Re: Equality and subclass axioms

At 16:24 +0000 11/25/00, Ian Horrocks wrote:
>I have heard some worries expressed about the effect of having
>equality (if and only if) axioms in an ontology. For example, we could
>state in our ontology that an object is a Triangle if and only if it
>is in the intersection of the classes Polygon and ThreeSidedThing (in
>this case it is sometimes said that being in the intersection of the
>classes Polygon and ThreeSidedThing is a necessary and sufficient
>condition for being a Triangle). The semantics of this axiom are that
>in every satisfying "interpretation" (i.e., in every model of the
>world that conforms to the structural constraint imposed by the
>axiom), the set of objects that are Triangles must be equal to the
>intersection of the set of objects that are Polygons and the set of
>objects that are ThreeSidedThings.
>As far as I can understand it, the worry is as to what will happen if
>another equality axiom w.r.t. Triangle is added to the ontology, e.g.,
>that an object is a Triangle if and only if it is in the intersection
>of the classes Polygon and ThreeAngledThing. This doesn't cause any
>problem: the set of objects that are Triangles must be equal to the
>intersection of the set of objects that are Polygons and the set of
>objects that are ThreeAngledThings, and from the transitivity of
>equality we can of course also infer that the intersection of the set
>of objects that are Polygons and the set of objects that are
>ThreeSidedThings is equal to the intersection of the set of objects
>that are Polygons and the set of objects that are ThreeAngledThings.
>As with other ontological axioms, these kinds of axiom give structure
>to the domain of discourse by restricting the set of valid models. Of
>course it is possible to restrict the set so tightly that some (or
>even all) classes are empty in all valid models, but this can happen
>with or without equality axioms. If I am allowed to add a plug for
>reasoning at this point, I would say that this is an example of how it
>can be useful as it makes it possible for a tool to draw the users
>attention to such an occurrence, which may indicate an error in the
>design of the ontology.
>Regards, Ian

  Here's my worry, which is controllable, but definitely worth 
thinking about.  You, being a rational person, come about with a rule 
that says triangles are three-sided things.  I, being an irrational 
person, have an axiom stating triangles are 4-sided.  We both use if 
and only if rules.  Some web crawler comes, scrapes both of our rules 
into the same knowledge base -- and then what happens?  Do we get two 
kinds of triangles or no kinds of triangles.  Obviously, no model can 
allow triangles that are  both  three and four-sided - so I would 
assume by throwing in my irrational axiom I've somehow "negated" 
yours.  Notice if we're not using If and only if, this problem 
doesn't come up (i.e. IF three sides and IF 4 sided can concurrently 
  I think in general we who are worrying about web semantics will 
either need something that tags axioms to their ontologies or 
something that allows very permissive semantics.  Notice that I'm not 
bothered by having something that uses the term triangle for two 
different things - because both intended extensions are in there (so, 
for a "search engine" we'd just be getting extra answers and we'd use 
some rules to eliminate the ones we don't like).  I'm more worried 
about something that let's someone whose beliefs differ from mine do 
something that causes my inferences to fail.
  Again, these situations are sure to come up on something as 
permissive as the net (imagine the pro-choice and anti-abortion folks 
trying to defeat each others' theorem provers) - but we need to 
decide at some point how to tag or mark or distinguish or something 
where axioms come from, or something like that.  This is clearly 
beyond RDF, but crucial to any really practical web semantics.
  -Jim H.

Prof. James Hendler		Program Manager
DARPA/ISO			703-696-2238 (phone)
3701 N. Fairfax Dr.		703-696-2201 (Fax)
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Received on Sunday, 26 November 2000 11:24:23 UTC