# Genus–differentia definition

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

**a genus**(or family): An existing definition that serves as a portion of the new definition; all definitions with the same genus are considered members of that genus.**the differentia**: The portion of the definition that is not provided by the genus.

For example, consider these two definitions:

*a triangle*: A plane figure that has 3 straight bounding sides.*a quadrilateral*: A plane figure that has 4 straight bounding sides.

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

*one genus*:*the genus for both a triangle and a quadrilateral*: "A plane figure"

*two differentiae*:*the differentia for a triangle*: "that has 3 straight bounding sides."*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 (384–322 BCE).^{[1]}

## Differentiation and Abstraction

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 square*: a quadrilateral that has interior angles which are all right angles, and that has bounding sides which all have the same length.

A part of that definition may be singled out (using parentheses here):

*a square*: (a quadrilateral that has interior angles which are all right angles), and that has bounding sides which all have the same length.

and with that part, an abstraction may be formed:

*a rectangle*: a quadrilateral that has interior angles which are all right angles.

Then, the definition of *a square* may be recast with that abstraction as its genus:

*a square*: a rectangle that has bounding sides which all have the same length.

Similarly, the definition of *a square* may be rearranged and another portion singled out:

*a square*: (a quadrilateral that has bounding sides which all have the same length), and that has interior angles which are all right angles.

leading to the following abstraction:

*a rhombus*: a quadrilateral that has bounding sides which all have the same length.

Then, the definition of *a square* may be recast with that abstraction as its genus:

*a square*: a rhombus that has interior angles which are all right angles.

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:

*a square*: a rectangle that is a rhombus.*a square*: a rhombus that is a rectangle.

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

*a square*: a rectangle and a rhombus.

or completely equivalently:

*a square*: a rhombus and a rectangle.

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

*a Definition*: a Genus_{1}that is a Genus_{2}and that is a Genus_{3}and that is a… and that is a Genus_{n-1}and that is a Genus_{n}, which has some non-genus Differentia.*a Definition*: a Genus_{2}that is a Genus_{1}and that is a Genus_{3}and that is a… and that is a Genus_{n-1}and that is a Genus_{n}, which has some non-genus Differentia.*a Definition*: a Genus_{3}that is a Genus_{1}and that is a Genus_{2}and that is a… and that is a Genus_{n-1}and that is a Genus_{n}, which has some non-genus Differentia.- …
*a Definition*: a Genus_{n-1}that is a Genus_{1}and that is a Genus_{2}and that is a Genus_{3}and that is a… and that is a Genus_{n}, which has some non-genus Differentia.*a Definition*: a Genus_{n}that is a Genus_{1}and that is a Genus_{2}and that is a Genus_{3}and that is a… and that is a Genus_{n-1}, which has some non-genus Differentia.

could be recast as:

*a Definition*: a Genus_{1}and a Genus_{2}and a Genus_{3}and a… and a Genus_{n-1}and a Genus_{n}, which has some non-genus Differentia.

## Structure

A genus of a definition provides a means by which to specify an *is-a relationship*:

- A square is a rectangle, which is a quadrilateral, which is a plane figure, which is a…
- A square is a rhombus, which is a quadrilateral, which is a plane figure, which is a…
- A square is a quadrilateral, which is a plane figure, which is a…
- A square is a plane figure, which is a…
- A square is a…

The non-genus portion of the differentia of a definition provides a means by which to specify a *has-a relationship*:

- A square has an interior angle that is a right angle.
- A square has a straight bounding side.
- A square has a…

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* (or

*more*) than any one of its predecessors, and a node with no successor is

**derived***a most differentiated*(or

*a most derived*) definition.

When a definition, *S*, is the tail of each of its successors (that is, *S* has at least one successor and each direct successor of *S* is a most differentiated definition), then *S* is often called *the species* of each of its successors, and each direct successor of

*S*is often called

*an*(or

**individual***an*) of the species

**entity***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:

*[the] John Smith*: a human that has the name 'John Smith'.

In this case:

- The whole definition is
*an individual*; that is,*[the] John Smith*is an individual. - The genus of
*[the] John Smith*(which is "a human") may be called synonymously*the species*of*[the] John Smith*; that is,*[the] John Smith*is an individual of the species*[a] human*. - The differentia of
*[the] John Smith*(which is "that has the name 'John Smith'") may be called synonymously*the identity*of*[the] John Smith*; that is,*[the] John Smith*is identified among other individuals of the same species by the fact that*[the] John Smith*is the one "that has the name 'John Smith'".

As in that example, the identity itself (or some part of it) is often used to refer to the entire individual, a phenomenon that is known in linguistics as a *pars pro toto synecdoche*.

## References

- ↑
Parry, William Thomas; Hacker, Edward A. (1991).
*Aristotelian Logic*. G - Reference,Information and Interdisciplinary Subjects Series. Albany: State University of New York Press. p. 86. ISBN 9780791406892. https://books.google.com/books?id=rJceFowdGEAC. Retrieved 8 Feb 2019. "Aristotle recognized only one method of real definition, namely, the method of*genus*and*differentia*, applied to defining real things, not words."

Original source: https://en.wikipedia.org/wiki/ Genus–differentia definition.
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