Summary of discussion of mereology

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Message-id: <9205152032.AA03426@cs.umn.edu>
Reply-To: cg@cs.umn.edu
Date: Fri, 15 May 92 16:30:07 EDT
From: sowa@watson.ibm.com
To: CG@cs.umn.edu, INTERLINGUA@ISI.EDU
Subject: Summary of discussion of mereology

I received a note that helped me to clarify and summarize the
discussion over the past couple of days:

> If you adopt a standard model that includes all sets and where part-of
> means subset, and if you add enough additional primitives (with
> intended meanings) so that one can define all mathematical concepts,
> then the resulting theory is A SYNTACTIC VARIANT of set theory.

I was only suggesting that model as a way of demonstrating that
mereology is at least as consistent as set theory.  It is not a
variant of set theory, but a variant of a proper part of set theory.

> If you do not do these things, then you have lost the ability to
> state definitions.

What David Lewis and others have shown is that the subtheory is rich
enough for many, if not most, of the definitions you need.

> Either you have lost definitions, or you have a syntactic variant of
> set theory.  Which is it?

You can consider mereology to be a subset of set theory and of many
other kinds of theories that are significantly different from set
theory.  By eliminating some of the things in set theory, you can
cut back to a structure that can be extended in new ways.

For example, you get set theory by adding the element-of operator.
But you can get a theory of mass terms by avoiding element-of and
adding different operators for measuring and subdividing.  An
operator that allows unlimited subdividing would be inconsistent
with set theory, but it is compatible with mereology.

In summary, I like mereology because

 1. It is simpler than set theory -- it's a syntactic variant
    of a subtheory of set theory.

 2. But it is powerful enough to support recursive definitions.

 3. It can be extended in ways that would be incompatible with set
    theory -- especially ways that are needed for mass terms.

 4. It has a nicer (non-Platonic) underlying ontology.

 5. The formal operations it supports seem to be more compatible
    than set theory with the operations required for NL semantics.

Since there are many research issues to be resolved, I wouldn't propose
mereology as a basis for the standards efforts I'm currently working on,
but I would suggest it as a promising topic for further research.

John Sowa