A definition states the meaning of a word using other words. This is sometimes challenging. Common dictionaries contain lexical descriptive definitions, but there are various types of definition – all with different purposes and focuses.
Adefinition is asemantic statement of the meaning of a term (aword,phrase, or other set ofsymbols).[1][2] Definitions can be classified into two large categories:intensional definitions (which try to give the sense of a term), andextensional definitions (which try to list the objects that a term describes).[3] Another important category of definitions is the class ofostensive 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.[4][a]
Inmathematics, 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 andaxioms form the basis on which all of modern mathematics is to be constructed.[5]
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 thedefiniendum, and the word, group of words, or action that defines it is called thedefiniens.[6] For example, in the definition"An elephant is a large gray animal native to Asia and Africa", the word "elephant" is thedefiniendum, and everything after the word "is" is thedefiniens.[7]
Thedefiniens is notthe meaning of the word defined, but is instead something thatconveys the same meaning as that word.[7]
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 ("This," [said while pointing to a large grey animal], "is an Asian elephant."); andprecising definitions, which reduce the vagueness of a word, typically in some special sense ("'Large', among female Asian elephants, is any individual weighing over 5,500 pounds.").[7]
Intensional definitions vs extensional definitions
Anextensional definition, also called adenotative definition, of a concept or term specifies itsextension. It is a list naming everyobject that is a member of a specificset.[3]
Thus, the "seven deadly sins" can be definedintensionally as those singled out byPope Gregory I as particularly destructive of the life of grace and charity within a person, thus creating the threat of eternal damnation. Anextensional 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).
Agenus–differentia definition is a type ofintensional definition that takes a large category (thegenus) and narrows it down to a smaller category by a distinguishing characteristic (i.e. thedifferentia).[8]
More formally, a genus–differentia definition consists of:
agenus (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.[6]
For example, consider the following genus–differentia definitions:
atriangle: A plane figure that has three straight bounding sides.
aquadrilateral: A plane figure that has four straight bounding sides.
Those definitions can be expressed as a genus ("a plane figure") and twodifferentiae ("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:
One important form of the extensional definition isostensive 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 whoAlice (an individual) is, by pointing her out to another; or what arabbit (a class) is, by pointing at several and expecting another to understand. The process of ostensive definition itself was critically appraised byLudwig Wittgenstein.[9]
Anenumerative definition of a concept or a term is anextensional definition that gives an explicit and exhaustive listing of all theobjects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets (and only practical for small sets).
Divisio andpartitio areclassical terms for definitions. Apartitio is simply an intensional definition. Adivisio is not an extensional definition, but an exhaustive list ofsubsets of a set, in the sense that every member of the "divided" set is a member of one of the subsets. An extreme form ofdivisio 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 listmembers, and notsubsets.[10]
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.[11]
The idea that a definition should state the essence of a thing led to the distinction betweennominal andreal essence—a distinction originating with Aristotle. In thePosterior Analytics,[12] 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 thequid nominis, or the "whatness of the name", and the underlying nature common to all the things it names, which they called thequid rei, or the "whatness of the thing".[13] The name "hobbit", for example, is perfectly meaningful. It has aquid nominis, but one could not know the real nature of hobbits, and so thequid rei of hobbits cannot be known. By contrast, the name "man" denotes real things (men) that have a certainquid 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 betweennominal andreal 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 orquid 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".[14]
More recentlyKripke's formalisation ofpossible world semantics inmodal logic led to a new approach toessentialism. Insofar as the essential properties of a thing arenecessary to it, they are those things that it possesses in all possible worlds. Kripke refers to names used in this way asrigid designators.
Ahomonym is, in the strict sense, one of a group of words that share the same spelling and pronunciation but have different meanings.[15] Thus homonyms are simultaneouslyhomographs (words that share the same spelling, regardless of their pronunciation)andhomophones (words that share the same pronunciation, regardless of their spelling). The state of being a homonym is calledhomonymy. Examples of homonyms are the pairstalk (part of a plant) andstalk (follow/harass a person) and the pairleft (past tense of leave) andleft (opposite of right). A distinction is sometimes made between "true" homonyms, which are unrelated in origin, such asskate (glide on ice) andskate (the fish), and polysemous homonyms, orpolysemes, which have a shared origin, such asmouth (of a river) andmouth (of an animal).[16][17]
Polysemy is the capacity for asign (such as aword,phrase, orsymbol) to have multiple meanings (that is, multiplesemes orsememes and thus multiplesenses), usually related by contiguity ofmeaning within asemantic field. It is thus usually regarded as distinct fromhomonymy, in which the multiple meanings of a word may be unconnected or unrelated.
In mathematics, definitions are generally not used to describe existing terms, but to describe or characterize a concept.[18] For naming the object of a definition mathematicians can use either aneologism (this was mainly the case in the past) or words or phrases of the common language (this is generally the case in modern mathematics). The precise meaning of a term given by a mathematical definition is often different from the English definition of the word used,[19] which can lead to confusion, particularly when the meanings are close. For example, aset is not exactly the same thing in mathematics and in common language. In some case, the word used can be misleading; for example, areal number has nothing more (or less) real than animaginary number. Frequently, a definition uses a phrase built with common English words, which has no meaning outside mathematics, such asprimitive group orirreducible variety.
In first-order logic definitions are usually introduced usingextension by definition (so using a metalogic). On the other hand,lambda-calculi are a kind of logic where the definitions are included as the feature of the formal system itself.
Authors have used different terms to classify definitions used in formal languages like mathematics.Norman Swartz classifies a definition as "stipulative" if it is intended to guide a specific discussion. A stipulative definition might be considered a temporary, working definition, and can only be disproved by showing a logical contradiction.[20] In contrast, a "descriptive" definition can be shown to be "right" or "wrong" with reference to general usage.
Swartz defines aprecising definition as one that extends the descriptive dictionary definition (lexical definition) for a specific purpose by including additional criteria. A precising definition narrows the set of things that meet the definition.
C.L. Stevenson has identifiedpersuasive 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.[21]
Arecursive definition, sometimes also called aninductive 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 definitionrecursive.
Each natural number has a unique successor, such that:
the successor of a natural number is also a natural number;
distinct natural numbers have distinct successors;
no natural number is succeeded by "0".
Nothing else is a natural number.
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. The second condition in the definition itself refers to natural numbers, and hence involvesself-reference. Although this sort of definition involves a form ofcircularity, it is notvicious, and the definition has been quite successful.
In the same way, we can defineancestor as follows:
A parent is an ancestor.
A parent of an ancestor is an ancestor.
Nothing else is an ancestor.
Or simply: an ancestor is a parent or a parent of an ancestor.
a reflection of current scientific knowledge.[22][23]
Certain rules have traditionally been given for definitions (in particular, genus-differentia definitions).[24][25][26][27]
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 speciesequus" 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, acirculus 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 termobscurum per obscurius. However, sometimes scientific and philosophical terms are difficult to define without obscurity.
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. For example, it appears difficult to define blindness in positive terms rather than as "the absence of sight in a creature that is normally sighted".
Given that anatural language such asEnglish contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely uponprimitive notions. If every term of everydefiniens must itself be defined, "where at last should we stop?"[28][29] A dictionary, for instance, insofar as it is a comprehensive list oflexical definitions, must resort tocircularity.[30][31][32]
Many philosophers have chosen instead to leave some terms undefined. Thescholastic philosophers claimed that the highest genera (called the tengeneralissima) cannot be defined, since a higher genus cannot be assigned under which they may fall. Thusbeing, unity and similar concepts cannot be defined.[25]Locke supposes inAn Essay Concerning Human Understanding[33] that the names of simple concepts do not admit of any definition. More recentlyBertrand Russell sought to develop a formal language based onlogical atoms. Other philosophers, notablyWittgenstein, rejected the need for any undefined simples. Wittgenstein pointed out in hisPhilosophical Investigations that what counts as a "simple" in one circumstance might not do so in another.[34] 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",[35] claiming instead that explanation of a term is only needed to avoid misunderstanding.
Locke andMill also argued thatindividuals 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.[36] This is not possible when no one else is acquainted with the particular thing that has "fallen under our notice".[37] Russell offered histheory of descriptions in part as a way of defining a proper name, the definition being given by adefinite description that "picks out" exactly one individual.Saul Kripke pointed to difficulties with this approach, especially in relation tomodality, in his bookNaming and Necessity.
There is a presumption in the classic example of a definition that thedefiniens can be stated. Wittgenstein argued that for some terms this is not the case.[38] The examples he used includegame,number andfamily. 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 afamily resemblance. For terms such as these it is not possible and indeed not necessary to state a definition; rather, one simply comes to understand theuse of the term.[b]
^Terms with the same pronunciation and spelling but unrelated meanings are calledhomonyms, while terms with the same spelling and pronunciation and related meanings are calledpolysemes.
^Stevenson, C.L.,Ethics and Language, Connecticut 1944
^abcdeMcPherson, M.; Arango, P.; Fox, H.; Lauver, C.; McManus, M.; Newacheck, P. W.; Perrin, J. M.; Shonkoff, J. P.; Strickland, B. (1998). "A new definition of children with special health care needs".Pediatrics.102 (1 Pt 1):137–140.doi:10.1542/peds.102.1.137.PMID9714637.S2CID30160426.
^This problem parallels thediallelus, but leads to scepticism about meaning rather than knowledge.
^Generallylexicographers 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] LexicographerSidney I. Landau's essay "Sexual Intercourse in American College Dictionaries" provides other examples of circularity in dictionary definitions. (McKean, p. 73–77)
^An exercise suggested byJ. 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 andG. J. Warnock. Oxford: Oxford UP, 1961. 1979.)
^In the game ofVish, players compete to find circularity in a dictionary.
^He continues: "Whereas an explanation may indeed rest on another one that has been given, but none stands in need of another – unlesswe 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
