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Alky
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 Re: Pop Quiz Riddle   « Reply #200 on: Feb 21st, 2006, 9:16am » Quote Modify

Hold on.

If the test is on tuesday, then they won't know the test is on friday thursday night. Only in the case where the test is on friday can they find out it is friday. If it is before friday then they obviously can't figure this out.

Maybe the problem we are facing is that we are assuming because we have "if the test is on friday then they will know it's on friday" is enough to show the test won't be on friday. If the test is on tuesday they won't see it coming. For example, to guess friday they would have to wait until thursday night (they only get one guess) so if it is any other day they lose.
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River Phoenix
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 Re: Pop Quiz Riddle   « Reply #201 on: Feb 21st, 2006, 5:10pm » Quote Modify

on Feb 21st, 2006, 3:28am, towr wrote:
 It isn't nonsense. It's actually a common problem you face in dynamic epistemic logic with an update operator. Certain updates don't lead to knowledge because believing the information leads to a contradiction. This is exactly what the students are facing.

I was quoting some things that others had said earlier in the thread, namely that since you reach a contradiction, then one of the assumptions must have been wrong. Are you saying that this is not the case? I am very interested to understand how the problem works; I can see that it all boils down to the issue that "belief in X" --> "not X", as you say. What other problems contain this dilemma? Does it relate to Godel's incompleteness theorem?
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 Re: Pop Quiz Riddle   « Reply #202 on: Feb 22nd, 2006, 12:54am » Quote Modify

on Feb 21st, 2006, 5:10pm, River Phoenix wrote:
 I was quoting some things that others had said earlier in the thread, namely that since you reach a contradiction, then one of the assumptions must have been wrong.
The important things is, is that the contradiction is in your knowledge/beliefs about the world, not in the state of the world you're in. Propositional logic doesn't cut it in this case, what you need is some form of modal logic. So instead of just stating what is the case, you can talk about what you believe is the case (which isn't necessarily the same thing). And next to that you need something to 'update' your knowledge/beliefs.

It now all depends on how you define these modal operators; typically if you define 'knowledge', K(X) ("I know that X"), then whatever you know must be true: K(X) -> X. With belief that's not necessarily the case, you might belief things that aren't the case. Further you know what you know, and know what you don't know: K(X) -> K(K(X)) and ~K(X) -> K(~K(X))
If you remember Rumsfeld's famous words, it makes sense when you write it out.
there are A,B,C such that K(K(A)) & K(~K(B)) & ~K(~K(C))
There are known knowns, known unknowns and unknown unknowns.

Anyway.. What causes the paradox lies in the other aspect, updating your knowledge. Allowing every update to be succesfull may cause your beliefs/knowledge to become inconsistent. So one solution is not to allow such update to succeed.
It's believing what the professor says that causes the problem, not whether what he says is true.

http://www.illc.uva.nl/Publications/Dissertations/DS-2003-01.text.pdf goes into dynamic epistemic logic in section 4.3, and probabilistic epistemic logic in 6 (where there's a very brief mention of unsuccesfull updates on page 104)

Quote:
 Are you saying that this is not the case? I am very interested to understand how the problem works; I can see that it all boils down to the issue that "belief in X" --> "not X", as you say.
More accurately it's "belief in X" --> "belief in not X"
Reality doesn't change because you belief something, but your belief may become inconsistent if you'll believe anything people say.
And since you don't want inconsistent beliefs, you shouldn't belief everything; not even if it's true because it may become false by believing it.

Quote:
 What other problems contain this dilemma?
I don't know more examples from the top of my head, at least not ones that are basicly the same. There's a version with a quiz, with 5 boxes one of which has a prive (i.e. just like 5 days, one of which has a surprise test). And there's the hangman's paradox (make the execution of the prisoner as much of a surprise as possible.)

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 Does it relate to Godel's incompleteness theorem?
I don't think so. It's only a problem if you define your logic in the wrong way; that's to say, 'wrong' if you think it's bad to belief contradictions. I'm not sure how consistent people's beliefs are in reality though, so it may not have to be a problem to belief contradictory things
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River Phoenix
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 Re: Pop Quiz Riddle   « Reply #203 on: Mar 6th, 2006, 2:43pm » Quote Modify

on Feb 22nd, 2006, 12:54am, towr wrote:
 I don't think so. It's only a problem if you define your logic in the wrong way; that's to say, 'wrong' if you think it's bad to belief contradictions. I'm not sure how consistent people's beliefs are in reality though, so it may not have to be a problem to belief contradictory things

Descartes argues "I think, therefore I am", by using the construction that A="I think", and "disbelief in A" -> A
Does this mean that his argument might be wrong, and it is simply his act of doubting that is fallacious?
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 Re: Pop Quiz Riddle   « Reply #204 on: Mar 7th, 2006, 1:02am » Quote Modify

on Mar 6th, 2006, 2:43pm, River Phoenix wrote:
 Descartes argues "I think, therefore I am", by using the construction that A="I think", and "disbelief in A" -> A Does this mean that his argument might be wrong, and it is simply his act of doubting that is fallacious?
I don't really see why there'd be a problem there. I mean, aside from that he might not be the one thinking.

What you have is:
Ex.T(x)
T(~Ex.T(x)) => Ey.T(y)
The implication is true. And the former follows from experience, he's aware that he thinks.

If he thinks that he thinks, than disbelieving that he thinks would be inconsistent, so logically he should believe he does think.
Whether that translates to knowledge is really more complicated. It's a bit odd to start with defining thought in such a way that you have to exist to do it; then go on to claim you think; and then to claim that therefor you exist.
For one thing, are you really doing what you defined as thought?
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 Re: Pop Quiz Riddle   « Reply #205 on: Jun 15th, 2006, 10:47am » Quote Modify

My opinion (just an opinion, I am not good with riddles and I do not claim it's worth to be called a solution):

If the definition of "surprise test" is "a test so that NOW you can not determine which day it is", then it can be on any day with 1/5 probability. Obviously this can not be the case, it would be too easy.

Let's focus on the problem definition.

Let's say now that the definition is "a test so that at any day before it you can not determine which day it is". Then, since the whole domain is discrete, there is no "intermediate" point at which you can say "there has been no test today", because after a day there is immediately the next day, and on Thursday you can not assume the test has not been given on the same Thursday until it is already Friday, thus nullifying the students' logic.

If the definition is "a test so that at any time before it you can not determine which day it is", then the problem definition is probably incorrect, or at least has little sense.
Time is continuous, while days are discrete. This causes the paradox, because the students can reason in the continuous domain while the test can only be placed on points in a discrete domain. Maybe that makes sense apparently, but I am inclined to consider it a poorly specified (and thus potentially flawed) problem.

Now, what about a totally continuous space (so that the test can happen in any instant in the continuum and the definition of "surprise" is the continuous one)? Would the students' logic be consistent in a really continuous domain?
Definitely not.
For any two given instants, there are uncountable infinite instants between them, unless they are the same instant. Therefore, you could say that the test can not take place on the last instant of your time interval, but you can not define the "previous" instant, and you cannot "propagate" the "can not happen" backwards.

I hope I explained my point clearly, even though I doubt it (sorry I am not english born and I struggle to find words sometimes).

A side note that might be interesting...

I think this pop quiz relates to a paradox in probability laws as well.

The students say that on Thursday evening they know that the test will be on Friday if the prof didn't give it on any other day.
Therefore we can say that the conditioned probability of the test being on Friday is 1. By "conditioned" I mean "assuming the statement that the test has not been on the previous days is true". However, that statement itself has a probability.

At the beginning, the probability of the test being given in any single day is 1/5 (one divided by the number of valid days).
On Monday evening, if the test has not been given, the probability it will happen on any other day is 1/4. However, the probability that the test has not been given on Monday is 4/5. Therefore, the probability of the test being given on any other day than Monday is still 1/5 (because it's equal to 1/4 * 4/5).

You can iterate for all the days - the probability stays the same. But shouldn't it change day by day?
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rmsgrey
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 Re: Pop Quiz Riddle   « Reply #206 on: Jun 15th, 2006, 2:27pm » Quote Modify

The riddle still works if you discretise everything: suppose the test would be given at noon (in the regular slot for the lesson) and guesses have to be made at 4pm each day (at the end of classes) so the week becomes 10 discrete and separate chunks of time (4pm the previous Friday, noon and 4pm Monday, noon and 4pm Tuesday, noon and 4pm Wednesday, noon and 4pm Thursday, and noon Friday).

Odd numbered time blocks are for guessing when the test will be; even numbered for finding out if you were right (or having the test)

If you start with the assumption that each day is equally likely at 1/5, then it's hardly surprising if valid manipulations bring you back to the conclusion that each day is equally likely at 1/5.
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chanklas
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 Re: Pop Quiz Riddle   « Reply #207 on: Jun 20th, 2006, 9:17pm » Quote Modify

Sorry if this duplicates someone else's post (I jumped from page 2 to page 9 on the forum, and at the end of page 9 it didn't look like someone posted this)...

...anyway, isn't the solution just that the professor knows the students are following their logic, which leads them to conclude that there can be no surprise test, so therefore when he DOES give one on Tuesday (or any day for that matter) it's sure to be a surprise?
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Icarus
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 Re: Pop Quiz Riddle   « Reply #208 on: Jun 22nd, 2006, 5:19pm » Quote Modify

Yes, this is why the students are surprised. But the whole point of this riddle is, why did the students' logic fail?

They had an apparently logical proof that there could be no test, because it would not be a surprise. Yet, they had a test and were surprised by it. So, what was wrong with their proof?

One other thing: You can't be count on reading the latest discussion to let you know anything about what has been mentioned before. If there was one continuous discussion in this thread with everybody cognizant of the whole topic, then you might expect to get an idea of how it went by this page alone. But then you would have just spoiled it for the next guy because you have commented without knowing what went on before, effectively introducing a new thread of conversation that isn't dependent on earlier replies. So the next guy is going to be misled.

I say "would have" because, of course, you are not the first to skip. The conversation has been very disjointed, and so the only way to know what has been brought up is to read through it.
 « Last Edit: Jun 22nd, 2006, 5:45pm by Icarus » IP Logged

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Roy42
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 Re: Pop Quiz Riddle   « Reply #209 on: Jul 25th, 2006, 12:19am » Quote Modify

If the whole point of the riddle was, why was the students logic flawed? the answer would be they were wrong. that's it. i belive the question posed was, what was the flaw in the students logic.
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 Re: Pop Quiz Riddle   « Reply #210 on: Jul 25th, 2006, 1:21pm » Quote Modify

The answer to why has multiple layers. Demanding that the answer be only the first layer, and the second layer is somehow an entirely different question (one which I also stated explicitly in my previous post), is ludicrous.
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 Re: Pop Quiz Riddle   « Reply #211 on: Aug 1st, 2006, 10:57am » Quote Modify

Having read some very interesing positions in this thread, here's my two cents:

- With respect to the definition of surprise and the argument that you can't announce an upcoming "surprise" quiz.  I believe this argument is false - there are many courses where, at the beginning of the term, the students learn that there will be surprise quizzes during the term.  Empirically speaking, when they happen, they still tend to be surprising.  (Even though the student's logic could actually be applied in a situation where multiple surprise quizzes are expected)
- The professor, in combining the announcement with the offer has put the students into a paradox where a surprise test is unavoidable.  With them confident each day that the test cannot be that day, the professor can surprise them as he pleases (since they're not expecting a test).  This includes Friday, as, since they would enter class on Friday confident that the test would have been predictable that day, they won't be expecting it and therefore be surprised.
- The students failed to predict the day it would be on - they said it couldn't be on any day.  Thus he did not have to cancel the surprise quiz.
- Who's to say that the surprise has anything to do with the timing?  Perhaps the subject matter was the surprise.
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Roy42
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 Re: Pop Quiz Riddle   « Reply #212 on: Aug 1st, 2006, 2:21pm » Quote Modify

True, he may have before the class, just rolled a die to decide what day he would make the quiz, possibly he also made six on the die an option to not have the quiz as the student predicted. But he probably rerolled, i mean, where's the fun in no quiz?!
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 Re: Pop Quiz Riddle   « Reply #213 on: Nov 20th, 2006, 12:24am » Quote Modify

I think the answer may be as simple as next week never comes and the professor opts to give the quiz on tuesday of the CURRENT week.
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rmsgrey
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 Re: Pop Quiz Riddle   « Reply #214 on: Nov 20th, 2006, 8:20pm » Quote Modify

on Nov 20th, 2006, 12:24am, Emmit1977 wrote:
 I think the answer may be as simple as next week never comes and the professor opts to give the quiz on tuesday of the CURRENT week.

The question of the riddle is not "how could the professor give a test?" but "what is wrong with the students logic that concludes that the test can't happen?" - the students' logic wouldn't be affected if the professor had said instead: "there will be a surprise test at one of these 5 times unless you manage to correctly predict which of these occasions the test would have been on"
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divisionst06
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 Re: Pop Quiz Riddle   « Reply #215 on: May 23rd, 2007, 12:16pm » Quote Modify

It's a paradox. The error to their logic is waiting until thursday to reply to the test. If the test is Thursday they are a day late with their response. The teacher makes the statement "GNice to see your thinking about it." Unless that's a varient in the riddle that would imply that more than one guess could be made. Thus, the error in their thinking would be trying to eliminate choices that are all valid rather than optimize their chances of getting the right answer.
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 Re: Pop Quiz Riddle   « Reply #216 on: May 29th, 2007, 9:47am » Quote Modify

I think I got it.
I got it.
But I may be wrong, so please feel free to attack my solution from any angle.

The professor says: "We're going to have a surprise quiz next week, but I'm not telling you what day... if you can figure out what day it will be on, I'll cancel the quiz." (format by me)

The use of the undeterminative "a" means that nothing prevents the professor from scheduling two surprise quizzes for next week (in fact the professor can schedule any number of quizzes he wishes, because they are surprise quizzes, so he is not obliged to tell the students).
If the professor can give two quizzes, the students' logic fails, because the assumption "if the quiz is on day X it can not be on any day after X" fails.
Note that the students have the possibility of canceling one quiz. However, they can not know which one of the two quizzes is the one whose date they have to guess in order to cancel it.
So they can not say "if the quiz was on Friday we would know by Thursday", because they can never know if the test given on, say, Tuesday, is the one added by the professor or the one they can actually cancel.

And it's all down to first-order logic. When I started writing the axioms of the problem, I realized that existance does not imply uniqueness.

And now, here's a more mathematical explanation:

--- Problem description ---

Value set:
T = {1,2,3,4,5}

If the students guess, then there can be no test on that day:
ForAll t in T: (ScheduledTest(t) AND StudentsSay(t) ) => NOT GivenTest(t)

If the students do not guess right, the test is given:
ForAll t1, t2 in T: ( (t1 != t2) AND StudentsSay(t1) AND ScheduledTest(t2) ) => GivenTest(t2)
In other words, if a quiz is scheduled on any day different from what the students said, that quiz will be given.

The students can guess only one day (this is expressed using a mathematical trick with equivalence to shorten the expression):
ForAll t1, t2 in T: ( StudentsSay(t1) AND StudentsSay(t2) ) => t1 == t2

The professor schedules a test next week:
Exists t in T: ScheduledTest(t)     <---- this is the crucial one. There could be more than one t.

There is a test on Tuesday:
GivenTest(2)

--- End of problem description ---

The students' assumption is based on another, simpler assumption:
ForAll t1 in T: ScheduledTest(t1) => ( ForAll t2 in T: (t2 > t1) => NOT ScheduledTest(t2) )

Which is clearly and evidently false.
If he wanted, the professor could say
ForAll t in T : ScheduledTest(t)
Which does not contradict any of the problem's axioms.

You may say that's cheating.
I think it's perfect first-order logic.
 « Last Edit: May 29th, 2007, 10:45am by shadebreeze » IP Logged
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 Re: Pop Quiz Riddle   « Reply #217 on: May 29th, 2007, 10:38am » Quote Modify

(Sorry, I may have skipped a passage which is not that easy)

The students' full assumption is "the test can't be on the last available day", and if you apply recursion you get to their conclusion.
Mathematically:

ForAll t1 in T : ( ( ForAll t2 in T : (t2 > t1) => NOT ScheduledTest(t2) ) AND ( ForAll t3 in T : (t3 < t1) => NOT GivenTest(t3) ) ) => NOT ScheduledTest(t1)

(for every t1, if no test is scheduled on any t2 greater than t1, and no test had been given on a t3 before t1, then no test can be scheduled on t1)

This works only if there is only one test scheduled for the week.
In that case, the students start from
SheduledTest(5)    <--- the test is on the last day
StudentsSay(5)     <--- it is not a surprise
And they have NOT GivenTest(1) AND NOT GivenTest(2) AND NOT GivenTest(3) AND NOT GivenTest(4) because there is only one test.
So both parts of the premise in the students' hypothesis are true, and they get a contradiction ( NOT ScheduledTest(5) ),  so ScheduledTest(5), which was their starting point, can't be true. And you can then use recursion.

It is immediately proven false when there are two scheduled tests or more and they are on different days, because the students can cancel only one of them. So, when they assume ScheduledTest(5) and StudentsSay(5) to start the recursion, there is at least another t so that GivenTest(t < 5).
So their assumption does not hold, and it is perfectly possible that ScheduledTest(5). The starting point of the recursion is removed, and the students' reasoning fails.

Here is an example:

ScheduledTest(2)
ScheduledTest(5)
StudentsSay(5)

Since the students can only say one day, the obvious conclusion is:
GivenTest(2)

Therefore,
... ForAll t3 in T : (t3 < t1) => NOT GivenTest(t3) ...
is false for t1 = 5, and they do not have the start of recursion.

 « Last Edit: May 29th, 2007, 11:01am by shadebreeze » IP Logged
towr
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 Re: Pop Quiz Riddle   « Reply #218 on: May 29th, 2007, 10:54am » Quote Modify

The students reasoning is wrong even if there is just one test.
Nothing fundamentally changes if the professor said "There will be exactly one surprise test next week, unless you can deduce on which day I plan to give it, in which case there won't be any test"
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 Re: Pop Quiz Riddle   « Reply #219 on: May 29th, 2007, 11:34am » Quote Modify

As many people have already said, this paradox is not clear on its terms: specifically, the notion of surprise.  Also, the fact that the teacher says that the test will be a surprise most likely alters the probability of the test being on day x.  For instance, if the teacher said that there would be a test the following week (without mentioning 'surprise'), distribution over the days would be more or less uniform.  However, once the teacher says that the test next week will be a surprise, the distribution changes for each day, assuming the teacher plans out which day the test will be on.  The teacher, though, could still roll a die to determine which day the test would be on, or any like action (picking a day out of a hat, etc.)
This information is not privy to the students or us.  So yes, they are correct in their logic that if they make it through Thrusday with no test, then the test will not be a surprise.  This logic shows that the students assume that the test can be on Friday.  Then the students dismiss Friday as a possibility, showing that they assume that the teacher picked the day of the test to be a surprise.  However, if the teacher had rolled a die or anything similar to choose the day of the test, the students' logic fails, and they cannot eliminate Friday as a possibility.
I think the paradox comes from these assumptions, coupled with an assumed tacit response from the teacher.  For if the students could honestly eliminate Friday as a possibility (and have their teacher agree that the test would not be on Friday), then their logic would work; eliminating the last day possible until there would be no days left.
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 Re: Pop Quiz Riddle   « Reply #220 on: May 29th, 2007, 2:02pm » Quote Modify

Uhm, so it's not that there is more than one planned quiz... interesting.

Yes the definition of surprise is difficult to express even mathematically...
My definition was:
ForAll t1, t2 in T: ( (t1 != t2) AND StudentsSay(t1) AND ScheduledTest(t2) ) => GivenTest(t2)
but since the students' reasoning works perfectly fine in first-order logic if you take the assumptions I wrote, I am inclined to think there is a flaw in those assumptions, and the definition of surprise seems to be the weakest. So I am planning to consider a more "literal" one, and see what comes out of it.

Furthermore, there is another assumption I have made, which is that if a test is given, it had been scheduled.
But that is not necessary, it suffices that the professor just gives the test one day.

In fact, if we define a surprise test this way:
ForAll t in T : ( ( NOT StudentsBelieveTestIsOn(t) ) AND GivenTest(t) ) <=> SurpriseTest(t)

Then, we could say that the students' logic builds a system where

ForAll t in T : NOT StudentsBelieveTestIsOn(t)

So the professor has all the freedom to give the test any day.
Basically, the students' logic can not alter reality, it only alters their belief (as it has been suggested already in this thread). But if the definition of surprise is based on belief, then the problem may be easier than we think.

I'll write down all the mathematical dribble and see what I can get out of it, but I think I may be on a good lead.
Otherwise, if I cannot use that definition, writing down the maths is just an excercise in logic... I'll do it anyway because it is fun, but I won't post it if somebody tells me now that that definition of surprise is wrong.
 « Last Edit: May 29th, 2007, 2:06pm by shadebreeze » IP Logged
divisionst06
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 Re: Pop Quiz Riddle   « Reply #221 on: May 30th, 2007, 5:06am » Quote Modify

Correct me if I'm wrong:

Surprise acts as a modifier of the word "test" not the date of the test. If the test is a surprise, by definition it could just as easily mean the materials the test covers have not been established, the materials on the test would be the surprise in this instance. So, if the date of the test is not the surprise then the error in their logic is that the date of the test is a "surprise." It's the difference between "There is a surprise test next week," and "There will be a test on a surprise date next week."
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 Re: Pop Quiz Riddle   « Reply #222 on: May 30th, 2007, 6:08am » Quote Modify

on May 30th, 2007, 5:06am, divisionst06 wrote:
 Correct me if I'm wrong:   Surprise acts as a modifier of the word "test" not the date of the test. If the test is a surprise, by definition it could just as easily mean the materials the test covers have not been established, the materials on the test would be the surprise in this instance. So, if the date of the test is not the surprise then the error in their logic is that the date of the test is a "surprise." It's the difference between "There is a surprise test next week," and "There will be a test on a surprise date next week."
It's an interesting interpretation; and the students would still not be able to deduce when the test is.
But in common vernacular "surprise test" does mean that the time is unknown, not that the content is known. A math teacher that gives a surprise test wouldn't given a test on english literature; no matter how surprising that would be.
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 Re: Pop Quiz Riddle   « Reply #223 on: May 30th, 2007, 6:36am » Quote Modify

I just came across that exploring multiple literal definitions of the word "surprise." It can mean to happen without warning or to happen without previously being established. Also, is there any way (provided we assign a value to each day of the week) in which tuesday would be the counter-example to their logic?
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 Re: Pop Quiz Riddle   « Reply #224 on: May 30th, 2007, 6:42am » Quote Modify

I'm not sure what you mean. If there is any test at all (tuesday or otherwise), it's a countermodel to their logic, because that told them there wouldn't be a test.
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