A professor walked into class on Monday and stated, "We
will have a suprise exam sometime this week. On the morning
of the exam you will not know that the exam will be that day."
A logic student reasoned with himself like this: "We can’t have
the exam on Friday, since, knowing that the exam must be this
week, we’d know that it would have to be that day. Since we
know it can’t be Friday, then we know it also can’t be Thursday,
since it we’d know on Thursday that it must be on that day because
it can’t be on Friday." This reasoning was extended all the way to
Tuesday. (The exam was not given on Monday.)
What’s wrong with this reasoning?
I believe I saw this in Raymond Smullyan’s "Forever Undecided," but
I don’t know if he created it. I’ve never seen a solution to this,
and would be interested to know if one exists.
—
% Randy Yates % "Rollin’ and riding and slippin’ and
%% Fuquay-Varina, NC % sliding, it’s magic."
%%% 919-577-9882 %
%%%% <ya…@ieee.org> % ‘Living’ Thing’, *A New World Record*, ELO
http://home.earthlink.net/~yatescr
Hi, Randy,
I read this problem in "Vicious circles and infinity: A panoply of
paradoxes" by Patrick Hughes; the book was published in 1975. (I can
lend it to you, if you’d like.) I think the main fallacy lies in
induction step ("Since we know it can’t be Friday, then we know it also
can’t be Thursday.") The student has set up a fallacy of the exluded
middle, or a false dilemma.
Assuming the class is held every day, on Wednesday after class, the
student would not know if the exam were to be Thurday or Friday. He
would be surpised if he were tested on Thursday. So his induction
fails.
There is another semantic problem in the definition of "surprise exam."
The prof is defining this as "an exam given this week on a day that I
will not reveal to you until the test crosses your desk." The student
is defining this as "an exam day that cannot be determined by my
logic."
-Rajah
Rajah says…
>Assuming the class is held every day, on Wednesday after class, the
>student would not know if the exam were to be Thurday or Friday.
On the contrary, he knows that it can’t be on Friday, according
to the rules given.
–
Daryl McCullough
Ithaca, NY
Randy Yates <ya…@ieee.org> writes:
> What’s wrong with this reasoning?
This has been discussed extensively in the group. From the most
recent exchange, I quote Tim Chow’s posting:
Newsgroups: rec.puzzles, sci.math, sci.logic
From: t…@lsa.umich.edu – Find messages by this author
Date: 08 Sep 2005 19:13:07 GMT
Local: Thurs, Sep 8 2005 8:13 pm
Subject: Updated unexpected hanging paradox bibliography
In 1998, I published an article in the American Mathematical Monthly on
the surprise examination or unexpected hanging paradox. Since then, I
have been trying to maintain an exhaustive bibliography on the paradox.
I have just updated it with 34 new entries; you can view it at either
http://alum.mit.edu/www/tchow/unexpected.pdf
or http://arxiv.org/abs/math.LO/9903160
- Hide quoted text — Show quoted text -
Torkel Franzen <tor…@sm.luth.se> writes:
> Randy Yates <ya…@ieee.org> writes:
>> What’s wrong with this reasoning?
> This has been discussed extensively in the group. From the most
> recent exchange, I quote Tim Chow’s posting:
> Newsgroups: rec.puzzles, sci.math, sci.logic
> From: t…@lsa.umich.edu – Find messages by this author
> Date: 08 Sep 2005 19:13:07 GMT
> Local: Thurs, Sep 8 2005 8:13 pm
> Subject: Updated unexpected hanging paradox bibliography
> In 1998, I published an article in the American Mathematical Monthly on
> the surprise examination or unexpected hanging paradox. Since then, I
> have been trying to maintain an exhaustive bibliography on the paradox.
> I have just updated it with 34 new entries; you can view it at either
> http://alum.mit.edu/www/tchow/unexpected.pdf
> or http://arxiv.org/abs/math.LO/9903160
Thank you, Torkel et al. I will evaluate.
—
% Randy Yates % "Midnight, on the water…
%% Fuquay-Varina, NC % I saw… the ocean’s daughter."
%%% 919-577-9882 % ‘Can’t Get It Out Of My Head’
%%%% <ya…@ieee.org> % *El Dorado*, Electric Light Orchestra
http://home.earthlink.net/~yatescr
Daryl McCullough wrote:
> Rajah says…
> >Assuming the class is held every day, on Wednesday after class, the
> >student would not know if the exam were to be Thurday or Friday.
> On the contrary, he knows that it can’t be on Friday, according
> to the rules given.
All the teacher has said is that he won’t know what day the exam is on.
So if the exam is on Friday, he won’t know in advance that it’s on
Friday. That’s all he’s entitled to conclude.
- Hide quoted text — Show quoted text -
> —
> Daryl McCullough
> Ithaca, NY
- Hide quoted text — Show quoted text -
Randy Yates wrote:
> A professor walked into class on Monday and stated, "We
> will have a suprise exam sometime this week. On the morning
> of the exam you will not know that the exam will be that day."
> A logic student reasoned with himself like this: "We can’t have
> the exam on Friday, since, knowing that the exam must be this
> week, we’d know that it would have to be that day. Since we
> know it can’t be Friday, then we know it also can’t be Thursday,
> since it we’d know on Thursday that it must be on that day because
> it can’t be on Friday." This reasoning was extended all the way to
> Tuesday. (The exam was not given on Monday.)
> What’s wrong with this reasoning?
> I believe I saw this in Raymond Smullyan’s "Forever Undecided," but
> I don’t know if he created it. I’ve never seen a solution to this,
> and would be interested to know if one exists.
> —
Quine has a good discussion of it in "On A Supposed Antinomy", in "The
Ways of Paradox."
- Hide quoted text — Show quoted text -
> % Randy Yates % "Rollin’ and riding and slippin’ and
> %% Fuquay-Varina, NC % sliding, it’s magic."
> %%% 919-577-9882 %
> %%%% <ya…@ieee.org> % ‘Living’ Thing’, *A New World Record*, ELO
> http://home.earthlink.net/~yatescr
Rupert says…
>Daryl McCullough wrote:
>> Rajah says…
>> >Assuming the class is held every day, on Wednesday after class, the
>> >student would not know if the exam were to be Thurday or Friday.
>> On the contrary, he knows that it can’t be on Friday, according
>> to the rules given.
>All the teacher has said is that he won’t know what day the exam is on.
In the usual telling of the story, the rules say that the test will
be a "surprise", which is interpreted to mean that on the day of the
test, it will be impossible for the students to deduce (from the
rules given, together with information about what has happened
before) that there will be a test on that day. These rules imply
that it is impossible to give a test on Friday.
–
Daryl McCullough
Ithaca, NY
- Hide quoted text — Show quoted text -
Daryl McCullough wrote:
> Rupert says…
> >Daryl McCullough wrote:
> >> Rajah says…
> >> >Assuming the class is held every day, on Wednesday after class, the
> >> >student would not know if the exam were to be Thurday or Friday.
> >> On the contrary, he knows that it can’t be on Friday, according
> >> to the rules given.
> >All the teacher has said is that he won’t know what day the exam is on.
> In the usual telling of the story, the rules say that the test will
> be a "surprise", which is interpreted to mean that on the day of the
> test, it will be impossible for the students to deduce (from the
> rules given, together with information about what has happened
> before) that there will be a test on that day. These rules imply
> that it is impossible to give a test on Friday.
Maybe that’s the source of the paradox. The rule asserts that it is
impossible for the students to deduce the information from the rules
given, where the rule itself is included. The rule is talking about
itself.
- Hide quoted text — Show quoted text -
> —
> Daryl McCullough
> Ithaca, NY
Rupert says…
>Maybe that’s the source of the paradox. The rule asserts that it is
>impossible for the students to deduce the information from the rules
>given, where the rule itself is included. The rule is talking about
>itself.
Yes, that certainly is the source of the paradox. It’s a self-referential
paradox like the Liar paradox. But the difference is that (at least in
some tellings) it turns to be both paradoxical and *true*.
–
Daryl McCullough
Ithaca, NY
Randy Yates wrote:
> A professor walked into class on Monday and stated, "We
> will have a suprise exam sometime this week. On the morning
> of the exam you will not know that the exam will be that day."
> A logic student reasoned with himself like this: "We can’t have
> the exam on Friday, since, knowing that the exam must be this
> week, we’d know that it would have to be that day. Since we
> know it can’t be Friday, then we know it also can’t be Thursday,
> since it we’d know on Thursday that it must be on that day because
> it can’t be on Friday." This reasoning was extended all the way to
> Tuesday. (The exam was not given on Monday.)
> What’s wrong with this reasoning?
The conditions given – it will be given and it will be a surprise –
are inconsistent and can be used to prove anything we want about when
the test will be given e.g. it won’t be given or it will be given on
exactly two days. We can cherry pick the proofs and look at the ones
that say it can’t be given on each day, or the proofs that say it
will be given on each day. But that is all simply the result of having
inconsistent conditions which allow us to prove anything.
It can’t be given as a surprise because we are able to prove it will
be given each day. It is expected each morning.
Say there are 3 days in a week and P / Q / R represent the propositions
"It will be given on day 1 / 2 / 3." Then it will be provable on
day 1 if P, on day 2 if ~P=>Q and on day 3 if (~P^~Q)=>R. To be a
surprise all must be false, so we have ~P, and ~(~P=>Q) i.e. ~Q, and
~((~P^~Q)=>R) i.e. ~R. So that the requirement of surprise means we
can’t give it. That is natural, since we can prove it every day so
any day that it might occur does not come as a surprise.
C-B
Is there a prize or not?
- Hide quoted text — Show quoted text -
> % Randy Yates % "Rollin’ and riding and slippin’ and
- Hide quoted text — Show quoted text -
Torkel Franzen wrote:
> Randy Yates <ya…@ieee.org> writes:
> > What’s wrong with this reasoning?
> This has been discussed extensively in the group. From the most
> recent exchange, I quote Tim Chow’s posting:
> Newsgroups: rec.puzzles, sci.math, sci.logic
> From: t…@lsa.umich.edu – Find messages by this author
> Date: 08 Sep 2005 19:13:07 GMT
> Local: Thurs, Sep 8 2005 8:13 pm
> Subject: Updated unexpected hanging paradox bibliography
> In 1998, I published an article in the American Mathematical Monthly on
> the surprise examination or unexpected hanging paradox. Since then, I
> have been trying to maintain an exhaustive bibliography on the paradox.
In his ambitious paper, Timothy Chow comes close to the truth about
solving the puzzle, but rejects it for an ill-formulated reason. On
page 6, he writes, "This statement [the teacher's announcement] is
self-contradictory. However, the teacher’s announcement appears to
be vindicated after the fact. This analysis appears to pin the blame
on the teacher’s announcement instead of the students, and surely
this cannot be correct."
The problem is that nowhere in this exhaustive survey is there
consideration of the question of whether multiple expectations (proofs
it will occur on multiple days) constitutes a lack of surprise for each
day (proof.) The students are right. You can prove it will occur
every day from the inconsistent announcement. Calling the announcement
true and therefore not the source of the paradoxical contradiction is
wrong.
If multiple expectations of the test are considered a lack of surprise,
then it is not a surprise because they expect (prove) it every day.
If multiple expectations are not considered a lack of surprise, then
the test is a surprise because the students expect (prove) it multiple
times and that does not constitute a lack of surprise.
C-B
> I have just updated it with 34 new entries; you can view it at either
Make this entry 35, Mr. Chow. : )
- Hide quoted text — Show quoted text -
> http://alum.mit.edu/www/tchow/unexpected.pdf
> or http://arxiv.org/abs/math.LO/9903160
I see this problem as a simple deductive system based on the following
three postulates:
(1) Day of exam (D) = M, Tu, W, Th or F.
(2) On D morning, D is still unknown.
(3) D /= M.
>From postulates 1 and 3, we may deduce that:
D = Tu, W, Th or F.
And from postulate 2 we may deduce that:
D /= last possible D, because on that morning
D would be known with certainty.
Therefore, D /= F, because F = last possible D.
But if D /= F, then Th = last possible D.
Therefore, D /= Th.
But if D /= Th, then W = last possible D.
Therefore, D /= W.
But if D /= W, then Tu = last possible D.
Therefore, D /= Tu.
Therefore, there is no day on which the exam is possible, which
conflicts with postulate 1.
One of the uses of a deductive system is to determine whether all the
postulates upon which the system is based are logically consistent. In
this particular system, the postulates are not logically consistent and
at least one of them must be false.
Gene Ledbetter
Gene Ledbetter wrote:
> I see this problem as a simple deductive system based on the following
> three postulates:
> (1) Day of exam (D) = M, Tu, W, Th or F.
> (2) On D morning, D is still unknown.
> (3) D /= M.
> there is no day on which the exam is possible, which
> conflicts with postulate 1.
But, they argue, it can be given on Wednesday and be a surprise.
C-B
- Hide quoted text — Show quoted text -
> One of the uses of a deductive system is to determine whether all the
> postulates upon which the system is based are logically consistent. In
> this particular system, the postulates are not logically consistent and
> at least one of them must be false.
> Gene Ledbetter