11.1.1 Excursus on multiple inheritance
The "is-a" relation
It is good to reflect on this ontology briefly.
Note: This page is advanced material.
Normally, we say that the upwards link in the ontology is an "is-a" relation.
This means that we have, in this ontology, committed ourselves to the "ontological commitment" that a Man is a Male, and a Woman is an Adult.
This interpretation could be critiqued. For example, if we were to add the "Sex" supertype to "Male" and "Female", and the "Age" supertype to "Juvenile" and "Adult", the beginning ontology would look like this:
Now what would happen if we added "Woman" underneath "Female" and "Adult"?
Now we would be making the ontological commitment that a Woman is an Age and is a Sex, whereas it might be more natural to say that a Woman is a Person who has an Age.
What is it "to be"?
This brings up fundamental issues of what it means "to be" something. No-one would doubt that a person's "Sex" is fundamental to that person's identity. Seen over time, the same is true for a person's "Age".
However, does this mean that the category Man "is an" Age, or "is a" Sex, in the same sense as the category Elephant "is a" Mammal?
Let us dig a little deeper. The fact that the category Elephant "is a" Mammal means that everything a Mammal is (in terms of attributes), an Elephant is also. The category "Elephant" is different from the category "Mammal" because of the fact, among other reasons, that is has more attributes than "Mammal", e.g., it has large ears and a trunk.
Thus the "is a" relation encodes a "loss of attributes" in the upwards going direction. When you go from "Elephant" to "Mammal", you could be taking about the same individual (e.g., the circus-elephant "Clyde"), but you are looking at this individual from the perspective of considering less attributes. This is because "Elephant" is everything a "Mammal" is, but has more attributes, as per its definition "by genus, species, and differentiae".
In the same way, it could be argued that "Man" is everthing "Male" is, and everything "Adult" is, but has more attributes.
However, we could end up with a contradiction in our ontology. Suppose we have a distinction in our ontology between "Physical" and "Abstract", and we have made the "ontological commitment" that this distinction is a dichotomy, i.e., nothing "Abstract" can be said to be "Physical" also. This is a standard assumption in ontology.
Suppose now that we have classified "Sex" (and therefore "Male" and "Female") as an Attribute, which we have classified somewhere beneath "Abstract". If we then wish to say that "Man" is a "Person", which belongs somewhere underneath "Object", which in turn is a subtype of "Physical", we are in trouble if we, at the same time, wish to say that "Man" is-a "Male".
This is because "Man" would then have supertypes both in the "Abstract" subiherarchy, and in the "Physical" subhierarchy.
A solution involving CGs
To avoid such confusion, it may be better to categorize Man as "Person" (in the "Physical" subhierarchy) and then define "Man" with a CG as follows:
This relates "Man" to "Male" and "Juvenile" in the way that seems most natural: Not with the "is-a" relation (saying that Man "is a" Male), but with the attribute relation in the definition.
Thus we have said that "Man" (species) is-a Person (genus) and its differentiae are that "Man" has two attributes: Male and Adult.
This sidesteps the issue of whether Man "is-a" Male or Adult completely, instead relating Man to Male and Adult by means of the "attr" relation.
First- and Second-order types
However, a further point of critique could be advanced against our solution: Are "Male" and "Female" really instances? Could one not say that they ought to be subtypes?
This is a long-standing debate in philosophy. It is similar to the question of whether "Red" is an instance of "Color", or a subtype of it.
One answer could be that "Red" is both a type and an instance. It is an instance of the type "Color", and as such belongs in a catalog of individuals. However, it is also a type in its own right, and as such belongs in a type hierarchy.
Therefore, types such as "Color" have been termed "Second-order types" because they have types as instances.
Something similar could be said for "Male" and "Female". They are both arguable types: Maybe they should be subtypes of "Attribute".
Yet they are also instances of the type "Sex". Therefore, "Sex" could also be termed a second-order type on a par with "Color".
However, Amine does not support second-order types, and therefore we choose to label "Male" and "Female" as individuals of type "Sex" rather than types.
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