Ed, (01)
I agree we need to be clear. (02)
> it is that one must be clear that:
> -- is-a -->
> has importantly different semantics from
> -- <term> -->
> where the <term> is anything other than "is-a". (03)
The possible semantic varieties of 'is-a' in natural language are quite well
known. (04)
It seems to me that you are dealing with the subsumption variety here (Man
is an Animal - all men are animals). (05)
Isn't there another variety that needs also to be considered - the
instantiation variety. Socrates is a man. (I wonder how the UML-ites will
diagram this?) (06)
While one can see subsumption as a relation between instances of terms
(Every instance of A is an instance of B) - one can also see it as a
relation between the terms. If one takes the second course, then these two
varieties of is-a share the property of being a relation between the terms
that are linked to rather than of instances of the term. For example, that
the class of men is a sub-type of the class of animals. (07)
This property is also shared by some of the other relations you mentioned -
union, intersection, etc. - though again these can be cashed out in terms of
their instances. (08)
You mentioned relations that are between instances/members of the relata -
e.g. A -- is part of --> 1..1 B. Won't the notation also need to deal with a
-- is a part --> b or (is-part-of a b) (where a is my hand and b my arm).
Here again the relation is relating the terms rather than instances of the
terms. (09)
Regards,
Chris (010)
> -----Original Message-----
> From: uom-ontology-std-bounces@xxxxxxxxxxxxxxxx [mailto:uom-ontology-
> std-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Ed Barkmeyer
> Sent: 13 July 2009 18:15
> To: Matthew West
> Cc: 'uom-ontology-std'
> Subject: Re: [uom-ontology-std] UoM ontology standard - a proposed
> program of work
>
> I wrote:
>
> >> (3) Matt doesn't mention the CDIF notation for
> "subtype"/subsumption.
> >> This is a foundational concept in OWL, and it is very important to
> >> modeling measurement concepts. In particular, every 'measurement
> unit'
> >> is_a 'quantity'. I would be wary of a notation like:
> >> measurement-unit -- is a --> quantity
>
> Matthew West wrote:
> > [MW] I do not recall that CDIF has a specific notation, so you can
> either do
> > what you have done here, or quite often I use the EXPRESS notation as
> an
> > extension.
>
> My point is that we need notation for subsumption, at least, and I
> think
> it useful to have a notation for unions and intersections and
> exclusions
> (all of which the ODM added to UML, for OWL). I also believe in
> ternary
> and occasionally quaternary relations, and I prefer to see some
> distinguishing notation for them. (The VIM has at least one ternary
> relationship in it: system-of-units appoints base-measurement-unit for
> quantity-kind.)
>
> >> because it makes the notation ambiguous. 'is_a' is a class-to-class
> >> relationship, rather than an instance-to-instance relationship (like
> >> 'part of'). It models an axiom, not just a relation. That is:
> >> A -- is part of --> B
> >> models a relation "is part of" whose domain is things that satisfy
> >> relation (class) A, and whose range is things that satisfy relation
> B -
> >> -
> >> a vocabulary item. It has two free variables. Whereas,
> >> A -- is a --> B
> >> models a proposition, a statement: Every A is a B. Formally,
> >> (forall x) (if (A x) (B x))
> >> It has no free variables. And the model asserts that proposition,
> >> making it an axiom.
> >> So I would object to overuse of the arrow notation, if it leads to
> such
> >> an ambiguity.
> >
> > [MW] I'd just like to push back on this a bit. I agree that you need
> to be
> > careful to distinguish between relationships between classes and
> > relationships that hold between members of classes. However, I had
> > understood subtype/supertype as being able to be seen in either way.
> So that
> > one way of seeing subtype/supertype is as an identity relationship
> between
> > members of the subtype and members of the supertype.
> >
> > Comments?
>
> This might be better discussed on the Forum exploder.
>
> In languages like CDIF and UML, specifying A and B to be "classes" or
> "entities", defines a relation with one free variable (A x) or (B x),
> where x is an arbitrary thing from the universe of discourse -- a
> discovered entity instance or object -- and the "result" of (A x) is
> either true or false -- the new x either is an A or it isn't. (It is
> possible that we don't know which, but there is an underlying
> epistemological assumption that exactly one of those statements is
> true,
> axiomatized as the "law of the excluded middle": (forall x)(OR (A x)
> (NOT (A x)).)
>
> And in those same languages: "A is-part-of B" introduces a relation
> (is-part-of a b)
> which maps an ordered pair of arbitrary things (a, b) to either true or
> false. There are some semantic scoping rules that have various
> interpretations, but let us assume that "is-part-of" is a unique
> relation that is specified to be meaningful only for class A. (The
> alternative is that it specifies a relationship "A:is-part-of", which
> is
> different from an "X:is-part-of" relationship for any other domain X.)
> That means that the specification implies an axiom:
> (forall a b)(if (is-part-of a b) (A a))
> in any such actual relationship, the first thing must be an A.
> And similarly, the second thing must be a B.
>
> Notably, this specification, of itself, does NOT say that every A is
> part of some B. In UML, I can add a notation, which we could represent
> as:
> A -- is part of --> 1..1 B
> and that adds two more axioms:
> (forall a)(if (A a) (exists b)(is-part-of a b))
> If a thing is an A then there is a b that it is part of. Each A
> is-part-of at least 1 B. And:
> (forall x b c)(if (and (is-part-of x b) (is-part-of x c))
> (= b c))
> If a thing x is a part of two things b and c, b and c must be the same
> thing. Each A is part of at most one B.
>
> So, following the above semantic interpretation patterns:
> A -- is-a --> B
> means (forall x y)(if (is-a x y) (A x))
> If for any two things x and y "x is-a y" is true, then x must be an A.
> The alternative interpretation, noted above, is that:
> A -->is-a --> B
> specifies a relationship "A:is-a", which is different from any other
> "is-a" relationship. I doubt that either of these is the
> interpretation
> we want for "is-a", which means that the notation doesn't have
> consistent semantics.
>
> And this goes further. If one chooses Matthew's SQL-like
> instance-to-instance identity relationship as the meaning, we have the
> further problem that
> A -- is-a --> B
> does not by itself say that every A is-a some B. We would need
> something like:
> A -- is-a --> 1..1 B.
> That is, we want to assert that the instance-to-instance relationship
> must exist. "pediatrician is-a doctor" is different from
> "pediatrician is-a-member-of preferred-provider-organization".
>
> This is why I am uncomfortable with such ad hoc conventions. It's not
> that one cannot agree to use: A -- is-a --> B to represent
> subsumption;
> it is that one must be clear that:
> -- is-a -->
> has importantly different semantics from
> -- <term> -->
> where the <term> is anything other than "is-a". And this violates what
> Bernd Wenzel calls the Law of Least Astonishment: it encourages false
> analogies and erroneous interpretations.
>
> -Ed
>
> --
> Edward J. Barkmeyer Email: edbark@xxxxxxxx
> National Institute of Standards & Technology
> Manufacturing Systems Integration Division
> 100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
> Gaithersburg, MD 20899-8263 FAX: +1 301-975-4694
>
> "The opinions expressed above do not reflect consensus of NIST,
> and have not been reviewed by any Government authority."
>
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