- From: pat hayes <phayes@ai.uwf.edu>
- Date: Thu, 15 Aug 2002 13:20:47 -0700
- 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 -- --------------------------------------------------------------------- IHMC (850)434 8903 home 40 South Alcaniz St. (850)202 4416 office Pensacola, FL 32501 (850)202 4440 fax phayes@ai.uwf.edu http://www.coginst.uwf.edu/~phayes
Received on Thursday, 15 August 2002 16:24:03 UTC