COMMENT ⊗   VALID 00002 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	ANOTHER FORMALIZATION OF THE WISE MAN PUZZLE
C00006 ENDMK
C⊗;
ANOTHER FORMALIZATION OF THE WISE MAN PUZZLE

	The "wise man puzzle" has served as a sample problem in studying
how to formalize knowledge of propositions by persons.  Our particular
goal is to show how it is possible to prove that someone does not know
a fact.

	The usual formulation of the puzzle is as follows:
%2A certain king wishes to determinehe smartest of his wise men, so he
paints a white spot on each of their foreheads, tells them that at least
one has a white spot and the others if any have black spots, and asks them
if they know the color of their spots after arranging them so that each
can see the foreheads of the others.  After a short time the wisest of
them announces that his spot is white.  How did he know?"%1

	The solution is that he reasoned that if his spot were black,
one of the others would have seen a white and a black and reasoned that
if his spot were black, the man with a white spot would have seen two
black spots and concluded that his spot was white on the basis of te
king's statement that at least one spot is white.  We are not ready to
formalize this problem as it stands, because the wisest man clearly has
to make assumptions about how fast his colleagues think.  Therefore, let
us suppose that wise men are not asked to speak up when ready but must
reply to the king's questions simultaneously without knowing the others'
answers.  In this version, the king asks the question three times, gets
"I don't know, sire" twice, and on the third asking, each replies that
his spot is white.

	Some years ago, I formalized knowledge of propositions in such
a way that one could prove that the wise men knew their spots were
white at the third asking, but that formalism was inadequate to prove
that they did not know the colors of their spots at the first and
second askings.  It turns out that proving a person does not know
a proposition is much more difficult than proving he does.  We now
think we understand the matter well enough to prove the "don't knows"
as well as the "knows".
	The formalism we shall use is a variant of that given in my Kyoto
lectures and discussed in by Hayashi, Igarashi, and Sato in various
versions.  The present treatment owes much to their results - as will
be apparent.