Another way to look at the types of word games that allow the paradox:
“When you first concieved the concept “suprise” (new information); it may not have been a suprise to you UNTIL now!
Suprise! That was actually a suprise, you just didn’t know it yet.
Astonishingly, it came up again, in a way you didn’t expect it to.
You must by definition be suprised NOW no matter what.”
-Justhink
i.e. “I bet that you could not have ever guessed that you were going to meet me, and that I was going to threaten a recursive function upon you; to such a degree that you have to disprove my existence in order to not be suprised by my decree.”
-Justhink
Daniel
But that’s the whole point. He can’t know on Thursday that the exam will be on Friday. He can’t know until Thursday is over. If the test is on Friday, then the day he knows and the day he gets the test are the same. In fact, that is true for any day of the week.
Rewording the paradox doesn’t help. You haven’t solved 2x=5 if you change the equation to 3x=5. Once again, Fasli’s approach is merely a different way of looking at the problem, not a different way of stating it.
Daniel
Look at her argument’s conclusion (which is proven to be true) one more time and meditate on it a bit:
CK[sub]G/sub [symbol]Þ[/symbol] CK[sub]G/sub
:eek:
There is another possibility that none have brought forward.
The teacher gives an exam on Monday.
Then, the teacher gives the surprise exam on any other day of the week. The children, thinking that Monday’s exam was the surprise one, would be completely surprised at a second exam.
Therefore, surprise is guaranteed without a messy paradox.
It looks like the definition of ‘surprise’ might play a role here. The way I first heard it was the way it’s described in the article by Hall (linked to above):
Fasli’s formulation seems slightly different :
It’s clear that the first statement of the problem disallows a surprise to occur on Thursday after class. In this case, Daniel’s rewording of the problem is justified, since all he did was reduce the number of potential test days (and I think we can say as long as the number of days > 1 it doesn’t matter), and mapped ‘Thursday class’ & ‘Thursday after class until Friday class’ to different days.
Whether or not the ‘common knowledge’ logic approach can be used to solve the first problem in the same way, I’m not sufficiently educated to say.
“”"""""""""There is another possibility that none have brought forward.
The teacher gives an exam on Monday.
Then, the teacher gives the surprise exam on any other day of the week. The children, thinking that Monday’s exam was the surprise one, would be completely surprised at a second exam.
Therefore, surprise is guaranteed without a messy paradox.""""""""
MY posts have been addressing the points directly related to variations of this idea. I fact I mentioned this exact scenario; amongst others which are accordingly related.
In general, the point raised was about the role that abstraction plays on the human capacity to be suprized vs. the ability for someone else to play covert ‘games’ with abstraction. shrug
I’m not known for clarity though…
-Justhink
Justhink wrote:
What do you mean exactly?
Hi Libertarian.
The way the paradox is set up, there is always intervening time in between the school days. In between the end of the school day on Thursday and the beginning of the school day on Friday, there are a number of hours during which the student may thin about the future and future events. He may make PREDICTIONS about the future. The paradox states that if during these intervening hours between Thursday and Friday, the student can know that the exam will be on Friday, the exam cannot be given on Friday.
The exam cannot be given on Friday whether the student knows about the Friday exam one minute before it is handed out. It does not matter how much time in advance he is able to know this- whether one day or one second.
Look at the paradox using the example of the 6 red and 1 black card. Clearly, there is a time interval between the turning over of each card, during which time someone may make predictions about the next card. So with the students, there is an interval between each school day. I tried to make this more clear by giving the student ALL of Thursday in which to contemplate what will happen on Friday.
The approach you are suggesting says that the surprise happens at the instant the school day on Thursday ends, as the student knows the exam will be Friday. But the exam does not get handed out at this time. There are intervening hours between the end of Thursday school day and the handing out of the exam on Friday. The exam does not occur on Thursday afternoon when the bell rings.
It seems that Fasli’s approach, if I am understanding it correctly (through your description), lumps together Thursday and Friday into one unit and does not allow any time interval between them. This is a different fact pattern from the one stated in the paradox.
Also, the surprise has to occur simultaneously with the moment the exam is handed out. This is because in the way the paradox is worded, the student CONCULUDES using the (false) logic that the exam cannot be given, so he is surprised when he sees that the exam is being passed out – he is surprised BECAUSE HE THOUGHT THE EXAM COULD NOT BE GIVEN AT ALL, (WHEN IT IS IN FACT POSSIBLE) – his surprise is that what he thought was impossible actually happened. But the student cannot have this kind of surprise at all on Thursday afternoon, since before he sees that the exam is passed out, there is nothing at all to contradict the conclusion he has reached. As a matter of fact, the fact that another day passed without the exam occurring should inductively strengthen his conviction that his conclusion – that the exam could not occur, was correct after all.
A bibliography taken from this link:
http://www.magnolia.net/~leonf/paradox/hanging.txt
D. O’Connor, “Pragmatic Paradoxes,” Mind 57:358-9, 1948.
L. Cohen, “Mr. O’Connor’s ‘Pragmatic Paradoxes,’” Mind 59:85-7, 1950.
P. Alexander, “Pragmatic Paradoxes,” Mind 59:536-8, 1950.
M. Scriven, “Paradoxical Announcements,” Mind 60:403-7, 1951.
D. O’Connor, “Pragmatic Paradoxes and Fugitive Propositions,” Mind 60:536-8,
1951
P. Weiss, “The Prediction Paradox,” Mind 61:265ff, 1952.
W. Quine, “On A So-Called Paradox,” Mind 62:65-7, 1953.
R. Shaw, “The Paradox of the Unexpected Examination,” Mind 67:382-4, 1958.
A. Lyon, “The Prediction Paradox,” Mind 68:510-7, 1959.
D. Kaplan and R. Montague, “A Paradox Regained,” Notre Dame J Formal Logic
1:79-90, 1960.
G. Nerlich, “Unexpected Examinations and Unprovable Statements,” Mind
70:503-13, 1961.
M. Gardner, “A New Prediction Paradox,” Brit J Phil Sci 13:51, 1962.
K. Popper, “A Comment on the New Prediction Paradox,” Brit J Phil Sci 13:51,
1962.
B. Medlin, “The Unexpected Examination,” Am Phil Q 1:66-72, 1964.
F. Fitch, “A Goedelized Formulation of the Prediction Paradox,” Am Phil Q
1:161-4, 1964.
R. Sharpe, “The Unexpected Examination,” Mind 74:255, 1965.
J. Chapman & R. Butler, “On Quine’s So-Called ‘Paradox,’” Mind 74:424-5, 1965.
J. Bennett and J. Cargile, Reviews, J Symb Logic 30:101-3, 1965.
J. Schoenberg, “A Note on the Logical Fallacy in the Paradox of the
Unexpected Examination,” Mind 75:125-7, 1966.
J. Wright, “The Surprise Exam: Prediction on the Last Day Uncertain,” Mind
76:115-7, 1967.
J. Cargile, “The Surprise Test Paradox,” J Phil 64:550-63, 1967.
R. Binkley, “The Surprise Examination in Modal Logic,” J Phil 65:127-36,
1968.
C. Harrison, “The Unanticipated Examination in View of Kripke’s Semantics
for Modal Logic,” in Philosophical Logic, J. Davis et al (ed.), Dordrecht,
1969.
P. Windt, “The Liar in the Prediction Paradox,” Am Phil Q 10:65-8, 1973.
A. Ayer, “On a Supposed Antinomy,” Mind 82:125-6, 1973.
M. Edman, “The Prediction Paradox,” Theoria 40:166-75, 1974.
J. McClelland & C. Chihara, “The Surprise Examination Paradox,” J Phil Logic
4:71-89, 1975.
C. Wright and A. Sudbury, “The Paradox of the Unexpected Examination,”
Aust J Phil 55:41-58, 1977.
I. Kvart, “The Paradox of the Surprise Examination,” Logique et Analyse 337-344, 1978.
R. Sorenson, “Recalcitrant Versions of the Prediction Paradox,” Aust J Phil
69:355-62, 1982.
D. Olin, “The Prediction Paradox Resolved,” Phil Stud 44:225-33, 1983.
R. Sorenson, “Conditional Blindspots and the Knowledge Squeeze: A Solution to
the Prediction Paradox,” Aust J Phil 62:126-35, 1984.
C. Chihara, “Olin, Quine and the Surprise Examination,” Phil Stud 47:191-9,
1985.
R. Kirkham, “The Two Paradoxes of the Unexpected Hanging,” Phil Stud
49:19-26, 1986.
D. Olin, “The Prediction Paradox: Resolving Recalcitrant Variations,” Aust J
Phil 64:181-9, 1986.
C. Janaway, “Knowing About Surprises: A Supposed Antinomy Revisited,” Mind
98:391-410, 1989.
I was just last night reading from “Sweet Reason” byu Tymoczko and Henle and they mention
Kaplan, David and Montague, Richard (1960). A Paradox Regained, Notre Dame Journal of Formal Logic 1:79-90. Reprinted in (Montague 1974).
I’m not known for clarity though…
“”"“What do you mean exactly?”"""
That was mildly funny!
I think all you have to do is read a vast majority of replies to my posts to determine what I meant by that statement.
-Justhink
actually, i’m pretty sure this isn’t even a paradox. as far as i know, the only paradoxes still considered “interesting” (by those who spend their lives playing with paradoxes) are Russell’s paradox and the other paradox mentioned by Russell, called berry’s paradox.
here’s what the exam paradox amounts to:
assumptions:
claim: these are contradictory assumptions, and can be used to prove any claim.
“proof”. already sketched out. the exam can’t be on friday, blah blah blah, so it can’t be on thursday…therefore the exam will not be a surprise. which is a contradiction.
since the assumptions are contradictory, and a contradiction implies everything, this pair of assumptions can be used to prove anything. trivially. (p -> ~p) -> x for all x. so it can be used to imply that the exam was not given, or that it was given and it was a surprise, or that it was not given, and that was a surprise, or what have you. even a contradiction.
by the definition of a proposition (that is, x -> x for all x),
(p -> ~p) -> (p -> ~p).
so it can even imply its own contradictory assumptions. so this isn’t really a paradox, just a fun logic trick to play on people.
-d x 2^n, for n=1.
I think you got it, Daniel. If the student believes that the teacher’s statements are completely true - it leads to paradox. But even a small doubt that the exam might not take place means that the exam will be a surprise no matter when it is given.
That’s a little unsatisfying, because the student might be 99.99% sure that the test will be given that day, but we still call it a surprise, since he wasn’t 100% sure.
Here’s what I find interesting: It’s the student’s belief that both statements cannot be true that makes it possible for both statements to actually be true.
Hi Ramanujan. You are a bit hasty in dismissing this paradox. You call the assumptions “contradictory” and you say that you can derive anything from a contradiction, such as p and - p. However, the assumptions, namely,
are in fact NOT contradictions, as demonstrated by the fact that the exam CAN happen. 1 and 2 are both true, as a matter of fact.
Moreover, I fail to see how a paradox that has generated so many articles (see above post by djb) and is one of the only paradoxes that has not unerversally accepted solution can be “uninteresting.”
You are correct in saying that Russell’s paradox is important. That paradox can be described in the following way:
Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself.
I don’t know however, whether this paradox is that interesting qua paradox, or whether its importance lies more in its import for set theory, logic, math, etc. You say the surprise exam paradox is merely a “trick” – well, that’s what a paradox always is - a well hidden trick. The paradox can be solved by curing exposing the trick.
Hi Heresiarch. Yes, I agree that this is the interesting thing about this paradox: both assumptions may be true (and necessary), yet we cannot know their truth.
Your point about the probability of the exam, in the studenty’s mind, is very interesting. I think the student’s estimation of the probability of the exam taking place can vary anywhere from close to zero to close to 100%. I think his probability estimation will depend on his beliefs about the teacher’s reliability, etc., whether the teacher is lying about the exam, etc. All we need to escape the paradox is that the student not be able to have 100% certain fore-knowledge of the exam.
I guess it really depends on your definition of “suprise.”
You see, I was in that class. No we weren’t suprised that that egotistical bastard had done this, it was in a long line of statements and actions whose sole purpose was to simultaneously deride us and elevate him. Yes we listened day after day about how if Pascal was alive and in the class he would be making important advancements in physics while we, the dolts we were, could barely handle the spoon feeding he was giving us. His stories about his intellectual battles with the evil Dr C- were a near daily occurance. We learned all about how the dean of sciences was an idiot. He even spelled out that this suprise exam was a paradox, but he didn’t think anyone would really understand it.
Now what lofty course did he teach?
Were we striving for the pricipal axis theorem in linear algebra? - No.
Were we plowing through the most neccessary Clasical Mechanics? - No.
Were we exploring the intricacies of Statistical and Thermal Physics? -No.
Were we studying his Nobel Prize winning work in Physics? -Hah!
The course was “Electronics for Engineers and Scientist.” Resistors, Capacitors, Diodes, etc–Mostly construction of simple devices.
So yeah, I guess we were surprised that he thought we’d be suprised.