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Re: comments on issue 5.19 (classes as instances) and 4.6 (equivalentTo)

From: pat hayes <phayes@ai.uwf.edu>
Date: Thu, 15 Aug 2002 13:20:47 -0700
Message-Id: <p05111b06b981ae21f1e0@[65.212.118.249]>
To: "Peter F. Patel-Schneider" <pfps@research.bell-labs.com>
Cc: www-webont-wg@w3.org

>[I found this note surprisingly difficult to write.  I may end up
>significantly revising it due to comments from the group.]
>
>
>Some preliminary discussion on classes as instances must preceed discussion
>of equivalentTo, because equivalence and identity for classes and
>individuals depends on how they are handled which is impacted by the stance
>on classes as instances.

Sorry I've been out of the loop lately, so these comments are late.

>References:
>
>[RDFMT] RDF Model Theory, http://www.w3.org/TR/rdf-mt/
>[D+OMT] A Model-Theoretic Semantics for DAML+OIL (March 2001)
>
>
>Issue 5.19: Classes as instances
>
>I see four stances that can be taken with respect to classes as instances.
>By the way, the same four stances exist with respect to properties as
>instances.  I will assume that any reasonable system takes the same stance
>for properties as instances that it takes for classes as instances.

I agree they tend to go together and there is no logical reason to 
distinguish them.

>
>1/ Super-strong: A super-strong stance on classes as instances says that a
>class is just something attached to an individual.  This is the stance
>taken in the new RDF model theory, where the extension of a class is
>defined from an individual, not from the name of the class.  This is shown
>in the definition of CEXT(x), which is defined on resources (individuals),
>not on names.  (Contrast this to IS, which gives meanings for URI refs as
>individual names by mapping them into resources.)

I wonder why you call this 'super-strong' It is in fact rather a weak 
stance, seems to me; it is what you get to by requiring a class 
semantics to be fully first-order. The key point is that the 
distinction between class individuals and class extensions allows two 
distinct classes to have the same extension; and that, in turn, 
blocks all potential inferences from anything to do with membership 
to any kind of class identity. Which is exactly why the resulting 
logic can be first-order.

>
>2/ Strong: A strong stance on classes as instances says that if two names
>denote the same individual then their meaning as classes must be the same.
>This stance is, of course, compatible with the super-strong stance, and, I
>think, has no observable difference from the super-strong stance.

It is indeed a consequence, yes.

>Consequence of the super-strong and strong stances:
>
>   E1 - if a and b are names that denote the same individual
>        then the class extensions of a and b are the same

If you say that a class *is* an individual with a class extension, 
then this consequence seems obvious. I suspect that you have a 
lingering tendency to think that classes cannot possibly actually BE 
individuals, so you draw a conceptual distinction between the 
individual and the class. Let me urge you to try the mental exercise 
of simply allowing that classes might actually be real things, and 
themselves be members of classes, have properties, and so on. The 
individual IS the class; the class extension is simply a technical 
semantic device used to allow non-well-founded constructions, such as 
rdfs:Class being an rdfs:Class.

It is important to grok the fact that in conventional (Tarskian) 
first-order semantics, the term 'individual' is not a sortal: it 
isn't a category of a certain 'kind' of 'thing'; it just means 'the 
things in the universe' (whatever they  happen to be). It simply 
doesn't make sense to *contrast* individuals with other, 
non-individual, things.  There is nothing in conventional model 
theory that says that things like classes, properties and so on 
cannot be in a first-order universe. In fact, there is an explicit, 
central, assumption in model theory that universes can be universe of 
anything; all that is required of a universe is that be a nonempty 
set, not that it be a set of any particular kind of things. 
First-order universes may contain classes and properties just as they 
might contain numbers, astronomical objects, people or pieces of 
cheese. What makes HOL higher-order is not that its universes contain 
things like sets and properties, but that they are *al*l required to 
contain *very large infinite numbers* of them, and these very strong 
requirements on the HOL universes go beyond what can be axiomatized 
in FOL. But its the cardinalities that make HO semantics 'higher', 
not the nature of things in the universe.

I would like to have written the RDF model theory (and the new KIF/CL 
model theory, which inspired it in some ways) more directly to 
reflect this central intuition, but that would have required basing 
it on an unconventional set theory (non-well-founded set theory). 
Anticipating that this would be too controversial for the current 
Webont.RDF community, not to mention the CL/KIF/CG  communities, I 
let myself be persuaded by Chris Menzel to use his elegant trick of 
having an explicit 'extension' mapping. But it is important to bear 
in mind that this is really only a formal trick; the intended meaning 
is that the individual here actually *is* the class. I should add 
that for Chris Menzel and several other 'serious ontology' people (I 
think our Chris Welty is among them), this construct has an 
additional advantage, in that it is compatible with a 'intensional' 
view of classes, where class identity is not inferable from identity 
of membership. This corresponds to a view of classes where a class is 
more like a concept or a category than a mathematical set; the 
classical example to illustrate the difference is 'human being' 
versus 'featherless biped'; the two concepts seem clearly distinct, 
yet considered as classes they happen to have the same members; two 
different classes with the same extension. People who worry about 
such matters (and they are legion) often like to claim that this 
reflects a difference between mere accidental identity and 
*necessary* identity, which is a path I personally prefer not to go 
down; but its popularity among ontologists, particularly those 
concerned with capturing the meanings of natural language concepts, 
might give us some reason to hope that this semantic framework will 
be useful, and certainly to provide an intuitive way of explaining 
what it is all about.

>(Strangely enough, having class and instance names disjoint does not
>totally preclude the strong stance.

Of course, thought it does introduce wholly unnecessary syntactic 
complexities. Its a bit like having wear a necktie: it hurts, plays 
no useful role in anything, and only adds to the cost; but many 
people feel that it makes everything look better.

>  It is possible to have a mapping from
>class names to individual names that serves to relate classes with
>individuals.  Of course, this approach has some differences from the
>regular strong stance, as the name mapping has to be used in appropriate
>places. )

Right. It is one way to see that the 'strong' stances are in fact 
rather weak, since they are expressively equivalent to an obvious 
first-order formulation (though much more compact and usable, and 
many people feel more 'natural')

>
>3/ Weak:  The weak stance on classes as individuals only says that every
>class has an associated individual.  There is no commitment that if two
>classes are associated with the same individual then their meaning as
>classes are the same.  One way of performing this association is to use the
>name of the class as the name of the individual.  This is the stance taken
>in the DAML+OIL model theory.
>
>In the weak stance E1 is not valid.

I rather fail to see the utility of this position. The only point of 
allowing classes to be individuals is that one can then put them into 
classes and predicate properties of them. If the 'association' is too 
weak to support this, then it plays no useful role.

>4/ None:  This stance denys any relationship between classes and instances
>besides the instance relationship.  Many DL-based representation systems
>have this stance.  This stance often comes along with a partitioning of the
>space names into separate subspaces for instances, classes, and
>properties.  Even if this is not the case, there is no relationship between
>a class and an instance that share the same name.

This is obviously semantically compatible with the 'strong' 
position(s); it amounts to a syntactic restriction on name use. This 
is the traditional restriction in first-order syntax, but it really 
isn't necessary. Its intellectual roots go back to Russel's 'layered' 
type theory solution to the Russel paradox; set theorists gave up on 
that as being unworkable in practice in the 1930s.

>
>Issue 4.6: equivalentTo
>
>There are several semantic relationships that have to be considered when
>investigating how to treat sameIndividualAs, sameClassAs, samePropertyAs,
>and equivalentTo.  The first three are rather easy to describe and do not
>depend on any of the stances taken on classes as individuals.

I would suggest cutting through this tangle and simply having what 
would be written in logic as equality. Call is SameAs. You can apply 
SameAs to anything; individuals, properties, classes, whatever. It 
means that two names are being used to refer to the same thing, is 
all. Obviously, if A and B are two names for the same class, that 
class has the members that it has whatever name you call it, and 
similarly for properties. In fact, Leibnitz' law applies: whatever 
expression you can write with one name, you can infer the same 
expression with the other name substituted.

What you cannot do, however, is to infer what would be written in CL as

(implies (forall (@x) (iff (P @x) (Q @x)) (SameAs P Q) )

but since the LHS wouldn't be expressible in OWL in any case, this 
doesn't even arise.

Pat

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Received on Thursday, 15 August 2002 16:24:03 GMT

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