From: Ian Horrocks <horrocks@cs.man.ac.uk>

Date: Sat, 25 Nov 2000 16:24:49 +0000 (GMT)

To: www-rdf-logic@w3.org

Message-ID: <14879.55411.77172.293725@localhost.localdomain>

Date: Sat, 25 Nov 2000 16:24:49 +0000 (GMT)

To: www-rdf-logic@w3.org

Message-ID: <14879.55411.77172.293725@localhost.localdomain>

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, IanReceived on Saturday, 25 November 2000 11:26:58 UTC

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