Reply to Pat Hayes & Chris Menzel
sowa@watson.ibm.com
Message-id: <9205140134.AA15663@cs.umn.edu>
Reply-To: cg@cs.umn.edu
Date: Wed, 13 May 92 21:33:25 EDT
From: sowa@watson.ibm.com
To: INTERLINGUA@ISI.EDU, CG@cs.umn.edu
Cc: FRANCONI@IRST.IT
Subject: Reply to Pat Hayes & Chris Menzel
Pat Hayes and Chris Menzel made some penetrating comments and
observations about mereology, and a few other people, who were afraid
to send their comments to the public lists, asked "What is mereology?"
Before getting to the comments by Pat and Chris, I'll give a summary
for the latter group (references at the end of this note):
1. Mereology is an alternative to set theory that avoids some of the
paradoxes and has some interesting properties that make it more
attractive as a basis for kn. representation, especially for NL.
In particular, it allows a nicer treatment of both plurals and
mass nouns than set theory.
2. Two people who independently developed variations of mereology
are Stanislaw Lesniewski, a logician working in Poland during the
1920s and 1930s, and Nelson Goodman, who developed his version
while working with Quine at Harvard in the 1930s. Since then,
many people have been using versions of mereology for various
purposes. Harry Bunt's book is probably the most accessible
version for NL applications.
3. Although the name "mereology" sounds abstruse, it is actually much
simpler than set theory. Whereas set theory has two distinct
operators, element-of and subset-of, mereology has only one basic
operator, part-of (the term "mereology" comes from the Greek "meros"
for "part").
In his note, Chris Menzel pointed out that element-of is the only
truly basic operator in set theory, since subset can be defined
in terms of element-of. That's true, but since part-of is
transitive, a similar definition in mereology would not yield
anything new -- the new operator would just be the same old part-of.
4. As a typical example, Nelson Goodman cited the political subdivision
of France into departments or provinces. When considered as sets,
the set of departments of France is very different from the set of
provinces of France. But in mereology, they would both be considered
identical -- the total territory of France.
5. As another example, mereology makes no distinction between an
individual and a singleton set nor between different ways of
building up sets by levels of nesting: {a,b,c} would be identical
to {a, {{{b}}, {c}}}. Goodman summarized that point in his slogan
"No distinction of individuals without distinction of content."
Goodman, by the way, used the term "individual" for everything
and said that individuals have other individuals as parts. Some
people use the term "collection"; Harry Bunt uses the term
"ensemble", although that is the French word for "set".
Now for some responses to Pat and Chris:
The first point is an answer to two different, but related objections.
Pat said,
> But notice that this also means that all collectives are similarly
> flattened. Thus in mereology, the collection of shoals of fish and
> the collection of fish are indistinguishable, and indeed are similarly
> indistinguishable from the collection of all fish flesh. It's all just
> the fishy part of the universe.
And Chris said,
> Note that part of set theory's appeal is that you can build models
> of mereology in set theory, but not vice versa. Mathematically, set
> theory is thus more powerful and more general. After all, as far as
> {\em modeling} goes, I should think we want the most powerful theory
> available. Just because we model a certain semantic or other
> phenomenon with sets doesn't mean that we have to say that that's
> what is really out there. We're only committed at most to saying
> that the phenomenon in question is {\em structurally} similar to the
> set theoretic model.
The common objection that both Pat and Chris are getting at is the fact
that a mereological collection (or individual in Goodman's terms) has
no "natural" breakdown into elements. In particular, the notion of
cardinality does not naturally arise in mereology, since there is
no unique decomposition -- i.e. the cardinality of France would be
different for sets of provinces or departments, but it would be
undefined in terms of mereology. I consider that an advantage of
mereology.
Instead of considering all the fish in the sea, let's just consider
five cats: C = {Yojo, Tigerlily, Muffy, Thothmes, Puff}. If you
consider C to be a set, it has a cardinality 5. But how many parts
does C have? At first glance, you might say that each subset of C
is a part, and therefore C would have 2**5 or 32 parts. But that is
only one way of dividing up C. We could also consider C a collection
of 5 nondetached subcollections of cat parts and get a much larger
answer. Or we might consider it a collection of molecules and get an
answer of 2**10**27, give or take a few hundred orders of magnitude.
Is that an objection to mereology? That depends on your point of view.
Pat continues with the following objection:
> For example, mereology makes no distinction between nouns and adjectives :
> there are no 'things' in mereology, only parts of the universal plenum
> which have certain properties. "Book" and "red" are exactly the same
> kind of entity. Maybe John's rebarbative notation has a way of making
> this distinction, but if it does then it goes beyond mereology
> and must be resting on some other ontological perspective.
I'm glad you asked, because I do happen to have a way of making that
distinction. But I don't know why you use the term "rebarbative",
which suggests some sort of disapproval. I thought that in an earlier
note you had agreed that predicate calculus was a rather rebarbative
notation for logic. For anything that you can barb in PC, I can rebarb
more elegantly in CGs. And if you have a less barbative notation of
your own, I'd like to see it.
In predicate calculus, if you like, let me introduce the function
count(t,i), where t is a type and i is an individual. This function
will produce an integer that gives the number of times the type t
appears in the individual i. For our collection of cats C, the
result of count(cat,C) is 5, which is exactly what you would get
for cardinality(C). But if you considered C to be a collection of
molecules, you would get count(molecule,C) to be about 10**27.
This distinction happens to be built into the ontology that underlies
the CG system. I would represent C above by the notation,
[CAT: {Yojo, Tigerlily, Muffy, Thothmes, Puff}]
which I would map into a formula like the following in KIF or
predicate calculus enriched with sets:
(Es)(set(s) & s={Yojo, Tigerlily, Muffy, Thothmes, Puff}
& (Ax)(element(x,s) -> cat(x))).
But if I were mapping the formula into predicate calculus enriched
with mereology, I would get something like the following:
(Es)(s={Yojo, Tigerlily, Muffy, Thothmes, Puff}
& (Ax)(partof(x,s) -> (Ey)(cat(y) & partof(y,s)
& (Ez)(nonempty(z) & partof(z,x) & partof(z,y))))).
Note that in set theory, we have to assume that there is a unique way
of dividing up the thing into elements, every one of which is a cat.
But in mereology, we have the weaker claim that every part of s has
a nonempty intersection with some cat in s.
This claim may sound too weak for something as cleanly divisible as
a set of cats. But suppose I told you that in my basement, I have a
pile of bottles to be recycled, a box of tools, and a case of wine.
How many things are there? I just mentioned three -- a pile, a box,
and a case -- but each of them contains other clearly distinguishable
lumps, and some of the lumps have other parts that are attached more
or less cleanly. Suppose the tool box has a two nuts and two bolts,
where one nut and bolt are loose, but another bolt has a nut screwed on.
Does that make 4 things or 3 things?
Or consider the collection of redwood trees in Muir Woods. What is
its cardinality? Are two trees that share a common root system one or
two? Does it make a difference if their trunks divide above or below
the ground level? And what if the ground level changes with erosion
or falling needles?
Set theory is much too rigid for modeling such things, but mereology
is much more appropriate. Set theory is poorly suited to dealing with
mass terms, but mereology is fine -- you can easily talk about the
same mass of water measured as 5 liters or 25 glassfuls.
The semantics of conceptual graphs, by the way, can be defined in
model theory on their own terms (which is philosophically preferable),
but I frequently define it by a mapping into predicate calculus for
pedagogical reasons (for people who were brainwashed in their youth).
But I hope that you notice that both PC formulas are more rebarbative
than the CG form. That brings us to the next couple of points:
> Notice also that this means that there is no way that any mathematical
> modelling can be done in mereology....
> Russell's paradox indeed does not arise in mereology. This however does> not b
> any means establish the consistency of mereology. To do so would require some
> formal account of how expressions in mereology acquired meaning, and this
> would need at least a model theory, and that needs set theory. This is
> precisely where mereology has failed to excite the imaginations of most
> philosophers, I think.
What excites the imagination depends primarily on the advertising hype.
A lot more philosophers (and especially philosophy students) get
excited by squishy things like existentialism than by any of the hard
topics like logic, model theory, set theory, mereology, etc.
But there has been a great deal of work done on the foundations of
mereology (see references below). Furthermore, you don't need sets for
doing model theory of FOL -- all you need is a collection of individuals
and a collection of relations among them. And if you need to distinguish
individuals, you introduce a predicate like cat(x) to help you do that.
> Apparently this kind of distinction can be made in Sowa graphs: then
> they must have their feet on something firmer than mereology, since
> this distinction [between sets and elements] is invisible there.
I didn't say that I can translate CGs into mereology, but into
predicate calculus enriched with mereology. The mereology gives
me the mechanism for collections and parts, but then I have to use
predicates like cat(x) or molecule(x) for distinguishing cats from
molecules. The advantage of this approach is that I don't assume
that the world comes already prepackaged in clearly distinguishable
elements. Instead, I start with undifferentiated stuff and introduce
predicates to make the distinctions I want. The predicates must also
be definable either implicitly by axioms or explicitly by mechanisms
of perception and movement (symbol grounding, if you will).
> What you call them IS relevant to the extent that it guides and claims
> your intuitions. This is especially important when you claim to be
> capturing natural meaning.
I'm sure that we both agree that ontology is extremely important.
But I claim to capture meaning in an ontology expressed in the concept
and relation types I use in the conceptual graphs, not in the structures
of the underlying definitional system. I prefer mereology for its
purely formal operations, which as I said above give me a more flexible
definitional system than set theory. See my papers on knowledge soup
for more reasons why I want that kind of flexibility (ref's below).
> 10. This is a very confusing essay. If Sowa's graphs are thought of as a
> formalism of some kind, then the distinction between upper and lower case
> symbols is of no deep significance: he is simply distinguishing between two
> different formal notations. But sometimes he seems to be placing his graphs on a
Yes, I consider CGs to be a formal system. You got my point, but
I was trying to address those people who have trouble distinguishing
different levels of formalisms.
> par with a natural language, so that their symbols (and presumably their syntax)
> have no technical meaning at all. In this case, we could simplify things
> tremendously by just having the English and the predicate calculus translations
> and removing these graphs from the picture altogether.
I have tried to explain in many of my papers why I believe that predicate
calculus is a rather poor language for formalizing NL semantics. I have
been developing CGs as a formal language whose structure is much closer
to the underlying semantic structures of natural languages. The question
of how close is an empirical issue to be tested by comparing translations
>From CGs and other formalisms to and from many different NL's. So CGs
have two aspects: their formal side as a system of logic, and their
empirical side as my best guess about the ideal structure for encoding
natural language semantics.
Now for some comments on Chris Menzel's comments:
>> 1. I believe that mereology has a more natural application to plurals
>> in ordinary language than set theory.
>
> This seems to be true, as witnessed especially by the excellent work
> of Godehard Link on plurals....
> This theory yields very elegant analyses of simple plural
> constructions, and more complex phenomena such as
> distributive/collective ambiguity ("Three professors graded fifteeen
> tests"), plural reflexive constructions ("The Smiths and the Joneses
> hate each other"), numerical definite descriptions ("The three men
> fought"), and others.
Yes, Link has done some good work. But I would quibble with the word
"elegant". He has been using Montague's formalism, which I consider
brilliant, but unwieldy. If Montague had been a programmer, he would
have been an outstanding hacker. But I think that people like Link
have been misguided by Montague's reputation into blindly following
his horribly unwieldy hacks. See my paper "Towards the Expressive
Power of NL", where I cite an example from one of Link's papers to show
how CGs make Montague-style notation look rebarbative to say the least.
A lot of linguists have been so rebarbed by Montague that they get
turned off by any form of logic whatever.
>> In mereology, such paradoxes do not arise, since every collection
>> is part of itself, and there cannot be a collection that is not
>> part of itself.
>
> There is a paper by Sobocinski called something like "Lesniewski's
> solution of the Antinomies of Russell and Burali-Forti," in which he
> lays out very clearly the solution to the paradoxes given by
> mereology's founder that you allude to here. I'm afraid I don't
> have the reference handy, but could scare it up if you (or anyone)
> is interested.
I have the book by Luschei (cited below), and I'll take a look there.
>>9. It is possible to translate any statement about sets, however
>> paradoxical, into either English or conceptual graphs. But to
>> do so, you have to introduce new concept types, such as SET.
>> The paradoxes do not arise from the basic structure of the system,
>> as they do in set theory.
>
> Again, most (or at least many) set theorist's would regard this
> remark as tendentious, to say the least. They would insist that,
> once we got clear about the set/class distinction, or, as G\"{o}del
> put it, the "mathematical" vs. the "logical" conception of set, the
> source of the set theoretic paradoxes was made clear. Which is not
> to say that no questions remain.
I was raised as a mathematician and was thoroughly imbued with the
set-theoretical (i.e. Georg-Cantor-style) way of thinking. But my
mathematical training also made me realize that there are infinitely
many formalisms that one can develop to describe almost any kind of
phenomena. Cantor-Frege-Peano-Whitehead-Russell-Tarski-Carnap-Montague
have certainly done great work. But I also believe that other great work
has been done in the Peirce-Lesniewski-Goodman traditions. The first
path has been fruitful for the foundations of mathematics, but it has
been much less successful for linguistics.
I think that many of the AI people who have been arguing against the
logicist position have been partly misguided, but they are also partly
correct in saying that predicate calculus with sets is a lousy language
for knowledge representation. I have been trying to show that there is
a version of logic that is just as formal, but a lot more natural.
John Sowa
________________________________________________________________________
Bunt, H. (1985) _Mass Terms and Model-Theoretic Semantics_,
Cambridge University Press, Cambridge.
Goodman, N. (1972) _Problems and Projects_, Bobbs-Merrill Co., New York.
Lewis, D. (1991) _Parts of Classes_, Basil Blackwell, Oxford.
Luschei, E. C. (1962) _The Logical Systems of Lesniewski_,
North-Holland Publishing Co., Amsterdam.
Sowa, J. F. (1990) "Crystallizing theories out of knowledge soup,"
in Z. W. Ras & M. Zemankova, eds., _Intelligent Systems: State of the
Art and Future Directions_, Ellis Horwood Ltd., London, pp. 456-487.
Sowa, J. F. (1991) "Towards the expressive power of natural languages,"
in J. F. Sowa, ed., _Principles of Semantic Networks_,
Morgan Kaufmann Publishers, San Mateo, CA, pp. 157-189.