Re: Smullyan's Quiz Problem
From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 12/03/04
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Date: Fri, 3 Dec 2004 20:16:01 +0000 (UTC)
In article <4d5e4663.0412031200.28d91b6b@posting.google.com>,
William Hughes <wpihughes@hotmail.com> wrote:
>magidin@math.berkeley.edu (Arturo Magidin) wrote in message news:<copu0q$k5t$1@agate.berkeley.edu>...
[.snip.]
>> >I agree with your analysis of the purely logical version of
>> >the paradox. The professor can be said to be asserting a contradiction.
>> >Thus, either the professor must be considered fallible, or
>> >we have a truth teller asserting a contradiction. In this case
>> >we can prove anything, (including the facts that a test on Wednesday
>> >is both expected and unexpected).
>> >
>> >However, I think that the paradox is much deeper than this. Consider
>> >four statements the professor could make:
>> >
>> > i) there will be an unexpected test tomorrow
>> > ii) there will be an unxepected test in the next three days
>> > iii) there will be an unexpected test next week
>> > iv) there will be an unexpected test this semester
>> >
>> >All four have the same logical structure. However, i) seems absurd,
>> >ii) seems questionable, iii) seems fine, iv) is completely unremarkable.
>> >A full resoution of the paradox must explain the difference.
>>
>> The paradox here is literally that they contradict our intuition.
>
>Agreed, the arguments are now informal, so "paradox" has
>exactly this meaning.
It should also be added that there is a fair amount of the
"Wilde"-like paradox in this; you know, the "The only thing worse than
being talked about is not being talked about" kind of statements,
since we argue that if we expect the test then we don't expect the
test (the old "I got you by not getting you" argument and variants
thereof).
>> But the only explanation needed is that ->our intuition is wrong<-.
>
>True, but the bald statement "our intuintion is wrong" is not
>very helpful. A full resolution of the paradox requires an
>explanation as to why our intuition is wrong
I think out intuition is wrong because we are making a lot of
unstated assumptions about these statements. As I noted: "there will
be an unexpected test tomorrow" seems absurd when we assume that
'unexpected' means something like 'I won't know exactly when', but
that knowing the ->day<- of the exam narrows the window sufficiently
for us to figure out "exactly when". The other three statements don't
lull us into that assumption because, prima facie, we have more than
one option and no information about which option we should take. But
that (i) if we know the day then we know when; and (ii) if we don't
know which day in advance then we don't know exactly when; are common
enough conclusions which are both unwarranted in this situation.
>>Just
>> because some of them "seem" absurd and others do not does not mean
>> that the statements actually are or are not wrong.
>>
>> These statements have a whole bunch of unstated assumptions. For
>> example, I disagree with you that (i) seems absurd prima facie. It
>> only becomes absurd if you assume that you know what time and what
>> context such a test would take place in.
>
>
>Yes, there are ways to make (i) sensible (e.g. An unxepected exam
>is one on a different colour of paper), however, these do not
>seem sensible to me.
I'm not even saying that. You can even make it sensible if
"unexpected" refers to "you won't know exactly when"; I teach at three
different hours; which hour will contain the exam?
> If you take (i) to mean (and this seems
>to me the most obvious meaning) "You will have an exam tomorrow but
>you don't know this", (i) is Bertram's paradox.
And here we are making another assumption about what "unexpected"
means. Does it mean, you do not expect an exam at all? Or does it
mean, you won't know exactly when? It is in the latter sense that the
Paradox of the Unexpected Hanging uses it, for example.
>> The reason (iv) do not seem
>> so absurd is that the latitude for those unknowns seems so much wider
>> that you cannot lull yourself into a false sense of knowledge about
>> the statement.
>>
>
>Indeed. However, both the professor and students agree
>that information has been communicated, and appear to agree on
>what has been communicated.
Which, as we know, in the real world is a virtual guarantee that it
is not true that what the professor meant to communicate is the
communication that the students have received... (-:
> So the question's are: Do the two
>parties actually agree on what has been communicated? and
>If the professor's statement is self contradictory, what should
>he have said?
Do you mean, how can he make a statement which communicates the same
information but is not self-contradictory? You may be begging the
question there, as in fact the point is that you cannot communicate
that information without also communicating a contradiction.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu
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