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

[Bernard Vatant]

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.=20

Let me try to sum it up, to see if I got it well.

Classes are not sets. OK. What are the differences?

- 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.

- Sets equality A =3D 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 =3D> PB, and
then PB =3D> PA. BTW this is using inclusion antisymmetry.=20
"PA =3D> PB" is equivalent to "A is a subset of B", so:=20
If A is a subset of B, and B is a subset of A, then A =3D 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.=20

Example: Let's assume that Murray has a total and exclusive relation of
friendship with cats.=20

	1. "Any cat is Murray's friend"
	2. "Any Murray's friend is a cat"

- A naive set representation of that defines

C =3D the set of all cats
F =3D the set of all Murray's friends

You infer that C =3D F from 1. and 2.=20

That does not make sense, actually, because the "set of all cats" is
actually impossible to define in extension: All living cats? All the
cats Murray is likely to meet in his life time? All the cats that have
ever been and will be? Is the Cheshire's cat in Alice an element of this
set? etc ...=20
Any definition of that kind is silly. So "the set of all cats" is a
non-sense. But the class "cat" makes sense, as makes sense the class
"Murray's friend".=20

Now, in any context where you gather specific instances of cats and
other living things in a room with Murray, you can define proper sets in
extension:

Ci =3D the set of all instances of "cat" in the room
Fi =3D the set of all instances of "Murray's friend" in the room

And infer from 1. and 2. that Ci =3D Fi.

But can you infer that classes "cat" and "Murray's friend" are the same?
Certainly not, because intensional definitions are clearly distinct. It
would be silly ro replace "cat" by "Murray's friend" in a zoological
taxonomy.=20

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.=20

NB: If you don't like the antisymmetry property in the form:=20

"If A is a subclass of B, and B is a subclass of A, then A =3D B"

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,=20
then B is not a subclass of A"

Well - I think that makes it for today :)

Bernard

| -----Original Message-----
| From: Murray Altheim [mailto:m.altheim@open.ac.uk]
| Sent: vendredi 14 f=E9vrier 2003 18:29
| To: Bernard Vatant
|=20
| Bernard Vatant wrote:
| > Murray
| >
| > [private]
| >
| > Let me have another try, starting otherwise:
| > Do we agree on the following definition of subclassing?
| >
| > "A is a subclass of B"
| > 		<=3D>
| > "If x is instance of A, then x is instance of B"
| >
| > If no, well, see below another try, not using instances.
|=20
| I wouldn't honestly think instances have anything to do with
| the definition of a class other than being examples of that
| class -- the class is an abstraction of those instances, but
| this seems akin to defining humans by looking at a bunch of
| them rather than simply having a definition that describes
| their common characteristics (bipedal, mammal, etc.).
|=20
| > If yes, subclassing is an order relation between classes, since
| > the three properties Reflexivity, Transitivity and Antisymmetry
| > can be inferred from that definition.
|=20
| I can see that those definitions can be inferred about the
| instances, i.e., about the order relation between the instances
| of the classes, but not about the classes themselves. "Superclass"
| and "subclass" are relationships between the classes, not the
| individuals. "Hominid" is a subclass of "Primate", but "John"
| is not a subclass of "man" (or of "primate"), he is an instance.
|=20
| > 1. Reflexivity:
| >
| > "If x is instance of A, then x is instance of A"
| >
| > 2. Transitivity:
| >
| > "If x is instance of A, then x is instance of B"
| > "If x is instance of B, then x is instance of C"
| >
| > Implies: "If x is instance of A, then x is instance of C"
| >
| > 3. Antisymmetry
| >
| > "If x is instance of A, then x is instance of B"
| > "If x is instance of B, then x is instance of A"
| >
| > Implies: "A=3DB"
| >
| > Or it that the one you don't agree with?
| >
| > Sets have nothing to do with it, seems to me.
|=20
| Classes or types are characteristics imposed upon sets, or upon
| sets of individuals. In either case, "class" is not a characteristic
| imposed on "John" or "Fido" (as individuals, though they can belong
| to a class), whereas "class" *is* a characteristic imposed on "man"
| or "dog".
|=20
| > | IOW, if I hear the word "class" or "type" we're not talking
members
| > | of sets (or "instances of classes"), we're talking the
relationship
| > | between classes.
| >
| > If you do not define the subclassing by instances, how do you define
| them?
| > I guess you use characteristic properties, right? So let's say class
| > A is characterized by Pa, class B is characterized by Pb ...
|=20
| Yes, by class definitions. "primate", "hominid", "vehicle", etc. are
| class definitions. I don't understand how you could define a class by
| instances. That would seem almost anecdotal. A class could be
| considered as a set of characteristics shared by a group of
| individuals, but the class definition itself is independent of any
| specific group of individuals.
|=20
| > The definition of subclassing is then
| >
| > "A is a subclass of B"	<=3D> "Pa =3D> Pb"
| >
| > ... seems to me that leads to the same conclusion than above ...
| >
| > So ... what did I miss?
|=20
| I dunno. We're either talking past each other or there's some
| other fundamental difference in our understanding of the language.
| That's why I included the definition of superclass and subclass
| from set theory in my previous message.
|=20
| Murray
|=20
| ......................................................................
| Murray Altheim                  <http://kmi.open.ac.uk/people/murray/>
| Knowledge Media Institute
| The Open University, Milton Keynes, Bucks, MK7 6AA, UK
|=20
|      "In Las Vegas Mr Gates also demonstrated a prototype
|       fridge magnet which can be programmed to receive traffic
|       reports, sports results and advertisements from local
|       restaurants using the same FM signal as the wristwatch."
|                                   -- The Guardian, 10 Jan 2003.