Definition
A definition is a semantic statement of the meaning of a term. Definitions can be classified into two large categories: intensional definitions, and extensional definitions. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.
In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what the mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed.
Basic terminology
In modern usage, a definition is something, typically expressed in words, that attaches a meaning to a word or group of words. The word or group of words that is to be defined is called the definiendum, and the word, group of words, or action that defines it is called the definiens. For example, in the definition "An elephant is a large gray animal native to Asia and Africa", the word "elephant" is the definiendum, and everything after the word "is" is the definiens.The definiens is not the meaning of the word defined, but is instead something that conveys the same meaning as that word.
There are many sub-types of definitions, often specific to a given field of knowledge or study. These include, lexical definitions, or the common dictionary definitions of words already in a language; demonstrative definitions, which define something by pointing to an example of it ; and precising definitions, which reduce the vagueness of a word, typically in some special sense.
Intensional definitions vs extensional definitions
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. 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.
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.
Classes of intensional definitions
A genus–differentia definition is a type of intensional definition that takes a large category and narrows it down to a smaller category by a distinguishing characteristic.More formally, a genus–differentia definition consists of:
- a genus : 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.
- a triangle: A plane figure that has three straight bounding sides.
- a quadrilateral: A plane figure that has four straight bounding sides.
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:
Thus, a "square" is a member of both genera : 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 is, by pointing her out to another; or what a rabbit 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 a 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.
''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, 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 subsets.Nominal definitions vs real definitions
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—a distinction originating with Aristotle. In the Posterior Analytics, he says that the meaning of a made-up name can be known without knowing what he calls the "essential nature" of the thing that the name would denote. 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". 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 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, 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 essence 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 that it possesses in all possible worlds. Kripke refers to names used in this way as rigid designators.
Operational vs. theoretical definitions
A definition may also be classified as an operational definition or theoretical definition.Terms with multiple definitions
Homonyms
A homonym is, in the strict sense, one of a group of words that share the same spelling and pronunciation but have different meanings. Thus homonyms are simultaneously homographs and homophones. The state of being a homonym is called homonymy. Examples of homonyms are the pair stalk and stalk and the pair left and left. A distinction is sometimes made between "true" homonyms, which are unrelated in origin, such as skate and skate, and polysemous homonyms, or polysemes, which have a shared origin, such as mouth and mouth.Polysemes
is the capacity for a sign to have multiple meanings, usually related by contiguity of meaning within a semantic field. It is thus usually regarded as distinct from homonymy, in which the multiple meanings of a word may be unconnected or unrelated.In logic, mathematics and computing
In mathematics, definitions are generally not used to describe existing terms, but to describe or characterize a concept. For naming the object of a definition mathematicians can use either a neologism or words or phrases of the common language. The precise meaning of a term given by a mathematical definition is often different from the English definition of the word used, which can lead to confusion, particularly when the meanings are close. For example, a set is not exactly the same thing in mathematics and in common language. In some case, the word used can be misleading; for example, a real number has nothing more real than an imaginary number. Frequently, a definition uses a phrase built with common English words, which has no meaning outside mathematics, such as primitive group or irreducible variety.In first-order logic definitions are usually introduced using extension by definition. On the other hand, lambda-calculi are a kind of logic where the definitions are included as the feature of the formal system itself.