From: mtaranto@shearson.com (Mark Taranto)
Subject: A note on Modal Logic that has nothing to do with Icky Sex
Organization: Yale CS Mail/News Gateway
Date: Thu, 17 Dec 1992 16:47:01 GMT


Like most of you, I skipped over most of the latest M Z thread on
the morality of various sexual practices.
 
But in the middle of one of the articles, a statement jumped out
at me.  It was something like "What has been shown is that my
understanding of category theory is better than your understanding
of modal logic."
 
I had to laugh.  This phrase set off a series of memories reminiscent
of a certain bite into a madeleine.  I've had experiences with both
areas of study.
 
Later, I started seeing lots of articles popping up which had appeals
to what goes on in "Possible Worlds."
 
For those who are not aware, Modal Logic is the study of logical
systems which add the "It is possible that" and the "It is necessary that"
operators to the logical language.
 
There are a number of different ways that people look at the field, but
in my experience they really break down into three different approaches.
 
The first approach is to look at a logical system with these operators
set up.  "Possibly" and "Necessarily" have very strict definitions.  These
systems of logic are interesting as logical models.  Several of them can
be shown to be complete (using an extension of Godel's completeness theorem
(not to be confused with Godel's incompleteness theorem for richer logical
systems)).  But I found these definitions too restrictive to use the
logical system for anything other than a mind game.
 
The second approach is to bring the structure up one level.  In this, we
start quantifying over possible worlds, and use arguments very much like
the ones in the sexual practice thread.  The definitions of "Possibly"
and "Necessarily" remain strict.  A statement is possible if there is
at least one possible world where it is true.  It is necessary if it
is true in all possible worlds.  Some statements (like the tautology
'X if and only if X') are necessarily true.
 
The problem that I have with this approach is in the idea of quantifying
over possible worlds.  I've seen lots of people define what they mean
by a possible world -- but I've never been comfortable with the idea
of quantifying over these possible worlds unless their definition is
so restrictive that the system reduces to one which is useless.
 
For those who wonder why I have a problem quantifying over possible worlds,
it is for the same reason that there are problems quantifying over the
set of all sets.  I see cardinality problems.  I read a paper where some
modal logicians tried to prove that there is a countable subset of
possible worlds which is dense in the set of all possible worlds.
Their logic was impeccable with one exception.  The collection of
possible worlds is not a set (for the same reason that there is no
set of all sets -- see Russell's Paradox).
 
I find any logical argument which depends on quantifying over possible worlds
to be dubious at best.
 
I mentioned that there is a third way to deal with modal operators.  This is
through Linguistics and the Philosophy of Language.  When we say "X is
possible" or "X is necessarily true" what do we really mean?  Part of the
problem is that we don't mean the same things.
 
Often it comes down to "X is possible" means that "I am unaware that
X is inconsistent with what I know."
 
I believe that this is a more fruitful avenue of study.
 
Mark
 
 
 
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