# 5.2 Methods of definition

## Introduction

It is important to understand the ways in which a type can be defined. In this section, we list four such ways and then treat each of them in some detail.

## Four types of definition

Types may be defined in a number of ways. Here are four of them:

• By extension
• By intension
• By axioms
• By reference to other types, with differentiae.

## By extension

When one defines a type by extension, one provides a list of references to all of the members of the group which the type defines. This method is only practical if the group of individuals is relatively small.

For example, the type JonesFamilyMember could comprise the individuals <Marilyn, Richard, Tom, Sue> if those were the names of the four members of the Jones family. The list <Marilyn, Richard, Tom, Sue> would be an extensional definition of the type JonesFamilyMember.

## By intension

When defining a type by intension, one lists the properties that define which individuals belong to the group and which do not. It can also be done by giving a general rule for finding out whether a particular individual is a member of the group. This is only practical if one can define the type by rules using some language.

For example, the type "Mammal" could be defined by intension with the rule "A Mammal is an Animal that is sexual, bears offspring by pregnancy, and breastfeeds the young".

As an example of a type defined by a general rule, consider the type "OddNumber". An odd number is a number that cannot be divided by two. If one tries to divide it by 2, there will always be 1 left over. Therefore, a rule for defining "OddNumber" by intension could be "If one divides the number by 2 and then gets 1 left over, then the number is odd. Otherwise, the number is not odd."

## By axioms

The third method, by axioms, is practical if we are dealing with an abstraction, e.g., mathematical entities. If you want to know more about axioms (not required), click here.

## By reference to other types with differentiae

The fourth method, when defining "by reference to other types by differentiae", is practical for many things. For example, if we have defined the type "Professor", we can define the type "FullProfessor" like this: "A FullProfessor is a Professor with the differentiae that he or she has a permanent employment contract and has been approved competent at the level of a full professor".

This method is actually the one being used when defining type hierarchies of concept types. This is done mostly by defining new types in terms of conceptual graphs. These conceptual graphs, in turn, are based on other types already defined. The relations between the concepts, as well as the referents of the concepts, show the differentiae. Therefore, when defining concept types using conceptual graphs, we are defining "by reference to other types with differentiae".

When defining by intension, one usually also employs this method. It can be difficult to define a type completely without making any reference to other types. However, for some types it can be done, and therefore "by intension" was listed apart from "by reference to other types".

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