Editing Definition (section)

==Intensional definitions vs extensional definitions==
{{Main|Intension|Extension (semantics)}}

An ''[[intensional definition]]'', also called a ''connotative'' definition, specifies the [[necessary and sufficient conditions]] for a thing to be a member of a specific [[Set (mathematics)|set]].<ref name=Lyons/> Any definition that attempts to set out the essence of something, such as that by [[Genus-differentia definition|genus and differentia]], is an intensional definition.

An ''[[extensional definition]]'', also called a ''denotative'' definition, of a concept or term specifies its ''[[extension (semantics)|extension]]''. It is a list naming every [[object (philosophy)|object]] that is a member of a specific [[Set (mathematics)|set]].<ref name=Lyons/>

Thus, the "[[seven deadly sins]]" can be defined ''intensionally'' as those singled out by [[Pope Gregory I]] as particularly destructive of the life of grace and charity within a person, thus creating the threat of eternal damnation. An ''extensional'' definition, on the other hand, would be the list of wrath, greed, sloth, pride, lust, envy, and gluttony. In contrast, while an intensional definition of "[[prime minister]]" might be "the most senior minister of a cabinet in the executive branch of parliamentary government", an extensional definition is not possible since it is not known who the future prime ministers will be (even though all prime ministers from the past and present can be listed).

===Classes of intensional definitions===
{{Main|Genus–differentia definition}}
A ''genus–differentia definition'' is a type of [[intensional definition]] that takes a large category (the ''genus'') and narrows it down to a smaller category by a distinguishing characteristic (i.e. the ''differentia'').<ref>Bussler, Christoph, and Dieter Fensel, eds. Artificial Intelligence: Methodology, Systems and Applications: 11th International Conference, AIMSA 2004: Proceedings. Springer-Verlag, 2004. p.6</ref>

More formally, a genus–differentia definition consists of:
* ''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 new definition that is not provided by the genus.<ref name=":0" />

For example, consider the following genus–differentia definitions:
* ''a [[triangle]]'': A plane figure that has three straight bounding sides.
* ''a [[quadrilateral]]'': A plane figure that has four straight bounding sides.

Those definitions can be expressed as a genus ("a plane figure") and two ''differentiae'' ("that has three straight bounding sides" and "that has four straight bounding sides", respectively).

It is also possible to have two different genus–differentia definitions that describe the same term, especially when the term describes the overlap of two large categories. For instance, both of these genus–differentia definitions of "square" are equally acceptable:
* ''a square'': a [[rectangle]] that is a [[rhombus]].
* ''a square'': a [[rhombus]] that is a [[rectangle]].

Thus, a "square" is a member of both genera (the plural of ''genus''): the genus "rectangle" and the genus "rhombus".

===Classes of extensional definitions===
One important form of the extensional definition is ''[[ostensive definition]]''. This gives the meaning of a term by pointing, in the case of an individual, to the thing itself, or in the case of a class, to examples of the right kind. For example, one can explain who ''Alice'' (an individual) is, by pointing her out to another; or what a ''rabbit'' (a class) is, by pointing at several and expecting another to understand. The process of ostensive definition itself was critically appraised by [[Ludwig Wittgenstein]].<ref>''Philosophical investigations'', Part 1 §27–34</ref>

An ''[[enumerative definition]]'' of a concept or a term is an ''[[extensional definition]]'' that gives an explicit and exhaustive listing of all the [[object (philosophy)|objects]] that fall under the concept or term in question. Enumerative definitions are only possible for finite sets (and only practical for small sets).

===''Divisio'' and ''partitio''===
''Divisio'' and ''partitio'' are [[Classics|classical]] terms for definitions. A ''partitio'' is simply an intensional definition. A ''divisio'' is not an extensional definition, but an exhaustive list of [[subset]]s of a set, in the sense that every member of the "divided" set is a member of one of the subsets. An extreme form of ''divisio'' lists all sets whose only member is a member of the "divided" set. The difference between this and an extensional definition is that extensional definitions list ''members'', and not ''subsets''.<ref>Katerina Ierodiakonou, "The Stoic Division of Philosophy", in ''Phronesis: A Journal for Ancient Philosophy'', Volume 38, Number 1, 1993, pp. 57–74.</ref>

===Nominal definitions vs real definitions===
{{Main|Essence}}
In classical thought, a definition was taken to be a statement of the essence of a thing. [[Aristotle]] had it that an object's essential attributes form its "essential nature", and that a definition of the object must include these essential attributes.<ref>[[Posterior Analytics]], Bk 1 c. 4</ref>

The idea that a definition should state the essence of a thing led to the distinction between ''nominal'' and ''real'' essence—a distinction originating with Aristotle. In the [[Posterior Analytics]],<ref>[[Posterior Analytics]] Bk 2 c. 7</ref> he says that the meaning of a made-up name can be known (he gives the example "goat stag") without knowing what he calls the "essential nature" of the thing that the name would denote (if there were such a thing). This led medieval logicians to distinguish between what they called the ''quid nominis'', or the "whatness of the name", and the underlying nature common to all the things it names, which they called the ''quid rei'', or the "whatness of the thing".<ref>. Early modern philosophers like Locke used the corresponding English terms "nominal essence" and "real essence".</ref> The name "[[hobbit]]", for example, is perfectly meaningful. It has a ''quid nominis'', but one could not know the real nature of hobbits, and so the ''quid rei'' of hobbits cannot be known. By contrast, the name "man" denotes real things (men) that have a certain ''quid rei''. The meaning of a name is distinct from the nature that a thing must have in order that the name apply to it.

This leads to a corresponding distinction between ''nominal'' and ''real'' definitions. A nominal definition is the definition explaining what a word means (i.e., which says what the "nominal essence" is), and is definition in the classical sense as given above. A real definition, by contrast, is one expressing the real nature or ''quid rei'' of the thing.

This preoccupation with essence dissipated in much of modern philosophy. [[Analytic philosophy]], in particular, is critical of attempts to elucidate the essence of a thing. [[Bertrand Russell|Russell]] described essence as "a hopelessly muddle-headed notion".<ref>''A History of Western Philosophy'', p. 210.</ref>

More recently [[Kripke semantics|Kripke's]] formalisation of [[possible world]] semantics in [[modal logic]] led to a new approach to [[essentialism]]. Insofar as the essential properties of a thing are ''necessary'' to it, they are those things that it possesses in all possible worlds. Kripke refers to names used in this way as [[rigid designator]]s.