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A definition is a statement of the meaning of a term (a
word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), meaning. In many languages, words also co ...

word
,
phrase In syntax In linguistics, syntax () is the set of rules, principles, and processes that govern the structure of Sentence (linguistics), sentences (sentence structure) in a given Natural language, language, usually including word order. The ter ...

phrase
, or other set of
symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. All (and ) is achieved th ...

symbol
s). Definitions can be classified into two large categories,
intensional definition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label ...
s (which try to give the sense of a term) and
extensional definition In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
s (which try to list the objects that a term describes).Lyons, John. "Semantics, vol. I." Cambridge: Cambridge (1977). p.158 and on. Another important category of definitions is the class of
ostensive definitionAn ostensive definition conveys the meaning of a term by pointing out examples. This type of definition is often used where the term is difficult to define verbally, either because the words will not be understood (as with children and new speakers ...
s, 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. Definitions and
axioms An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
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, among many others, 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," aid while pointing to a large grey animal "is an Asian elephant."''); and precising 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."'').


Intensional definitions vs extensional definitions

An ''
intensional definition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label ...
'', also called a ''connotative'' definition, specifies the
necessary and sufficient conditions In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
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 In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
'', also called a ''denotative'' definition, of a concept or term specifies its '' extension''. It is a list naming every
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at any particular time or pl ...
that is a member of a specific set. Thus, the "
seven deadly sins The seven deadly sins, also known as the capital vices or cardinal sins, is a grouping and classification of vices within Christian teachings, although they are not mentioned in the Bible. Behaviours or habits are classified under this cate ...
" can be defined ''intensionally'' as those singled out by
Pope Gregory I Pope Gregory I ( la, Gregorius I; – 12 March 604), commonly known as Saint Gregory the Great, was the bishop of Rome A bishop is an ordained, consecrated, or appointed member of the Clergy#Christianity, Christian clergy who is generally ent ...

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 A prime minister or a premier is the head of the cabinet Cabinet or The Cabinet may refer to: Furniture * Cabinetry, a box-shaped piece of furniture with doors and/or drawers * Display cabinet, a piece of furniture with one or more transpar ...
" 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).


Classes of intensional definitions

A genus–differentia definition is a type of
intensional definition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label ...
that takes a large category (the genus) and narrows it down to a smaller category by a distinguishing characteristic (i.e. the differentia). More formally, a genus–differentia definition consists of: # a
genus Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank In biological classification In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure ...
(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. For example, consider the following genus–differentia definitions: * ''a
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The b ...

triangle
'': A plane figure that has three straight bounding sides. * ''a
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and t ...

quadrilateral
'': A plane figure that has four straight bounding sides. Those definitions can be expressed as a genus ("a plane figure") and two differentiae ("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: * ''a square'': a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
that is a
rhombus In plane Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's m ...

rhombus
. * ''a square'': a
rhombus In plane Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's m ...

rhombus
that is a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
. Thus, a "square" is a member of both genera (the plural of ''genus''): the genus "rectangle" and the genus "rhombus".


Classes of extensional definitions

One important form of the extensional definition is ''
ostensive definitionAn ostensive definition conveys the meaning of a term by pointing out examples. This type of definition is often used where the term is difficult to define verbally, either because the words will not be understood (as with children and new speakers ...
''. 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'' (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 Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationalit ...

Ludwig Wittgenstein
. An ''
enumerative definition An enumerative definition of a concept or term is a special type of 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 ...
'' of a concept or a term is an ''
extensional definition In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
'' 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 in fact only practical for relatively small sets).


''Divisio'' and ''partitio''

''Divisio'' and ''partitio'' are
classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea *Classical architecture, architecture derived from Greek and ...

classical
terms for definitions. A ''partitio'' is simply an intensional definition. A ''divisio'' is not an extensional definition, but an exhaustive list of
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

subset
s 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 Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

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 The ''Posterior Analytics'' ( grc-gre, Ἀναλυτικὰ Ὕστερα; la, Analytica Posteriora) is a text from Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher an ...
, 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 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 Hobbits are a fictional race of people in the novels of J. R. R. Tolkien. About half average human height, Tolkien presented hobbits as a variety of humanity, or close relatives thereof. Occasionally known as halflings in Tolkien's writings, ...
", 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 (men) 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 (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 Analytic philosophy is a branch and tradition of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy o ...
, 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 A possible world is a complete and consistent way the world is or could have been. They are widely used as a formal device in logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously ...
semantics in
modal logic Modal logic is a collection of formal systems originally developed and still widely used to represent statements about Linguistic modality, necessity and possibility. The basic Unary operation, unary (1-place) modal operators are most often interpr ...
led to a new approach to
essentialism Essentialism is the view that objects have a set of attributes that are necessary to their identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group e ...
. 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 designatorIn modal logic and the philosophy of language In analytic philosophy, philosophy of language investigates the nature of language A language is a structured system of communication used by humans, including speech (spoken language), gestur ...
s.


Operational vs. theoretical definitions

A definition may also be classified as an
operational definition An operational definition specifies concrete, replicable procedures designed to represent a construct. In the words of American psychologist S.S. Stevens (1935), "An operation is the performance which we execute in order to make known a concept." ...
or
theoretical definition A theoretical definition defines a term in an academic discipline, functioning as a proposal to see a phenomenon in a certain way. A theoretical definition is a proposed way of thinking about potentially related events. Theoretical definitions cont ...
.


Terms with multiple definitions


Homonyms

A
homonym In linguistics, homonyms, broadly defined, are words which are homographs (words that share the same spelling, regardless of pronunciation) or homophones (words that share the same pronunciation, regardless of spelling), or both. For example, acco ...
is, in the strict sense, one of a group of words that share the same spelling and pronunciation but have different meanings.homonym
''Random House Unabridged Dictionary'' at dictionary.com
Thus homonyms are simultaneously
homograph A homograph (from the el, ὁμός, ''homós'', "same" and γράφω, ''gráphō'', "write") is a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with ...
s (words that share the same spelling, regardless of their pronunciation) ''and''
homophone A homophone () is a word that is pronouncedPronunciation is the way in which a word or a language is spoken. This may refer to generally agreed-upon sequences of sounds used in speaking a given word or language in a specific dialect ("correct p ...
s (words that share the same pronunciation, regardless of their spelling). The state of being a homonym is called ''homonymy''. Examples of homonyms are the pair ''stalk'' (part of a plant) and ''stalk'' (follow/harass a person) and the pair ''left'' (past tense of leave) and ''left'' (opposite of right). A distinction is sometimes made between "true" homonyms, which are unrelated in origin, such as ''skate'' (glide on ice) and ''skate'' (the fish), and polysemous homonyms, or
polysemes Polysemy ( or ; from grc-gre, πολύ-, , "many" and , , "sign") is the capacity for a word or phrase to have multiple meanings, usually related by contiguity of meaning within a semantic field. Polysemy is thus distinct from homonymy—or h ...
, which have a shared origin, such as ''mouth'' (of a river) and ''mouth'' (of an animal).


Polysemes

Polysemy Polysemy ( or ; from grc-gre, πολύ-, , "many" and , , "sign") is the capacity for a word or phrase to have multiple related meanings. Polysemy is thus distinct from homonymy—or homophone, homophony—which is an accidental similarity betwee ...
is the capacity for a
sign A sign is an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at ...
(such as a
word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), meaning. In many languages, words also co ...

word
,
phrase In syntax In linguistics, syntax () is the set of rules, principles, and processes that govern the structure of Sentence (linguistics), sentences (sentence structure) in a given Natural language, language, usually including word order. The ter ...

phrase
, or
symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. All (and ) is achieved th ...

symbol
) to have multiple meanings (that is, multiple semes or
sememe __NOTOC__ A sememe () is a semantic Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference Reference is a relationship between objects in which one object designates, or acts as a means by whic ...
s and thus multiple
senses Sense relates to any of the systems and corresponding organs involved in sensation, i.e. the physical process of responding to Stimulus (physiology), stimuli and providing data for perception. During sensation, sense organs collect stimuli for Tran ...
), usually related by contiguity of
meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discussed in philosophy * Meaning (non-linguistic), a general ter ...
within a
semantic fieldIn linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include pho ...
. It is thus usually regarded as distinct from
homonymy In linguistics, homonyms, broadly defined, are words which are homographs (words that share the same spelling, regardless of pronunciation) or homophones (words that share the same pronunciation, regardless of spelling), or both. For example, acco ...
, in which the multiple meanings of a word may be unconnected or unrelated.


In logic and mathematics

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 A neologism (; from Greek νέο- ''néo-'', "new" and λόγος ''lógos'', "speech, utterance") is a relatively recent or isolated term, word, or phrase that may be in the process of entering common use, but that has not yet been fully accepted ...
(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 than 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
has nothing more (or less) real than an
imaginary number An imaginary number is a complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
. Frequently, a definition uses a phrase built with common English words, which has no meaning outside mathematics, such as
primitive group In mathematics, a permutation group ''G'' Group action, acting on a non-empty finite set ''X'' is called primitive if ''G'' acts transitive action, transitively on ''X'' and ''G'' preserves no nontrivial Partition_of_a_set, partition of ''X'', wher ...
or
irreducible variety In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commuta ...
.


Classification

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. In contrast, a "descriptive" definition can be shown to be "right" or "wrong" with reference to general usage. Swartz defines a ''
precising definition A precising definition is a definition that contracts or reduces the scope of the lexical definition of a term for a specific purpose by including additional criteria that narrow down the set of things meeting the definition. For example, a diction ...
'' 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 identified ''
persuasive definition A persuasive definition is a form of stipulative definition which purports to describe the true or commonly accepted meaning of a term, while in reality stipulating an uncommon or altered use, usually to support an argument for some view, or to crea ...
'' 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.


Recursive definitions

A
recursive definition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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 Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics Linguistics is the science, scientific study of language. It e ...

recursive
. # All other things are excluded from the set For instance, we could define a
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
as follows (after
Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
): # "0" is a natural number. # 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. Notice that the second condition in the definition itself refers to natural numbers, and hence involves
self-reference The ancient symbol Ouroboros, a dragon that continually consumes itself, denotes self-reference. Self-reference occurs in natural language, natural or formal languages when a Sentence (linguistics), sentence, idea or Well-formed formula, formu ...
. Although this sort of definition involves a form of circularity, it is not
vicious Vicious may refer to: Music * Johnny Vicious, American house DJ, producer and remixer * Sid Vicious (1957–1979), punk rock musician * Vicious (rapper), Jamaican-American rapper and reggae artist active in the 1990s * Vicious (Nasty Idols album), ...
, and the definition has been quite successful. In the same way, we can define
ancestor An ancestor, also known as a forefather, fore-elder or a forebear, is a parent A parent is a caregiver of the offspring In biology, offspring are the young born of living organism, organisms, produced either by a single organism or, in the ...

ancestor
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.


In medicine

In medical dictionaries,
guideline A guideline is a statement by which to determine a course of action. A guideline aims to streamline particular processes according to a set routine or sound practice. Guidelines may be issued by and used by any organization (governmental or priv ...
s and other consensus statements and
classification Classification is a process related to categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such as Object (philosophy), objects, eve ...
s, definitions should as far as possible be: *simple and easy to understand, preferably even by the general public; *useful clinically or in related areas where the definition will be used; *specific (that is, by reading the definition only, it should ideally not be possible to refer to any other entity than that being defined); *measurable; *a reflection of current scientific knowledge.


Problems

Certain rules have traditionally been given for definitions (in particular, genus-differentia definitions).Joyce, Ch. X #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. #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".


Fallacies of definition


Limitations of definition

Given that a
natural language In neuropsychology Neuropsychology is a branch of psychology. It is concerned with how a person's cognition and behavior are related to the brain and the rest of the nervous system. Professionals in this branch of psychology often focus on ...
such as
English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading lan ...

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 Scholastic may refer to: * a philosopher or theologian in the tradition of scholasticism * Scholastic (Notre Dame publication), ''Scholastic'' (Notre Dame publication) * Scholastic Corporation, an American publishing company of educational material ...
claimed that the highest genera (called the ten ''generalissima'') cannot be defined, since a higher genus cannot be assigned under which they may fall. Thus
being In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, ...

being
, unity and similar concepts cannot be defined. supposes in ''
An Essay Concerning Human Understanding ''An Essay Concerning Human Understanding'' is a work by John Locke John Locke (; 29 August 1632 – 28 October 1704) was an English philosopher and physician, widely regarded as one of the most influential of Enlightenment thin ...
'' that the names of simple concepts do not admit of any definition. More recently
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British , , , , , , , , and .Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003 Throughout his life, Russell considered himself a , a and ...
sought to develop a formal language based on logical atoms. Other philosophers, notably
Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationalit ...

Wittgenstein
, rejected the need for any undefined simples. Wittgenstein pointed out in his ''
Philosophical Investigations ''Philosophical Investigations'' (german: Philosophische Untersuchungen) is a work by the philosopher Ludwig Wittgenstein. The book was published posthumously in 1953. Wittgenstein discusses numerous problems and puzzles in the fields of semanti ...
'' 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 Mill may refer to: Science and technology * Mill (grinding) * Milling (machining) * List of types of mill * Mill, the arithmetic unit of the Analytical Engine early computer * Textile manufacturing, Textile mill * Steel mill, a factory for the manu ...
also argued that
individuals An individual is that which exists as a distinct wikt:entity , entity. Individuality (or self-hood) is the state or quality of being an individual; particularly (in the case of humans) of being a person unique from other people and possessing one ...
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 The theory of descriptions is the philosopher Bertrand Russell's most significant contribution to the philosophy of language. It is also known as Russell's theory of descriptions (commonly abbreviated as RTD). In short, Russell argued that the ...
in part as a way of defining a proper name, the definition being given by a
definite description In formal semantics (natural language), formal semantics and philosophy of language, a definite description is a denotation, denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun. The definite description is ''pr ...
that "picks out" exactly one individual.
Saul Kripke Saul Aaron Kripke (; born November 13, 1940) is an American philosopher and logician in the analytic tradition. He is a Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emeritus professor at Pr ...

Saul Kripke
pointed to difficulties with this approach, especially in relation to
modality Modality may refer to: Humanities * Modality (theology), the organization and structure of the church, as distinct from sodality or parachurch organizations * Modality (music), in music, the subject concerning certain diatonic scales * Modalities ...
, 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.''Philosophical Investigations 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 Family resemblance (german: Familienähnlichkeit, link=no) is a philosophical idea made popular by Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian Austrian may refer to: * Austri ...
. 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

*
Analytic proposition Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic can also have the following meanings: Natural sciences Chemistry * ...
*
Circular definition A circular definition is a definition A definition is a statement of the meaning of a term (a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with s ...
*
Definable setIn mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
*
DefinitionismDefinitionism (also called the classical theory of concepts) is the school of thought in which it is believed that a proper explanation of a theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the res ...
*
Extensional definition In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
* Fallacies of definition *
Indeterminacy Indeterminacy or underdeterminacy may refer to: * Indeterminacy in computation (disambiguation) * Aleatoric music and indeterminacy in music. * Statically indeterminate *Indeterminacy (literature) a literary term * In set theory and game theory, the ...
*
Intensional definition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label ...
* Lexical definition *
Operational definition An operational definition specifies concrete, replicable procedures designed to represent a construct. In the words of American psychologist S.S. Stevens (1935), "An operation is the performance which we execute in order to make known a concept." ...
*
Ostensive definitionAn ostensive definition conveys the meaning of a term by pointing out examples. This type of definition is often used where the term is difficult to define verbally, either because the words will not be understood (as with children and new speakers ...
*
Ramsey–Lewis methodThe Ramsey–Lewis method is a method for defining terms found in theory, theoretical frameworks (such as in scientific theory, scientific theories), credited to Frank P. Ramsey and David Lewis (philosopher), David Lewis. By using this method, a set ...
*
Semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another o ...
*
Synthetic propositionSynthetic things are composed of multiple parts, often with the implication that they are artificial. In particular, 'synthetic' may refer to: Science * Synthetic chemical or compound, produced by the process of chemical synthesis As a topic of ...
*
Theoretical definition A theoretical definition defines a term in an academic discipline, functioning as a proposal to see a phenomenon in a certain way. A theoretical definition is a proposed way of thinking about potentially related events. Theoretical definitions cont ...


Notes


References

*
(full text of 1st ed. (1906))

(worldcat)(full text of 2nd ed. (1916))
* (full text
vol 1vol 2
* * * * *


External links


Definitions
Stanford Encyclopedia of Philosophy Gupta, Anil (2008)
Definitions, Dictionaries, and Meanings, Norman Swartz 1997
*Guy Longworth (ca. 2008
"Definitions: Uses and Varieties of"
= in: K. Brown (ed.): ''Elsevier Encyclopedia of Language and Linguistics'', Elsevier.
Definition and Meaning
a very short introduction by Garth Kemerling (2001). {{Authority control Definition, Philosophical logic Philosophy of language Semantics Linguistics terminology Mathematical terminology Concepts in logic Lexicography Meaning (philosophy of language)