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A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols).[1][2] Definitions can be classified into two large categories, intensional definitions (which try to give the sense of a term) and extensional definitions (which try to list the objects that a term describes).[3] 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.[4][a]

In mathematics, 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.[5] Definitions and axioms form the basis on which all of modern mathematics is to be constructed.[6]

Polysemy is the capacity for a sign (such as a word, phrase, or symbol) to have multiple meanings (that is, multiple Polysemy is the capacity for a sign (such as a word, phrase, or symbol) to have multiple meanings (that is, multiple semes or sememes and thus multiple senses), 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 and mathematics

In mathematics, definitions are generally not used to describe existing terms, but to

In mathematics, definitions are generally not used to describe existing terms, but to describe or characterize a concept.[19] For naming the object of a definition mathematicians can use either a neologism (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,[20] 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 (or less) 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.

Classification

Authors have used different

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.[21] In contrast, a "descriptive" definition can be shown to be "right" or "wrong" with reference to general usage.

Swartz defines a precising 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 def

Swartz defines a precising 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 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.[22]

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:

  1. At least one thing is stated to be a member of the set being defined; this is sometimes called a "base set".
  2. 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.
  3. <

    For instance, we could define a natural number as follows (after Peano):

    1. "0" is a natural number.
    2. 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".
    3. Nothing else is a natural number.

    So "0" will have exactly one successor, which for conveni

    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:

    1. A parent is an ancestor.
    2. A parent of an ancestor is an ancestor.
    3. Nothing else is an ancestor.

    Or simply: an ancestor is a parent or a parent of an ancestor.

    In the same way, we can define ancestor as follows:

    Or simply: an ancestor is a parent or a parent of an ancestor.

    In medicineIn medical dictionaries, guidelines and other consensus statements and classifications, definitions should as far as possible be:

    • simple and easy to understand,[23] preferably even by the general public;[24]
    • Certain rules have traditionally been given for definitions (in particular, genus-differentia definitions).[25][26][27][28]

      1. A definition must set out the essential attributes of the thing defined.
      2. Definitions should avoid circularity. To define a horse as "a member of the species equus" would convey no information whatsoever. For this reason, Locking[specify] 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.
      3. 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).
      4. 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.
      5. A definition should not be negative where it can be positive. We should not define "wisdom" as

        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?"[29][30] A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity.[31][32][33]

        Many philosophers have chosen instead to leave some terms undefined. The scholastic philosophers 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, unity and similar concepts cannot be defined.[26] Locke supposes in An Essay Concerning Human Understanding[34] 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.[35] 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",[36] 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.[37] This is not possible when no one else is acquainted with the particular thing that has "fallen under our notice".[38] Russell offered his theory of descriptions in part as a way of defining a proper name, the definition being given by a scholastic philosophers 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, unity and similar concepts cannot be defined.[26] Locke supposes in An Essay Concerning Human Understanding[34] 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.[35] 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",[36] 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.[37] This is not possible when no one else is acquainted with the particular thing that has "fallen under our notice".[38] 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.[39] 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.[b]