A genus–differentia definition

Definition

A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...

is a type of intensional definition

Intensional definition

In logic and mathematics, an intensional definition gives the meaning of a term by specifying all the properties required to come to that definition, that is, the necessary and sufficient conditions for belonging to the set being defined....

, and it is composed of two parts:

1. a genus
   Genus
   In biology, a genus is a low-level taxonomic rank used in the biological classification of living and fossil organisms, which is an example of definition by genus and differentia...
   
   (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.
2. the differentia
   Differentia
   In Scholastic logic, differentia is one of the predicables. It is that part of a definition which is predicable in a given genus only of the definiendum....
   
   : The portion of the new definition that is not provided by the genera.

For example, consider these two definitions:

• a triangle
  Triangle
  A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
  
  : A plane figure that has 3 straight bounding sides.
• a quadrilateral
  Quadrilateral
  In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
  
  : A plane figure that has 4 straight bounding sides.

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

1. one genus: A plane figure.
2. two differentiae:
   • the differentia for a triangle: that has 3 straight bounding sides.
   • the differentia for a quadrilateral: that has 4 straight bounding sides.

Differentiation and Abstraction

This 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

Abstraction

Abstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....

; 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
  Square (geometry)
  In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
  
  : 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
  Square (geometry)
  In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
  
  : (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
  Rectangle
  In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square 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
  Square (geometry)
  In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
  
  : 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
  Square (geometry)
  In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
  
  : (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
  Rhombus
  In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...
  
  : 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
  Square (geometry)
  In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
  
  : 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 definition expressed as having  genera can be recast as at least  equivalent definitions, each of which has just one genus. Thus, the following:

• a Definition: a Genus₁ and a Genus₂ and a Genus₃ and a ... and a Genus_n-1 and a Genus_n that has some non-genus Differentia.

could be recast as:

• a Definition: a Genus₁ that is a Genus₂ and that is a Genus₃ and that is a ... and that is a Genus and that is a Genus that has some non-genus Differentia.
• a Definition: a Genus₂ that is a Genus₁ and that is a Genus₃ and that is a ... and that is a Genus and that is a Genus that has some non-genus Differentia.
• a Definition: a Genus₃ that is a Genus₁ and that is a Genus₂ and that is a ... and that is a Genus and that is a Genus that has some non-genus Differentia.
• ...
• a Definition: a Genus that is a Genus₁ and that is a Genus₂ and that is a Genus₃ and that is a ... and that is a Genus that has some non-genus Differentia.
• a Definition: a Genus that is a Genus₁ and that is a Genus₂ and that is a Genus₃ and that is a ... and that is a Genus that has some non-genus Differentia.

Structure

In other words, a genus of a definition provides a means by which to specify an is-a relationship

Is-a

In knowledge representation, object-oriented programming and design, is-a or is_a or is a is a relationship where one class D is a subclass of another class B ....

:

• 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

Has-a

In database design and object oriented program architecture, has-a is a relationship where one object "belongs" to another object , and behaves according to the rules of ownership. In simple words, has-a relationship in an object is called a member field of an object...

:

• A square has interior angles which are all right angles.
• A square has bounding sides which all have the same length.
• A square has 4 straight bounding sides.
• ...

When a system of definitions is constructed with genera and differentiae, the definitions can be thought of as nodes forming a hierarchy

Hierarchy

A hierarchy is an arrangement of items in which the items are represented as being "above," "below," or "at the same level as" one another...

 or—more generally—a directed acyclic graph

Directed acyclic graph

In mathematics and computer science, a directed acyclic graph , is a directed graph with no directed cycles. That is, it is formed by a collection of vertices and directed edges, each edge connecting one vertex to another, such that there is no way to start at some vertex v and follow a sequence of...

; a node that has no predecessor

Directed graph

A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,...

 is a most general definition; each node along a directed path is more differentiated (or more derived) than any one of its predecessors, and a node with no successor

Directed graph

A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,...

 is a most differentiated (or a most derived) definition.

When a definition, S, is the tail

Directed graph

A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,...

 of each of its successors (that is, S has at least one successor and each direct successor

Directed graph

A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,...

 of S is a most differentiated definition), then S is often called the species

Species

In biology, a species is one of the basic units of biological classification and a taxonomic rank. A species is often defined as a group of organisms capable of interbreeding and producing fertile offspring. While in many cases this definition is adequate, more precise or differing measures are...

of each of its successors, and each direct successor of S is often called an individual

Individual

An individual is a person or any specific object or thing in a collection. Individuality is the state or quality of being an individual; a person separate from other persons and possessing his or her own needs, goals, and desires. Being self expressive...

(or an entity

Entity

An entity is something that has a distinct, separate existence, although it need not be a material existence. In particular, abstractions and legal fictions are usually regarded as entities. In general, there is also no presumption that an entity is animate.An entity could be viewed as a set...

) 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

Identity (philosophy)

In philosophy, identity, from , is the relation each thing bears just to itself. According to Leibniz's law two things sharing every attribute are not only similar, but are the same thing. The concept of sameness has given rise to the general concept of identity, as in personal identity and...

of that individual. For instance, consider the following definition:

• [the] Mfwitten: a Wikipedia user that has the account name 'Mfwitten'.

In this case:

• The whole definition (to which one may refer with the term Mfwitten or the Mfwitten) is an individual; that is, Mfwitten is an individual.
• The genus of Mfwitten (which is "a Wikipedia user") may be called synonymously the species of Mfwitten; that is, Mfwitten is an individual of the species [a] Wikipedia user.
• The differentia of Mfwitten (which is "that has the account name 'Mfwitten'") may be called synonymously the identity of Mfwitten; that is, Mfwitten is identified among other individuals of its species by the fact that Mfwitten is the one "that has the account name 'Mfwitten'".

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

Linguistics

Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....

 as a pars pro toto

Pars pro toto

Pars pro toto is Latin for "a part for the whole" where the name of a portion of an object or concept represents the entire object or context....

 synechdoche.

Examples

This can be clarified with a hackneyed example. Suppose we wanted to define the phrase human being. Following the ancient Greeks (Socrates and his successors) and modern biologists, we say that human being is a species and that each individual person is a member of the species human being. So we ask what the genus, or general category, of the species is; the Greeks (but not the biologists) would say that the genus is animal. What is the differentia of the species, that is, the distinguishing characteristic of human being that other animals do not have? The Greeks said it is rationality; thus, Aristotle said, A human being is a rational animal.

However, the use of the genus–differentia definition is by no means restricted to science. Rather, it is the natural thing to do if you are to explain the meaning of a particular word to someone. With this, the "classical" type of definition (definitio fit per genus proximum et differentiam specificam), you use the copula (is, are) after the definiendum (just as if you were using an equals sign in a mathematical equation) and then go on to explain the definiendum by using the appropriate generic term plus those characteristics specific to the thing you are describing, thereby narrowing down the meaning until the definiendum can no longer be confused with anything else; here are some examples from everyday life:

• A paperweight is a small, heavy object which is placed on papers to prevent them from being scattered.

  paperweight—definiendum

  object—generic term

  small but heavy, placed on papers, reason why—differentiae specificae

• Homesickness
  Homesickness
  Homesickness is the distress or impairment caused by an actual or anticipated separation from the specific home environment or attachment objects....
  
  is the feeling of unhappiness you may experience when you are away from home and miss your home and your family very much.

• Subtitles are the printed translation that you can read at the bottom of the screen when you are watching a foreign film.

• In film and broadcasting, a soundbite
  Soundbite
  In film and broadcasting, a sound bite is a very short piece of a speech taken from a longer speech or an interview in which someone with authority or the average "man on the street" says something which is considered by those who edit the speech or interview to be the most important point...
  
  is a very short piece of footage taken from a longer speech or interview in which someone with authority says something which is considered by those who edit the speech or interview to be a most important point.

• A mosque
  Mosque
  A mosque is a place of worship for followers of Islam. The word is likely to have entered the English language through French , from Portuguese , from Spanish , and from Berber , ultimately originating in — . The Arabic word masjid literally means a place of prostration...
  
  is a building, often with high towers and domes, where Muslims go to worship.