From: zeleny@husc10.harvard.edu (Michael Zeleny)
Newsgroups: rec.arts.books,sci.philosophy.tech
Subject: Re: A note on Modal Logic that has nothing to do with Icky Sex
Date: 17 Dec 92 20:45:30 GMT
Article-I.D.: husc3.1992Dec17.154532.18618
Organization: The Phallogocentric Cabal

In article <9212171647.AA08318@cfdev1.shearson.com> 
mtaranto@shearson.com (Mark Taranto) writes:

>Like most of you, I skipped over most of the latest M Z thread on
>the morality of various sexual practices.

I beg to differ, -- most of us have followed the thread in question
quite assiduously.  Watch those personal pronouns!

>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.

If you want to make this into a meaningful criticism, you will have to
be more elaborate.  As it stands, your statement is surely in error,
as witnessed by any number of working logics of time and computation.

>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).

This is known as Kaplan's Paradox, and its conclusive dismissal will
be found on pages 104--8 of David Lewis' book _On the Plurality of
Worlds_.  Note that Lewis' treatment is essentially independent of any
assumption of modal realism.  Also note that Russell's reasoning does
not in any way vitiate the possibility that the collection of all
possible worlds is a set, given a reasonable approach to semantics.
Finally, logicians quantify over all sets all the time; that's what
the notion of a proper class is for.

>I find any logical argument which depends on quantifying over possible worlds
>to be dubious at best.

As Gene Ward Smith might say, not an argument.

>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.

Sounds like emotivism to me.  Too bad it got so discredited; it would
be nice to have a license to go around, referring solely to your
private beliefs.

>Mark

cordially,
mikhail zeleny@husc.harvard.edu
"Le cul des femmes est monotone comme l'esprit des hommes."