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

[Martin Bryan]

Bernard/Murray

I'd always understood that, as Murray quoted:

> Peirce notes that the
> class relation "does not contain any individuals at all. It
> only contains general conditions which _permit_ the determination
> of individuals."

though I would paraphrase it to state "classes define the set of properties
that can be used to determine whether or not a particular instance may
legitimately be added to the 'set of instances of members of the class'."

But then we come to the claim:

> > - 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.

But can't we check class equality by checking that the set of properties
needed to identify x as an A is the same set  of properties needed to
identify x as a B? What happens if the set of properties needed to identify
x as a C is a subset of the set of properties needed to identify x as either
a B or an A? Is C a "subclass" of A and B, and are both B and A valid
superclasses of C? It was questions such as these I was hoping you would get
onto.

The problem is that many subsets of properties are shared. In such cases is
the differentiators that are more important than the shared properties. Let
me try to give a simplified example, similar to the simple ones give
earlier:

Pet mammals: Properties: legs=4, ears=2, eyes=2
Dogs: Properties: legs=4, ears=2, eyes=2, sound emitted = bark ....
Cats: Properties: legs=4, ears=2, eyes=2, sound emitted = meow ....
Cow: Properties: legs=4, ears=2, eyes=2, sound emitted = moo ...

Now the three-legged one-eared furry ball that has just gone past my window
would not fit into any of these classes. The fact that he meowed as he went
past is theoretically irrelevant. The fact that he eats dog-food as readily
as cat food does not differentiate him any more than the fact he was in the
same field as the cow. Is the cow really a pet mammal? What exactly
determines whether or not one of the instances of animal types is also an
instance of pet mammals?

> > - 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 am not sure this is true. The definition of oneOf is "the means to define
a class via direct enumeration of its members". To me this says that "if
property x of instance y has oneOf the values specified for the class then
it is a member of the class, otherwise it isn't". Surely this is not
definition by extension but definition by example. You don't define the
members of the class, only the property values that must be exhibited by
members of the class.

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

Where is this OWL specifications? As OWL only defines class and subclass I
cannot see how this loop can be created within OWL.

Martin Bryan