12.7 Designing the Properties 307
compelling as the others mentioned above. If there is some good reason
for not using subclasses (such as exceptions or extrinsic properties), then
the introduction of new properties is not a sufficient reason for using sub-
classes. On the other hand, having additional kinds of properties is not
a requirement for using subclasses. There are many taxonomies, includ-
ing biological taxonomies, where subclasses do not introduce any new
properties.
Possibly the most subtle issue concerning subclasses is the issue of how
the instances act or are acted upon. The classic example of the resulting con-
fusion is the question of whether square is a subclass of rectangle. From a
logical point of view, it seems obvious that squares are a proper subset (and
therefore subclass) of rectangles. However, according to one view of cog-
nition and concepts, objects can only be defined by specifying the possible
ways of acting on them (Indurkhya 1992). For instance, Piaget showed that
the childconstructsthe notion of object permanence in terms of his or her
own actions (Piaget 1971).
Accordingly, the concept square, when defined to be the set of all squares
without any actions on them, is not the same as the concept square in which
the objects are allowed to be shrunk or stretched. In fact, it has been found
that children’s concepts of square and rectangle undergo several transfor-
mations as the child’s repertoire of operations increases (Piaget and Inhelder
1967; Piaget et al. 1981). This suggests that one should model “squareness” as
a property value of theRectangleclass, called something likeisSquare,
which can be either true or false.
In general, concepts in the real world, which ontologies attempt to model,
do not come in neatly packaged, mind-independent hierarchies. There are
many actions that can potentially be performed on or by objects. The ones
that are relevant to the purpose of the ontology can have a strong affect on
how the ontology should be designed (Baclawski 1997b). For still more ex-
amples of how complex our everyday concept hierarchies can be, see (In-
durkhya 2002; Lakoff 1987; Rosch and Lloyd 1978).
12.7.2 Domain and Range Constraints
It is sometimes convenient to think of a property as being analogous to a
mathematical function. A mathematical function maps each element of a
domainto an element of arange. For example, the square-root function has the
