Is subclassing "strict order" or is it reflexive? RE: [sc34wg3] New SAM PSIs
Bernard Vatant
sc34wg3@isotopicmaps.org
Fri, 14 Feb 2003 17:50:41 +0100
Murray
I propose to continue that debate in private, because I think it's =
becoming irrelevant to this forum, already very busy.
But of course if some folks want to be in cc of this debate, just ask :)
_____________________________________
Bernard Vatant
Senior Consultant - Knowledge Engineering
www.mondeca.com
_____________________________________
| -----Original Message-----
| From: sc34wg3-admin@isotopicmaps.org [mailto:sc34wg3-
| admin@isotopicmaps.org] On Behalf Of Murray Altheim
| Sent: vendredi 14 f=C3=A9vrier 2003 15:46
| To: sc34wg3@isotopicmaps.org
|=20
| Bernard Vatant wrote:
| > Does Murray need a little help from the maths'teacher? :))
|=20
| Much appreciated. I hope you can bear with me. I'll necessarily
| repeat myself in hopes of being clear (which sometimes has the
| opposite effect I know).
|=20
| > Well, what is the type of class-subclass relation?
|=20
| Before we begin, I'd like to clarify the conversation. I note that
| the XTM 1.0 Specification calls this a superclass-subclass =
relationship,
| whereas the SAM calls it supertype-subtype. I will assume "class" and
| "type" are synonyms for the purpose of this discussion.
|=20
| In what domain are we talking (which is the real issue here, IMO)?
| Are we talking strictly set theory, or are we talking taxonomy? How
| are these different? (I.e., are the relationships between type labels
| in set theory the same as the inter-class relationships in
| taxonomies? How are these different? I'll try to demonstrate...).
|=20
| On the one hand, and the error I think OWL is making, is that it
| seems to be talking about *members* of sets. So, if we have a set A
| and a set B whose members are all the same, they are identical sets.
| Eg., if we have 100 hominids and 100 primates, and they are the same
| 100 individuals, we can consider the two sets identical. We cannot,
| and I repeat, cannot, necessarily therefore consider the classes of
| primates and hominids as identical. They are not.
|=20
| A type hierarchy or taxonomy is strictly talking about the
| relationships between types or classes, not their instances. This is
| the point about extension and intension that Lars Marius brought
| up.
|=20
| This may be interpreted one of two ways. First, as a relationship
| between sets of individuals, secondly as essentially an argument
| about differences in labels, saying, for example, that "primate"
| in English and "primas" in German refer to the same class, or that
| two different labels for a specific star refer to the same star;
| we're not actually talking about a *difference* in classes, merely
| their labels, that within one context or domain a difference in
| label is not a difference in actual class, i.e., it does not
| constitute a different class but merely a different label for the
| same class.
|=20
| So, if further down the line we learn more information, allowing
| us to determine that A and B refer to the same set, A and B are
| merely alternate labels or designations for the same set, not
| different classes.
|=20
| > -- An "order" relation, noted <=3D herafter, with the three =
following
| > properties ...
| >
| > 1. Transitivity: IF (a <=3D b) and (b <=3D c) THEN (a <=3D c)
| > 2. Reflexivity: (For all x) (x <=3D x)
| > 3. Antisymmetry: IF (a <=3D b) and (b <=3D a) THEN (a =3D b)
| >
| > -- Or a "strict order" relation, noted < hereafter, with the =
following
| > properties ...
| >
| > 1. Transitivity: IF (a < b) and (b < c) THEN (a < c)
| > 2. (For all x) (x < x) is false
| > 3. IF (a < b), THEN (b < a) is false
| >
| > Note that a "strict order" relation is *not* an "order" relation -
| speak
| > about consistency in maths vocabulary :(
|=20
| I don't think this is about ordering at all.
|=20
| But if I'm correct and we're talking taxonomies here and not members
| of sets, which also seems to be the more demanded relationship type
| for web-based ontologies (I don't think we want to consider such
| things as trucks<=3Dvehicles or primates>=3Dhominids in terms of sets =
but
| rather their taxonomic class relations), then all this talk of
| transitivity and reflexivity seems misplaced.
|=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.
|=20
| > I understand, Murray, you consider class-subclass relation as a =
"strict
| > order", and do not understand how it can be considered an "order".
|=20
| The transitivity of set membership is different than the relations
| between their types or classes. I searched across my available books
| and found the closest I could to what I believe you're talking about
| -- let's make sure we agree on some basic terms (here's an excerpt
| we can perhaps use as a baseline):
|=20
| "The type hierarchy is a partial ordering defined over the set
| of type labels. The symbol =3D designates the ordering. Let s, t,
| and u be type labels:
|=20
| * If s<=3Dt, then s is called a subtype of t; and t is called a
| supertype of s, written t>=3Ds.
| * If s<=3Dt and s!=3Dt, then s is called a proper subtype of t, =
written
| s<t; and t is called a proper supertype of s, written t>s.
| * If s is a subtype of t and a subtype of u (s<=3Dt and s<=3Du), =
then
| s is called a common subtype of t and u.
| * If s is a supertype of t and a supertype of u (s>=3Dt and =
s>=3Du),
| then s is called a common supertype of t and u.
|=20
| In AI, the type hierarchy supports the inheritance of properties
| from supertypes to subtypes of concepts."
|=20
| What I hear you saying is that the first bullet item above allows
| s<=3Dt and t>=3Ds, therefore s=3Dt is possible. This is what I believe =
OWL
| is stating. The error here is that a type hierarchy is a higher order
| relation not between the individual classes, but between classes of
| individuals -- this is the error I believe is being made in OWL.
|=20
| And this would suggest that the subject heading of this message is
| simply an incorrect question, that superclass-subclass is a class
| relationship, not a relation between set members.
|=20
| [...]
| > That's what "checking for loops" means, for OWL class-subclass like =
for
| > any order relation: same issue e.g. with "whole-part" and same way =
to
| > deal with it.
| >
| > Does that help ?
|=20
| Well, I had about a half hour discussion with Enrico Motta (our
| director, who was on the WebOnt WG), and I still came away thinking
| this is all wrong. So I'm pigheaded (this isn't news to some folks).
|=20
| Even if there is some mathematically-logical sense within which this
| is true, which I think I've demonstrated is not the case (unless
| we're talking about the relationships between classes of individuals
| and *not* the relationships between classes (ie., taxonomy), I still
| think this decision would be a mistake, since such "subtleties" are
| likely lost on 99% of the buying public (ie., developers and
| implementors).
|=20
| Based solely on the "principle of least surprise" I believe most
| people would intuitively disbelieve that a class can simultaneously
| have both a superclass and a subclass relationship to another class,
| and absent an example discussing *individuals* I can't think of an
| example relating classes where this is true that doesn't bring
| extensionality/intensionality into play, or simply a matter of
| labelling (eg., "human" and "homo sapiens" are the same class*),
| not the relationship between classes.
|=20
| Pity the world that has to deal with this stuff when it hits
| the Web: is porn queen A the same as porn queen B? :-)
|=20
| Murray
|=20
| * which isn't strictly true anyway -- one is a colloquial name, the
| other the zoological taxonomic (genus-species) identifier; they're
| identifiers from different domains.
| ......................................................................
| 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.
|=20
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