Re: Smullyan's Quiz Problem

From: Daryl McCullough (daryl_at_atc-nycorp.com)
Date: 12/03/04 

Date: 3 Dec 2004 04:14:12 -0800

William Hughes says...

>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 unexepected 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.

I think that if someone says "there will be a surprise test sometime
in the next week", he usually only means that there is no way *now*
to figure out what day it will occur on. People don't usually mean
the stronger claim that, even on the day of the test, you won't be
able to figure out that there will be a test. Obviously, on the last
day of the week, you can reason that if you haven't had a test yet,
you're going to have it today.

However, if someone says "there will be a surprise test tomorrow",
the weak interpretation of "surprise" is not available.

The strong interpretation of "surprise" is paradoxical, while the
weak interpretation is not.

As someone pointed out, the strong interpretation is not actually
paradoxical, once you realize that just because somebody says something
doesn't make it true. So, if the teacher announces that there will
be a surprise test, one possibility is that he is lying---either
there won't be a test, or it won't be a surprise. If you consider
the *possibility* that the teacher is lying, then you can't logically
deduce anything from the fact that he said there will be a surprise
test, and so on the day of the test, it *will* be a surprise. 

--
Daryl McCullough
Ithaca, NY