Genus-differentia

A genus–differentia definition is a type of intensional definition, and it is composed of two parts:

For example, consider these two definitions:

Those definitions can be expressed as one genus and two differentiae:

The process of producing new definitions by extending existing definitions is commonly known as differentiation (and also as derivation). 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 (using parentheses here):

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.

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 (and possibly no differentia, as in the following):

or completely equivalently:

More generally, a collection of ${\displaystyle n>1}$ equivalent definitions (each of which is expressed with one unique genus) can be recast as one definition that is expressed with ${\displaystyle n}$ genera. Thus, the following:

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