# Equality and subclass axioms

From: Ian Horrocks <horrocks@cs.man.ac.uk>
Date: Sat, 25 Nov 2000 16:24:49 +0000 (GMT)

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, Ian
```
Received on Saturday, 25 November 2000 11:26:58 UTC

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