Genus–differentia definition


A genus–differentia definition is a type of intensional definition, and it is composed of two parts:
  1. a genus : An existing definition that serves as a portion of the new definition; all definitions with the same genus are considered members of that genus.
  2. the differentia: The portion of the definition that is not provided by the genus.
For example, consider these two definitions:
Those definitions can be expressed as one genus and two differentiae:
  1. one genus:
  2. * the genus for both a triangle and a quadrilateral: "A plane figure"
  3. two differentiae:
  4. * the differentia for a triangle: "that has 3 straight bounding sides."
  5. * the differentia for a quadrilateral: "that has 4 straight bounding sides."
The use of genus and differentia in constructing definitions goes back at least as far as Aristotle.

Differentiation and Abstraction

The process of producing new definitions by extending existing definitions is commonly known as differentiation. The reverse process, by which just part of an existing definition is used itself as a new definition, is called abstraction; the new definition is called an abstraction and it is said to have been abstracted away from the existing definition.
For instance, consider the following:
A part of that definition may be singled out :
and with that part, an abstraction may be formed:
Then, the definition of a square may be recast with that abstraction as its genus:
Similarly, the definition of a square may be rearranged and another portion singled out:
leading to the following abstraction:
Then, the definition of a square may be recast with that abstraction as its genus:
In fact, the definition of a square may be recast in terms of both of the abstractions, where one acts as the genus and the other acts as the differentia:
Hence, abstraction is crucial in simplifying definitions.

Multiplicity

When multiple definitions could serve equally well, then all such definitions apply simultaneously. Thus, a square is a member of both the genus [a] rectangle and the genus [a] rhombus. In such a case, it is notationally convenient to consolidate the definitions into one definition that is expressed with multiple genera :
or completely equivalently:
More generally, a collection of equivalent definitions can be recast as one definition that is expressed with genera. Thus, the following:
could be recast as:
A genus of a definition provides a means by which to specify an is-a relationship:
The non-genus portion of the differentia of a definition provides a means by which to specify a has-a relationship:
When a system of definitions is constructed with genera and differentiae, the definitions can be thought of as nodes forming a hierarchy or—more generally—a directed acyclic graph; a node that has no predecessor is a most general definition; each node along a directed path is more differentiated than any one of its predecessors, and a node with no successor is a most differentiated definition.
When a definition, S, is the tail of each of its successors, then S is often called the species of each of its successors, and each direct successor of S is often called an individual of the species S; that is, the genus of an individual is synonymously called the species of that individual. Furthermore, the differentia of an individual is synonymously called the identity of that individual. For instance, consider the following definition:
In this case:
As in that example, the identity itself is often used to refer to the entire individual, a phenomenon that is known in linguistics as a pars pro toto synecdoche.