Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Path: utzoo!linus!decvax!yale-com!leichter
From: leichter@yale-com.UUCP (Jerry Leichter)
Newsgroups: net.math
Subject: Re: surprize quiz
Message-ID: <1994@yale-com.UUCP>
Date: Mon, 12-Sep-83 15:50:12 EDT
Article-I.D.: yale-com.1994
Posted: Mon Sep 12 15:50:12 1983
Date-Received: Mon, 12-Sep-83 19:23:47 EDT
References: ihuxr.627
Lines: 59

The "surprise quiz" problem has often been stated in terms of a prisoner,
who is told that he will be executed at dawn on some morning the following
week, but will not be able to know the evening before that he will die
the following morning; in fact, if had does know, and can prove it to the
judge, he will be granted a pardon.

W V Quine published a paper, the name of which I don't know, analyzing this
problem quite nicely, calling it a pseudo-paradox.  I think his disolution
of the paradox is quite nice.  It rests on the meaning of my "knowing"
something.  Analysis of what it means to know X (by Quine and others)
shows that it has 3 components:

	X is true;
	I believe X to be true;
	I have a good reason for my belief that X is true.

The classic example that shows that all three conditions are needed is
the following:  I look in a field and see sheep.  Do I know there are
sheep there?  (a) suppose that there are actually pictures of sheep,
but not sheep.  Then it isn't reasonable for me to say that I know
there are sheep because there aren't...  One goes through a number
of related examples, and ultimately looks at a case in which I see
pictures of sheep, and behind the pictures are real sheep, which I
can't see.  Now, the first two conditions hold, but not the third -
and in fact I'd rather not say I know there are sheep.

How does this apply to the exam/prisoner problem?  Suppose I am the
teacher/judge, you the student/prisoner.  The evening before the last
day, you come to me and say:  I know the exam/hanging is tomorrow.
Well, do you KNOW this?  I intend to test/hang you tomorrow, so
what it is in fact true that you will be tested/die; apparently you
believe it.  But what is your REASON for this belief?  Well, you
took my first statement:

	You will be tested/be hanged this week.

and assumed it to be true; from this, you concluded that tomorrow
is the big day.  However, I also made another statement:

	You won't know the day before that you will be tested/hanged.

Your alleged "proof", the basis of your claim of knowledge, makes this
statement false!  What gives you any particular reason to prefer one
statement over the other?  If you are going to use my statements as
starting points for a PROOF, you either take both as givens, or you
make no assumptions at all about their veracity.  You cannot give a
meaningful proof by picking and choosing which hypotheses to accept and
which to reject.

Hence, your proof fails.  You are in an interesting position:  You will
be tested/hanged tomorrow; you believe it; it's true; but you cannot,
by the very nature of the situation, KNOW it to be true!

Notice that if I had said:  You won't BELIEVE the day before that you
will be tested/hanged - you have an out:  You just tell me you believe
every single night, and you are quite safe.
							-- Jerry
				decvax!yale-comix!leichter leichter@yaele
\\\								   @yale