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A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols).[a] The term to be defined is the definiendum. The term may have many different senses and multiple meanings. For each meaning, a definiens is a cluster of words that defines that term (and clarifies the speaker's intention).
A definition will vary in aspects like precision or popularity. There are also different types of definitions with different purposes and focuses (e.g. intensional, extensional, descriptive, stipulative, and so on).
A chief difficulty in the management of definitions is the necessity of using other terms that are already understood or whose definitions are easily obtainable or demonstrable (e.g. a need, sometimes, for ostensive definitions).
In formal languages like mathematics, a "stipulative" definition guides a specific discussion. A stipulative definition can only be disproved by showing a logical contradiction. A stipulative definition might be considered a temporary, working definition. On the other hand, a "descriptive" definition can be shown to be "right" or "wrong" with reference to general usage.
A precising definition extends the descriptive dictionary definition (lexical definition) of a term for a specific purpose by including additional criteria, which narrow the set of things that meet the definition.
C.L. Stevenson has identified persuasive definition as a form of stipulative definition which purports to state the "true" or "commonly accepted" meaning of a term, while in reality stipulating an altered use (perhaps as an argument for some specific belief). Stevenson has also noted that some definitions are "legal" or "coercive" – their object is to create or alter rights, duties, or crimes.
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.
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 would be a list of the seven. In contrast, while 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 is not possible since it is not known who future prime ministers will be.
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 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.
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 are classical terms for definitions. A partitio is simply an intensional definition. A divisio is not an extensional definition, but 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.
A genus–differentia definition is a type of intensional definition, and it is composed by two parts:
For example, consider these two definitions:
Those definitions can be expressed as a genus and two differentiae:
When multiple definitions could serve equally well, then all such definitions apply simultaneously. For instance, given the following:
both of these definitions of "square" are equally acceptable:
Thus, a "square" is a member of both the genus "rectangle" and the genus "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):
or completely equivalently:
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 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 "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 one could not know the real nature of hobbits, even if there were such things, and so the real nature or 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 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. 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.
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:
So "0" will have exactly one successor, which for convenience can be called "1". In turn, "1" will have exactly one successor, which could be called "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.
In the same way, we can define ancestor as follows:
Or simply: an ancestor is a parent or a parent of an ancestor.
A working definition is chosen for an occasion and may not fully conform with established or authoritative definitions. Not knowing of established definitions would be grounds for selecting or devising a working definition. Or it refers to a definition being developed; a tentative definition that can be tailored to create an authoritative 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 rely upon primitive notions. 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 a higher genus cannot be assigned under which they may fall. Thus being, unity and similar concepts cannot be defined. 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 to avoid misunderstanding.
Locke and Mill also argued that individuals cannot be defined. Names are learned 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.
In medical dictionaries, definitions should to the greatest extent possible be:
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