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Definition
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A definition is a statement of the meaning of a word or phrase. The term to be defined is known as the definiendum (Latin: what is to be defined). The words which define it are known as the definiens (Latin: what defines).
Stipulative definitions
A definition may either describe the meaning that a term bears in general use, providing a descriptive definition, or that which the speaker intends to impose upon it, a stipulative definition.
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A definition is a statement of the meaning of a word or phrase. The term to be defined is known as the definiendum (Latin: what is to be defined). The words which define it are known as the definiens (Latin: what defines).
Stipulative definitions
A definition may either describe the meaning that a term bears in general use, providing a descriptive definition, or that which the speaker intends to impose upon it, a stipulative definition. A definition may be used to introduce a new term, or to prescribe a new meaning to a term which is already in use.
A descriptive definition can be shown to be "right" or "wrong" by comparison to usage, but a stipulative definition cannot, although it may lead to contradictions.
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.
Intension and extension
An intensional definition, also called a connotative 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.
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. Actually, 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.
Definition by genus and differentia
Traditionally, a definition consists of the genus (the family) of thing to which the defined thing belongs, and the differentia (the distinguishing feature which marks it off from other members of the same family). Thus 'triangle' is defined as 'a plane figure (genus) bounded by three straight sides (differentia).
Rules for definition by genus and differentia
Certain rules have traditionally been given for this particular type of definition.
- A definition must set out the essential attributes of the thing defined.
- Definitions should avoid circularity. To define a horse as 'a member of the species
equus would convey no information whatsoever. For this reason, Locke adds that a definition of a term must not consist 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.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).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).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
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.
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, 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 the so-called 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".
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:
- At least one thing is stated to be a member of the set being defined; this is sometimes called a "base set".
- 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.
- All other things are excluded from the set
For instance, we could define natural number as follows (after Peano):
- "0" is a natural number.
- 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".
- 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 is 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?" A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity.
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. Locke supposes in An Essay Concerning Human Understanding 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. 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", 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. This is not possible when no one else is acquainted with the particular thing that has "fallen under our notice". 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. 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.
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