Definition

A definition is a passage that explains the meaning of a term (a word, phrase or other set of symbols), or a type of thing. The term to be defined is the definiendum (plural definienda). A term may have many different senses or meanings. For each such specific sense, a definiens (plural definientia) is a cluster of words that defines it.

A chief difficulty in managing definition is the need to use other terms that are already understood or whose definitions are easily obtainable. The use of the term in a simple example may suffice. By contrast, a dictionary definition has additional details, typically including an etymology showing snapshots of the earlier meanings and the parent language.

Like other words, the term definition has subtly different meanings in different contexts. A definition may be descriptive of the general use meaning, or stipulative of the speaker's immediate intentional meaning. For example, in formal languages like mathematics, a 'stipulative' definition guides a specific discussion. A descriptive definition can be shown to be "right" or "wrong" by comparison to general usage, but a stipulative definition can only be disproved by showing a logical contradiction [3].

A precising definition extends the descriptive dictionary definition (lexical definition) of a term for a specific purpose by including additional criteria that narrow down the set of things meeting the definition.

C.L. Stevenson has identified persuasive definition as a form of stipulative definition which purports to describe the "true" or "commonly accepted" meaning of a term, while in reality stipulating an altered use, perhaps as an argument for some specific view.

Stevenson has also noted that some definitions are "legal" or "coercive", whose object is to create or alter rights, duties or crimes.[1]

Contents

Intension and extension

An intensional definition, also called a coactive definition, specifies the necessary and sufficient conditions for a thing being a member of a specific set. Any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition.

An extensional definition, also called a denotative definition, of a concept or term specifies its extension. It is a list naming every object that is a member of a specific set.

So, for example, an intensional definition of 'Prime Minister' might be the most senior minister of a cabinet in the executive branch of government in a parliamentary system. An extensional definition would be a list of all past, present and future prime ministers.

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. So you can explain who Alice (an individual) is by pointing her out to me; or what a rabbit (a class) is by pointing at several and expecting me to 'catch on'. The process of ostensive definition itself was critically appraised by Ludwig Wittgenstein.[2]

An enumerative definition of a concept or term is an extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets and only practical for relatively small sets.

Divisio and partitio

Divisio and partitio are classical terms for definitions. A partitio is simply an intensional definition. A divisio is not an extensional definition. Divisio is an exhaustive list of subsets 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 sets.[3]

Definition by genus and differentia

The various levels of the scientific classification system. Life Kingdom Phylum Order Family Genus Species
The hierarchy of biological classification's eight major taxonomic ranks, which is an example of definition by genus and differentia. Intermediate minor rankings are not shown.

A new definition can be composed by two parts:

  1. 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, and a definition can be composed of multiple genera (more than one genus).
  2. the differentia: The portion of the new definition that is not provided by the genera.

For example, consider these two definitions:

Those definitions can be expressed as a genus and 2 differentiae:

  1. a genus: A plane figure.
  2. 2 differentiae:
    • the differentia for a triangle: bounded by 3 straight sides.
    • the differentia for a quadrilateral: bounded by 4 straight sides.

Continuing the process of differentiation:

Importantly, differentiae can include genera. For instance, consider the following:

This definition could be recast as follows:

In other words, a genus of a definition provides a means by which to specify an is-a relationship, and the non-genus portions 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 predecessors is a most general definition; each node along a directed path is more differentiated (or more derived) than its predecessors, and a node with no successors is a most differentiated (or a most derived) definition. When a definition, S, is the tail of all of its successors (that is, S has at least one successor and all of the direct successors of S are most differentiated definitions), then S is often called a species and each of its direct successors is often called an individual or an entity; the differentia of an individual is called an identity. For instance:

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 synechdoche.

Rules for definition by genus and differentia

Certain rules have traditionally been given for this particular type of definition.[4][5][6]

  1. A definition must set out the essential attributes of the thing defined.
  2. Definitions should avoid circularity. To define a horse as 'a member of the species equus' would convey no information whatsoever. For this reason, Locking adds that a definition of a term must not comprise of terms which are synonymous with it. This would be a circular definition, a circulus in definiendo. Note, however, that it is acceptable to define two relative terms in respect of each other. Clearly, we cannot define 'antecedent' without using the term 'consequent', nor conversely.
  3. The definition must not be too wide or too narrow. It must be applicable to everything to which the defined term applies (i.e. not miss anything out), and to nothing else (i.e. not include any things to which the defined term would not truly apply).
  4. The definition must not be obscure. The purpose of a definition is to explain the meaning of a term which may be obscure or difficult, by the use of terms that are commonly understood and whose meaning is clear. The violation of this rule is known by the Latin term obscurum per obscurius. However, sometimes scientific and philosophical terms are difficult to define without obscurity. (See the definition of Free will in Wikipedia, for instance).
  5. A definition should not be negative where it can be positive. We should not define 'wisdom' as the absence of folly, or a healthy thing as whatever is not sick. Sometimes this is unavoidable, however. We cannot define a point except as 'something with no parts', nor blindness except as 'the absence of sight in a creature that is normally sighted'.

Essence

Main article: 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.[7]

The idea that a definition should state the essence of a thing led to the distinction between nominal and real essence, originating with Aristotle. In a passage from the Posterior Analytics,[8] he says that we can know the meaning of a made-up name (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 'whatness of the name', and the underlying nature common to all the things it names, which they called the quid rei or 'whatness of the thing'. (Early modern philosophers like Locke used the corresponding English terms 'nominal essence' and 'real essence'). The name 'hobbit', for example, is perfectly meaningful. It has a quid nominis. But we could not know the real nature of hobbits, even if there were such things, and so we cannot know the real nature or quid rei of hobbits. 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 thing must have in order that the name apply to it.

This leads to a corresponding distinction between nominal and real definition. 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. Russell described it as "a hopelessly muddle-headed notion".[9]

More recently 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 it possesses in all possible worlds. Kripke refers to names used in this way as rigid designators.

Recursive definitions

A recursive definition, sometimes also called an inductive definition, is one that defines a word in terms of itself, so to speak, albeit in a useful way. Normally this consists of three steps:

  1. At least one thing is stated to be a member of the set being defined; this is sometimes called a "base set".
  2. All things bearing a certain relation to other members of the set are also to count as members of the set. It is this step that makes the definition recursive.
  3. All other things are excluded from the set

For instance, we could define natural number as follows (after Peano):

  1. "0" is a natural number.
  2. Each natural number has a distinct successor, such that:
    • the successor of a natural number is also a natural number, and
    • no natural number is succeeded by "0".
  3. Nothing else is a natural number.

So "0" will have exactly one successor, which for convenience we can call "1". In turn, "1" will have exactly one successor, which we would call "2", and so on. Notice that the second condition in the definition itself refers to natural numbers, and hence involves self-reference. Although this sort of definition involves a form of circularity, it is not vicious, and the definition has been quite successful.

Limitations of definition

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or leave some terms undefined. If every term of every definiens must itself be defined, "where at last should we stop?"[10][11] A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity.[12][13][14]

Many philosophers have chosen instead to leave some terms undefined. The scholastic philosophers claimed that the highest genera (the so-called ten generalissima) cannot be defined, since we cannot assign any higher genus under which they may fall. Thus we cannot define being, unity and similar concepts.[5] Locke supposes in An Essay Concerning Human Understanding[15] that the names of simple concepts do not admit of any definition. More recently Bertrand Russell sought to develop a formal language based on logical atoms. Other philosophers, notably Wittgenstein, rejected the need for any undefined simples. Wittgenstein pointed out in his Philosophical Investigations that what counts as a "simple" in one circumstance might not do so in another.[16] He rejected the very idea that every explanation of the meaning of a term needed itself to be explained: "As though an explanation hung in the air unless supported by another one",[17] claiming instead that explanation of a term is only needed when we need to avoid misunderstanding.

Locke and Mill also argued that we cannot define individuals. We learn names by connecting an idea with a sound, so that speaker and hearer have the same idea when the same word is used.[18] This is not possible when no one else is acquainted with the particular thing that has "fallen under our notice".[19] Russell offered his theory of descriptions in part as a way of defining a proper name, the definition being given by a definite description that "picks out" exactly one individual. Saul Kripke pointed to difficulties with this approach, especially in relation to modality, in his book Naming and Necessity.

There is a presumption in the classic example of a definition that the definiens can be stated. Wittgenstein argued that for some terms this is not the case.[20] The examples he used include game, number and family. In such cases, he argued, there is no fixed boundary that can be used to provide a definition. Rather, the items are grouped together because of a family resemblance. For terms such as these it is not possible and indeed not necessary to state a definition; rather, one simply comes to understand the use of the term.

See also

Notes

  1. ↑ Stevenson, C.L., Ethics and Language, Connecticut 1944
  2. ↑ Philosophical investigations, Part 1 §27-34
  3. ↑ Katerina Ierodiakonou, "The Stoic Division of Philosophy", in Phronesis: A Journal for Ancient Philosophy, Volume 38, Number 1, 1993 , pp. 57-74.
  4. ↑ Copi 1982 pp 165-169
  5. ↑ 5.0 5.1 Joyce, Ch. X
  6. ↑ Joseph, Ch. V
  7. ↑ Posterior Analytics, Bk 1 c. 4
  8. ↑ Posterior Analytics Bk 2 c. 7
  9. ↑ A history of Western Philosophy, p. 210
  10. ↑ Locke, Essay, Bk. III, Ch. iv, 5
  11. ↑ This problem parallels the diallelus, but leads to scepticism about meaning rather than knowledge.
  12. ↑ Generally lexicographers seek to avoid circularity wherever possible, but the definitions of words such as "the" and "a" use those words and are therefore circular. [1] [2] Lexicographer Sidney I. Landau's essay "Sexual Intercourse in American College Dictionaries" provides other examples of circularity in dictionary definitions. (McKean, p. 73-77)
  13. ↑ An exercise suggested by J. L. Austin involved taking up a dictionary and finding a selection of terms relating to the key concept, then looking up each of the words in the explanation of their meaning. Then, iterating this process until the list of words begins to repeat, closing in a “family circle” of words relating to the key concept.
    (A plea for excuses in Philosophical Papers. Ed. J. O. Urmson and G. J. Warnock. Oxford: Oxford UP, 1961. 1979.)
  14. ↑ In the game of Vish, players compete to find circularity in a dictionary.
  15. ↑ Locke, Essay, Bk. III, Ch. iv
  16. ↑ See especially Philosophical Investigations Part 1 §48
  17. ↑ He continues: "Whereas an explanation may indeed rest on another one that has been given, but none stands in need of another - unless we require it to prevent a misunderstanding. One might say: an explanation serves to remove or to avert a misunderstanding - one, that is, that would occur but for the explanation; not every one I can imagine." Philosophical Investigations, Part 1 §87, italics in original
  18. ↑ This theory of meaning is one of the targets of the private language argument
  19. ↑ Locke, Essay, Bk. III, Ch. iii, 3
  20. ↑ Philosophical Investigations

References

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