Is subclassing "strict order" or is it reflexive? RE: [sc34wg3] New SAM PSIs

[Murray Altheim]

Bernard Vatant wrote:
> Hello Murray
> 
> Following various requests (Lars Marius, Sam, Martin ...), I'm back to
> the public forum for this issue. At the end is the private exchange
> copy. Along with your reply to the list, and with a little help from the
> week-end, I think I caught what you are about, and well, on most of it,
> I agree. 

Likewise, I spent some time this weekend reading over a number of
texts on the subject and think I've figured out what's going on.
I was somewhat buoyed by the notion that our discussion has covered
a territory that has been the subject of much intense debate over
the years, and that there has been confusion in the language among
the "experts" such as Russell, Peano, Cantor, Boole, Peirce, Quine,
etc.  I don't feel so bad if I was confused too.  :-)

> Let me try to sum it up, to see if I got it well.
> 
> Classes are not sets. OK. What are the differences?

Agreed.

> - Sets can be defined in extension (an exhaustive list of elements) or
> by some characteristic property (how to check if some element belongs or
> not to the set). This last form of definition is close to the
> intensional definition of a class, but even if the set is defined by a
> property, its definition in extension is always assumed to exist, even
> if you do not know how to build it actually, for example in uncountable
> sets, where you have to assume hard stuff like the choice axiom. All
> those issues have been solved by Godel, Cohen, and others, more than 50
> years ago, and led to the various flavors of set theory, with or without
> choice axiom, etc.

A problem with the concept of "class", i.e., a designation of a collective
entity, is that it's very difficult to tie down how it is connected to
the entity. Avoiding nominalistic arguments (as I believe Peirce would,
when he stated that there simply is no logical connection between a
name and its referent other than the one *we* put on it, which would
hold for classes as well), it seems that some jump to the extensionalist/
intensionalist approach to classes, which is incorrect and irrelevant.

Another issue that arose historically is that any characteristic
property used to define a set (the idea that one could define a class
intensionally is I believe different), is that any characteristic
property is itself composed of a divisible set of characteristic
properties (ad infinitum? likened to the frame problem?), which in effect
are the sum of all statements about the reality used to represent the
"universe of discourse". When you say that these issues were "solved"
by a *variety* of flavours I'll take that with a grain of salt. :-)

But I'm getting off track a bit here...

> - Sets equality A = B can be proven either by extension, checking "one
> by one" that any element of A belongs to B, and vice-versa, or by
> proving the equivalence of characteristic properties. This equivalence
> proof is generally obtained by proving separately that PA => PB, and
> then PB => PA. BTW this is using inclusion antisymmetry. 
> "PA => PB" is equivalent to "A is a subset of B", so: 
> If A is a subset of B, and B is a subset of A, then A = B.
> 
> Whatever the method, extensive of by property, the proof is about the
> elements (who belongs actually to the set?). If equality is proven by
> extension, characteristic properties are proven equivalent, and
> vice-versa.
> 
> - Classes are not defined in extension. That is where the notion
> introduced by OWL of "enumerated classes" (owl:oneOf) is quite strange,
> I agree. Classes definition is intensional. But instantiation of a class
> in a given context can define a set. 

I think you're being charitable. Not "strange" but wrong, and quite
wrong. And while instantiation of a class *in a given context*
can define a set, that set itself is then defined extensionally.
This doesn't impact the class definition.

I'm not comfortable with the statement "classes definition is
intensional" because that conflates set constitution with class
definition with no logically demonstrable bridge across the gap
between reality and semiotic, which has always been the problem.

Extension and intension are aspects of sets, not of classes. I think
at the beginning of this we had confusion due to the conflation of
sets and classes (or of collections and classes). The concept of a
"superclass-subclass loop" is itself fallacious -- it operates at
the wrong level.

This can only make sense as a superset-subset loop, which I'll
address below.                     ^^^    ^^^

[Bernard's example on Murray and cats...]

> So far, I eventually agree with Murray's view, and that some things in
> OWL should be revisited from that viewpoint.
> 
> Now back to the initial issue. What are the consequences of that for
> "class-subclass" properties? Seems to me that it does not affect the
> fact that "class-subclass" has to be considered an order relationship
> between classes *in a given ontology* (large or strict, depending if you
> accept reflexivity or not, which is not really an issue, as said above).
> Any implementation of classification should be able to check integrity
> of the order properties - especially if classes are defined as topics in
> a topic map. IMO this has nothing to do with instances nor the previous
> distinction between sets and classes. 
> 
> NB: If you don't like the antisymmetry property in the form: 
> 
> "If A is a subclass of B, and B is a subclass of A, then A = B"

There's only two interpretations of this that can be correct.

   1. The statement was misworded, and what was really meant was

       If A is a subset of B, and B is a subset of A, then A = B

      This indicates set membership, not class designation. So
      that if the membership of set A is the same as the membership
      of set B, then set A has identity with set B.

or

   2. That "A" and "B" are class designations (i.e., names of
      classes) and that if "A" and "B" refer to the same
      class then they are simply different designations for
      the same class, i.e., just different names for the same
      thing, by "intension" as you might say. Therefore, the
      statements "A is a subclass of B" and "B is a subclass
      of A" are logically incorrect, non sequitors.

> It can be expressed by a contrapositive equivalent form (more natural,
> if not more simple)
> 
> "If A is distinct of B, and A is a subclass of B, 
> then B is not a subclass of A"

I agree with this, which in fact supports point #2 above. If we add
to this its complement

   "If A is distinct of B, and A is a superclass of B,
   then B is not a superclass of A"

then we have the two statements necessary to disprove the existence
of "superclass-subclass loops", since they are posited upon the
notion that two classes can be equivalent at the same time as being
distinct (a "superclass" and "subclass" are not equivalency relations
but distinctions).

I believe OWL should revisit the issue and simply rename
their conundrum as a "superset-subset loop". The current
text doesn't make sense.

If you have a copy of "Studies in the Logic of Charles Sanders
Peirce" (Houser, Roberts, Van Evra, eds. Indiana Univ. Press 1997),
these essays are particularly helpful:

   "Peirce's Philosophical Conception of Sets"
    Randall R. Dipert, p.53.

If anyone is unclear on the definition of "class" this is a good
discussion of its history, as well as fairly clear on language.

   "Peirce and Russell: The History of a Neglected 'Controversy'"
    Benjamin S. Hawkins, Jr., p.111.

The latter discusses at some length concepts of "class" and the
nominalist/realist argument. Sections 3.5 to 3.7 on Peirce's
and Russell's definitions of "class", "being", "extension" and
"intension" are very interesting, and I'd have to say I strongly
disagree with Russell's extensionalist view of classes in the
same way as did Peirce, and in fact Peirce makes the same claim
as I was making on Friday (on p.133; "PM" is "Principia Mathematica"):

    The distinction [between ∈ and the relation of whole and
    part between classes] is the same," Russell observes in PM,
    as that between the relation of individual to species and
    that of species to genus, between the relation of Socrates
    to the class of Greeks and the relation of Greeks to men"
    (Russell 1903[1956]:19). Peirce responds by remarking about
    relations species infima and species subiicibilis that "They
    are not formally distinct except for the relation of identity."

I.e., (noting the quotes) if "A" and "B" are not formally
distinct designations of a class, they are synonyms and in
extension always refer to the same class. Therefore, A cannot
be a superclass or subclass of B, B cannot be a superclass or
subclass of A.

Additionally, although a bit off-topic, Peirce notes that the
class relation "does not contain any individuals at all. It
only contains general conditions which _permit_ the determination
of individuals." (6.185) For those interested, Nicola Guarino
has a web page concerned with ontology, mereology (study of
part-whole relations), and other topics germain to this discussion
at

    http://ontology.ip.rm.cnr.it/Publications.html

I would specifically recommend "Identity and subsumption",
"Supporting Ontological Analysis of Taxonomic Relationships",
and "Formal Ontology, Conceptual Analysis and Knowledge
Representation", though most everything I've read has been
helpful.

I think this Horse is Dead. Elvis has left the building.

Murray

......................................................................
Murray Altheim                  <http://kmi.open.ac.uk/people/murray/>
Knowledge Media Institute
The Open University, Milton Keynes, Bucks, MK7 6AA, UK

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