shared by every member of a group. It isn’t just knowledge of a fact, but knowledge of knowledge of a fact. And knowledge of knowledge of knowledge of a fact. And so on.

In the case of the Surprise Exam, for example, the Common Knowledge is not just knowledge by each student of the facts in the teacher’s announcement, but knowledge by each student that all other students know the same fact, and knowledge by each student that all other students know that all other students know the fact. And so on.

That is likely what made Common Knowledge occur as a way to approach the problem. There is a recursion in the paradox, and there is a recursion in Common Knowledge.

Prior self-referential treatments ignored the doxastic context, but the doxastic context is important because knowledge, or what is and isn’t known, is the actual topic of the paradox.

Cutting to the chase, what the treatment proves is that there will indeed be both a surprise and test, although the two need not happen at the same time. For example, if the test is on Friday, then the surprise will be on Thursday at the end of class. That’s because “there will be a test” is Common Knowledge.

23

Nightime wrote:

And yet, the following week, the teacher gives his exam and the students are surprised. I think this paradox uses the Tarskian truth model: T("[symbol]f[/symbol]") [symbol]Û f[/symbol]

24

Just glancing over this; a suprize exam seems possible only one of two ways IMO.

1.) Don’t give an exam
2.) Abstract the exam through subconscious behavior utilized in the primary exam; or apply misdirection with the primary exam; while administering the exam over the course of the entire week unbeknownst to the test takers.

Regardless; it seems a matter of jumping the system by running it one layer up or one layer back. Once that type of pattern is absorbed into ‘suprise’, I don’t think people can necessarily be suprized. After ‘creativity’ itself is mapped; then suprise becomes regularity or predictability - after this becomes equalized and mapped; throwing a single die at the beginning of the week?

It becomes increasingly more difficult over time to suprise people; unless you start to physically alter their memory through surgery.

One side effect of the ‘terrirtory redux’ phenomenon seems to be the perception of apathy in relation to it. “People don’t care about precision anymore; they already know that things happen; that something happens.”

It’s almost as if people are considered damaged goods at this point (for certain functions to be sure); which has some interesting speculations.

However; the idea of imposing suprise upon ones self seems to become a tangible goal, as a retirement of sorts, a vacation, an ‘eden’; once precision itself is reduced.

At this point; one can simulate or design a suprise for themselves - mapping them on one side; getting a quick fix for them on the other. Tasting different forms of suprise; like one would explore a buffet.

It seems that suprise must be self-imposed after a certain point though. A person can literally have suprised mapped:

“If I’m not dead, what is suprising?”

-Justhink

25

Ahhhggghh… stayin’ up late…

One aspect of the mind which makes this issue slightly more complex; is that the mind can actually abstract suprize and still run it at the same time. And I’m not talking about acting here.

The mind can literally run; “Hmm… I’m suprized again; go figure… not very suprising shrug”

This wraps around the consent issue I was noting above; yet with a different twist. Most individuals would apply a quasi-consent on allowing suprize to be an automated system for several scenarios. These would be utilized for social mores.

The other set would be set aside for ‘fight or flight’ situations; where suprise isn’t always considered the best option.

Tweaking with automated systems like this can have some very adverse effects unless you really have these systems down pat!

You could be disabling a standard system of projection that a person is looking for, so that they will not be aggressive towards you. By disabling this for all situations of ‘fight or flight’; you may be doing more harm than good to that degree.

It is generally considered optimal to always simulate the suprise response by default; but to code different levels of awareness for different situations. This allows the standard to not be breeched; while giving you the element of suprise.

Something incidentally I’m completely against; but there you go.

-Justhink

26

The smart pupil’s logic is self defeating. When I participated in that other thread, the example that made it all clear to me was to simplify it down to this:

The teacher says:
[ul]1.I will administer the test on Monday.
2.On Monday, you will be surprised when I administer the test.[/ul]
The smart pupil thinks that this is impossible, because of course if the test is administered on Monday that is what he has been told will occur, and so he will not be surprised. As a result, come Monday, he is surprised when he is administered the test.

Likewise, in the OP, the smart pupil, having concluded that the teacher cannot administer the test on any day, is surprised on Tuesday, precisely because he is defeated by his own self defeating logic.

This ties in with what Libertarian is saying. In the OP, the smart pupil concludes that the test cannot be given on the Friday, because on the Friday all options are gone, and a test on Friday cannot be a surprise. If Thursday afternoon is reached and no exam has occurred, the smart pupil will be surprised because he has concluded that the test cannot be given on Friday, and yet that is what is going to happen.

27

Princhester:

You are basically just saying that the smart student’s logic is self defeating. But we already know that - the fact that he is surprised by the exam proves it. The question is, WHAT was actually wrong with his logic?

The flaw in his logic is that his premise (there will be a surprise exam) is incompatible with his conclusion (there will not be a surprise exam). His argument is A -> Not A. And he concludes that Not A is true. But of course this is incorrect, because if the conclusion denies the premise you used to get that conclusion, you have to throw the argument out.

Actually, to be more specific his premises are A) I know there will an exam, and B) I know it will be a surprise. Now it should be quite obvious what the flaw is. His premises contradict each other! Imagine a student on Friday morning who knows there will be an exam, and knows it will be a surprise. There is no way both his premises can be true.

Even though it is true that there will be an exam, and it is true that it will be a surprise, it is NOT true that the student knows that there will be an exam and knows it will be a surprise.

Well, thats one interpretation, but I think the surprise and the exam have to happen on the same day. Also, you are relying on the student being fooled by his faulty logic.

The base of this whole paradox is that the students cannot know that there will be an exam and it will be a surprise. Even though it is true, they cannot know that it is true. The teacher knows it is true. You know it is true. But the students cannot. Even in your example, you say the students are surprised because the exam will be on Friday. But how do they know the exam will be on Friday? Because they were told it would be. But weren’t they also told it would be a surprise? So in order to conclude that the exam is on Friday, they would contradict one of their premises. The end result is that the students cannot know that their premises are true.

That is why I don’t think it is common knowledge that there will a test and it will be a surprise. In fact the students definitely do not know that there will be a test and it will be a surprise. They are the very people that cannot know, even though it will in fact happen.

It’s like if I were to say “Princhester cannot know that this sentence is true” then Princhester would be the one person who could not know that it was true, even if it is true.

28

This sentence is false.

29

Tracer has illustrated the difference between a statement and a proposition. Logic deals with propositions only.

30

Daniel Shabasson:

I just realized that my solution is almost the same as yours from the OP! I didn’t just copy you, I promise! :slight_smile:
I think the problem is changed a little if you think of it as a stack of red cards with one black card. In that case, when your premises contradicted each other on the last card, you would know which was wrong. If you got to the last card, you would know it was black and you would not be surprised. Therefore the teacher could not put the black card last. However, on the second to last card you would have the same problem as before. You would have a premise that tells you that you know the black card is next, and you would have a premise that tells you that you don’t know. You would have to doubt your premises, and thus you would be surprised.

So, the exam can be given on any day, but the black card can be placed in any posistion except last.

31

Daniel Shabasson:

I just realized that my solution is almost the same as yours from the OP! I didn’t just copy you, I promise! :slight_smile:
I think the problem is changed a little if you think of it as a stack of red cards with one black card. In that case, when your premises contradicted each other on the last card, you would know which was wrong. If you got to the last card, you would know it was black and you would not be surprised. Therefore the teacher could not put the black card last. However, on the second to last card you would have the same problem as before. You would have a premise that tells you that you know the black card is next, and you would have a premise that tells you that you don’t know. You would have to doubt your premises, and thus you would be surprised.

So, the exam can be given on any day, but the black card can be placed in any posistion except last.

32

Is this going to be on the test?
[sub]I think I’m gonna have to drop this class.[/sub]

33

Nightime, you state:

By “surprise”, I mean that the students cannot predict the day before the xam that the exam will definitely be the next day.

Nightime, you do seem to be arguing for the same solution as I am. However, you say your example with the 6 red and 1 black cards is a bit different in that the black card cannot be last, while, the exam can be given on Friday. However, I thing the two paradoxes are exactly analogous. I would argue that the black card can be last and still be a surprise for the same reasoning. There is no difference in the two paradoxes.

Libertarian, you wrote:

According to the way the paradox is set up, if the students can predict with certaintly that the exam will be the next day, then it is not a surprise. The way the paradox is set up, the students can get through all of Thursday and then think about Friday - the next day. If any student has justified true belief that the test will occur on Friday, then the exam cannot be a surprise on Friday.

Princhester, your solution to the paradox seems like mine, rather than the one that Libertarian is suggesting. Yes, the exam can be given on Friday as the student’s logic is “self-defeating” in that it does not allow him to come to a rational belief as to whether the exam will be given or not. The sudent comes to believe that the exam may not be given at all. So the exam can be given and he will not be able to predict it with certainty.

34

Daniel wrote:

Look again.

The days are school days, not calendar days. When the day ends Thursday, Friday has already begun.

The whole point of the Common Knowledge approach is that it proves that the students must be surprised (it is in fact “common knowledge”). Even if they reason that the test cannot possibly be given on any day, as in the Friday countdown, they will still be surprised when it is, well, given after all.

Incidentally, Maria Fasli is one of the world’s preeminent computer scientists, and is a top tier logician and philosopher of science. Why not give her treatment another look?

35

Hi Libertarian. If I understand you correctly, the surprise occurs at the moment school lets out on Thursday because at that point (assuming the exam has not yet been given) the students know that the exam will be on Friday, rather than Thursday. The surprise occurs a tthe moment they realize the test will be a Friday test, as they had not predicted that prior to the end of the Thursday school day, during which time they had been in complete suspense until the very last minute, when the bell finally rang.

Under this reading, the surprise happens on Thursday, while the exam takes place on Friday. So the day of the exam and the day of the surprise do not have to be the same.

Now, the way I set up the paradox (and the way I have always seen it worded), the exam cannot count as a surprise if at ANY moment before the teacher walks into the classroom and hands out the test, the student can KNOW the test will take place on that day, i.e., he has a justified true belief the exam will occur, then the exam does not count as a surprise according to the stipultated rules of the paradox. So if two minutes before class on Friday morning, any student can say “the exam has to be today” and this is a justified true belief, then the exam does not count as a surprise.

The approach you are talking about does describe a kind of surprise that happens on Thursday afternoon at the final school bell - it is no different than the surprise that the students would have at the end of the school day Monday, for then they would immediately know that the test will not be on Monday, but will instead occur on Tues, Wed, Thurs, or Fri. Moreover, this kind of surprise is no different from the surprise I a soldier in battle feels every morning when he wakes up and finds out that again, he is still alive for one more day. But it is not the kind of surprise that I have described in the paradox.

I hope I have not completely misconstrued what you were arguing.

36

Hi again Libertarian. If you like, we can re-phrase the paradox in the following way:

First, we get ride of the word “surprise”, since this is a vague thing - we can be surprise in many different ways.

Also, to avoid any confusion about when Friday school day begins and the Thursday school day ends, let’s suppose the exam can be given only on Monday, Wednesday, or Friday (Tues and Thurs are eliminated as possible test days).

So we have the paradox worded thusly:

A teacher announces he will give an exam on Mon, Wed, or Friday of the upcoming week. If on either Tuesday or Thursday, any student can (know for certain) i.e. have the justified true belief, that the exam will take place on a sepcific day, the exam cannot take place on that day.

So if on Thursday, any student can know that the Exam will be on Friday, the exam cannot be given on Friday.

The solution I have proposed is simply that the student cannot form a justified true belief about the exam having to be on Friday, as one cannot justifiable believe both that the exam must be given on Friday and that the exam cannot be given on Friday.

I realize now that this solution I advocate is exactly the one proposed by Doris Olin - I thought I diverged a bit from her thinking but I think now I do not.

37

Hi again Libertarian. If you like, we can re-phrase the paradox in the following way:

First, we get ride of the word “surprise”, since this is a vague thing - we can be surprise in many different ways.

Also, to avoid any confusion about when Friday school day begins and the Thursday school day ends, let’s suppose the exam can be given only on Monday, Wednesday, or Friday (Tues and Thurs are eliminated as possible test days).

So we have the paradox worded thusly:

A teacher announces he will give an exam on Mon, Wed, or Friday of the upcoming week. If on either Tuesday or Thursday, any student can (know for certain) i.e. have the justified true belief, that the exam will take place on a sepcific day, the exam cannot take place on that day.

So if on Thursday, any student can know that the Exam will be on Friday, the exam cannot be given on Friday.

The solution I have proposed is simply that the student cannot form a justified true belief about the exam having to be on Friday, as one cannot justifiable believe both that the exam must be given on Friday and that the exam cannot be given on Friday.

I realize now that this solution I advocate is exactly the one proposed by Doris Olin - I thought I diverged a bit from her thinking but I think now I do not.

38

Hi again Libertarian. If you like, we can re-phrase the paradox in the following way:

First, we get ride of the word “surprise”, since this is a vague thing - we can be surprise in many different ways.

Also, to avoid any confusion about when Friday school day begins and the Thursday school day ends, let’s suppose the exam can be given only on Monday, Wednesday, or Friday (Tues and Thurs are eliminated as possible test days).

So we have the paradox worded thusly:

A teacher announces he will give an exam on Mon, Wed, or Friday of the upcoming week. If on either Tuesday or Thursday, any student can (know for certain) i.e. have the justified true belief that the exam will take place on a specific day, the exam cannot take place on that day.

So if on Thursday, any student can know that the Exam will be on Friday, the exam cannot be given on Friday.

The solution I have proposed is simply that the student cannot form a justified true belief about the exam having to be on Friday, as one cannot justifiable believe both that the exam must be given on Friday and that the exam cannot be given on Friday.

I realize now that this solution I advocate is exactly the one proposed by Doris Olin - I thought I diverged a bit from her thinking but I think now I do not.

39

I’m not understanding where you are concluding that the ‘smart’ person must be suprised if they are told that they will be suprised that a test will be given on Monday, when it actually is given on Monday.

To some degree I attempted to illustrate this with one of my two conditions: Do not give the exam (ever).

By abstracting this one layer back; you are effectivly embedding the exam into them; until such time as it is given. This acts as a floating state of ‘exam embodyment’ by the issue of a command from another person.

This is considered “anything you do must be the exam on Monday; and only in the way that will suprise you.”

Instead of the test itself being used, the teacher is abstracting “suprise” one later back… they are embedding it INTO the pupil - as a floating state of perspective for judgement.

The pupil is recursively being defined AS suprise until such time until the abstraction snaps back out from the actual issuance of a suprise. This falls into the ability to render glory from zero, or something from nothing – to that degree.

-Justhink

40

The talking seems to focus on throwing out surprise as a necessity; (i.e. It is always implied that the student will be suprised - now how does the student actually figure out which day it is)

Suprise is assumed without evidence. Even if the student accurately predicts the exam; they may be suprised by anything thirty years down the road; at which point the teacher chimes in “See, I told you you’d be suprised by the exam on Monday!”

Suprise is being embedded into the student, regardless of evidence. Anything which issues a suprise response is being abstracted as necessarily being an exam. By stating the day “Monday”; one is entertaining an audience by performing a magic trick so to speak.

-Justhink