Definition

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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. Blacks-Law-Dictionary.jpg
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.

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] [lower-alpha 1]

Contents

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. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. [5] In computing, definitions can be used as logic programs

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. [6] 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. [7]

The definiens is not the meaning of the word defined, but is instead something that conveys 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."); 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."). [7]

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. [3] 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. [3]

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 (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 that takes a large category (the genus) and narrows it down to a smaller category by a distinguishing characteristic (i.e. the differentia). [8]

More formally, a genus–differentia definition consists of:

For example, consider the following genus–differentia definitions:

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:

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 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 (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. [9]

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 (and in fact only practical for relatively small 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. [10]

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. [11]

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, [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 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". [13] 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 (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, in particular, is critical of attempts to elucidate the essence of a thing. Russell described essence as "a hopelessly muddle-headed notion". [14]

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. [15] Thus homonyms are simultaneously homographs (words that share the same spelling, regardless of their pronunciation) and homophones (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, which have a shared origin, such as mouth (of a river) and mouth (of an animal). [16] [17]

Polysemes

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, mathematics and computing

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 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 from the English definition of the word used, [19] 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.

In first-order logic definitions are usually introduced using extension 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.

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. [20] 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 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. [21]

Recursive definitions

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. All other things are excluded from the set

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 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 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.

Logic programs

Logic programs can be understood as sets of recursive (and non-recursive) definitions. [22] [23] For example, the following Prolog and Datalog program (and database) provides intensional definitions of the parent_child , grandparent_child and ancestor_descendant relations, as well as partial, extensional definitions of the mother_child and father_child relations. The definition of the ancestor_descendant relation is recursive:

mother_child(elizabeth,charles).father_child(charles,william).father_child(charles,harry).parent_child(X,Y):-mother_child(X,Y).parent_child(X,Y):-father_child(X,Y).grandparent_child(X,Y):-parent_child(X,Z),parent_child(Z,Y).ancestor_descendant(X,Y):-parent_child(X,X).ancestor_descendant(X,Y):-ancestor_descendant(X,Z),ancestor_descendant(Z,Y).

Here :- represents if and , represents and.

In Prolog, definitions, which have the form conclusion:-conditions, are treated as goal-reduction procedures, using backward reasoning to reduce goals that unify with the conclusion to subgoals that correspond to the associated, instantiated conditions. In Datalog, the same definitions are typically used to reason forwards to derive instances of the conclusion from facts that unify with the conditions.

Definitions in Prolog have the expressive power of Turing machines. For example, here is a Prolog program that implements the Euclidean algorithm, using gcd(A,B,C) to "define" C as the greatest common divisor of A and B:

gcd(A,A,A).gcd(A,B,C):-A>B,gcd(A-B,B,C).gcd(A,B,C):-B>A,gcd(A,B-A,C).

Backward reasoning using SLD resolution turns the definition into the Euclidean algorithm:

To find the gcd C of two given numbers A and B: If A = B, then C = A. If A > B, then find the gcd of A-B and B, which is C. If B > A, then find the gcd of A and B-A, which is C. 

The standard definition of greatest common divisor can also be written and executed (inefficiently) in Prolog:

gcd(A,B,C):-divides(C,A),divides(C,B),forall((divides(D,A),divides(D,B)),D=<C).divides(C,Number):-between(1,Number,C),0isNumbermodC.

See also Algorithm = Logic + Control.

In medicine

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

Problems

Certain rules have traditionally been given for definitions (in particular, genus-differentia definitions). [26] [27] [28] [29]

Fallacies of definition

Limitations of 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?" [30] [31] A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity. [32] [33] [34]

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. [27] Locke supposes in An Essay Concerning Human Understanding [35] 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. [36] 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", [37] 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. [38] This is not possible when no one else is acquainted with the particular thing that has "fallen under our notice". [39] 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. [40] 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. [lower-alpha 2]

See also

Notes

  1. Terms with the same pronunciation and spelling but unrelated meanings are called homonyms, while terms with the same spelling and pronunciation and related meanings are called polysemes.
  2. Note that one learns inductively, from ostensive definition, in the same way, as in the Ramsey–Lewis method.

Related Research Articles

<span class="mw-page-title-main">Algorithm</span> Sequence of operations for a task

In mathematics and computer science, an algorithm is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes and deduce valid inferences, achieving automation eventually. Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus".

Fallacies of definition are the various ways in which definitions can fail to explain terms. The phrase is used to suggest an analogy with an informal fallacy. Definitions may fail to have merit, because they: are overly broad, use obscure or ambiguous language, or contain circular reasoning; those are called fallacies of definition. Three major fallacies are: overly broad, overly narrow, and mutually exclusive definitions, a fourth is: incomprehensible definitions, and one of the most common is circular definitions.

A genus–differentia definition is a type of intensional definition, and it is composed of two parts:

  1. 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.
  2. the differentia: The portion of the definition that is not provided by the genus.

In any of several fields of study that treat the use of signs—for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language—an intension is any property or quality connoted by a word, phrase, or another symbol. In the case of a word, the word's definition often implies an intension. For instance, the intensions of the word plant include properties such as "being composed of cellulose ", "alive", and "organism", among others. A comprehension is the collection of all such intensions.

Logic programming is a programming, database and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical form, representing knowledge about some problem domain. Computation is performed by applying logical reasoning to that knowledge, to solve problems in the domain. Major logic programming language families include Prolog, Answer Set Programming (ASP) and Datalog. In all of these languages, rules are written in the form of clauses:

Prolog is a logic programming language that has its origins in artificial intelligence and computational linguistics.

A connotation is a commonly understood cultural or emotional association that any given word or phrase carries, in addition to its explicit or literal meaning, which is its denotation.

<span class="mw-page-title-main">Circular definition</span> Self-referential description of meaning

A circular definition is a type of definition that uses the term(s) being defined as part of the description or assumes that the term(s) being described are already known. There are several kinds of circular definition, and several ways of characterising the term: pragmatic, lexicographic and linguistic. Circular definitions are related to Circular reasoning in that they both involve a self-referential approach.

A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. For instance the sentence "The sky is blue" denotes the proposition that the sky is blue. However, crucially, propositions are not themselves linguistic expressions. For instance, the English sentence "Snow is white" denotes the same proposition as the German sentence "Schnee ist weiß" even though the two sentences are not the same. Similarly, propositions can also be characterized as the objects of belief and other propositional attitudes. For instance if one believes that the sky is blue, what one believes is the proposition that the sky is blue. A proposition can also be thought of as a kind of idea: Collins Dictionary has a definition for proposition as "a statement or an idea that people can consider or discuss whether it is true."

In computer science, declarative programming is a programming paradigm—a style of building the structure and elements of computer programs—that expresses the logic of a computation without describing its control flow.

In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same.

Intuitionistic type theory is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to predicative versions. However, all versions keep the core design of constructive logic using dependent types.

In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.

Predicable is, in scholastic logic, a term applied to a classification of the possible relations in which a predicate may stand to its subject. It is not to be confused with 'praedicamenta', the scholastics' term for Aristotle's ten Categories.

<i>Topics</i> (Aristotle) Works by Aristotle

The Topics is the name given to one of Aristotle's six works on logic collectively known as the Organon. In Andronicus of Rhodes' arrangement it is the fifth of these six works.

<span class="mw-page-title-main">Haecceity</span>

Haecceity is a term from medieval scholastic philosophy, first coined by followers of Duns Scotus to denote a concept that he seems to have originated: the irreducible determination of a thing that makes it this particular thing. Haecceity is a person's or object's thisness, the individualising difference between the concept "a man" and the concept "Socrates". In modern philosophy of physics, it is sometimes referred to as primitive thisness.

A sign relation is the basic construct in the theory of signs, also known as semiotics, as developed by Charles Sanders Peirce.

Deductive lambda calculus considers what happens when lambda terms are regarded as mathematical expressions. One interpretation of the untyped lambda calculus is as a programming language where evaluation proceeds by performing reductions on an expression until it is in normal form. In this interpretation, if the expression never reduces to normal form then the program never terminates, and the value is undefined. Considered as a mathematical deductive system, each reduction would not alter the value of the expression. The expression would equal the reduction of the expression.

In knowledge representation, a class is a collection of individuals or individuals objects. A class can be defined either by extension, or by intension, using what is called in some ontology languages like OWL. According to the Type–token distinction, the ontology is divided into individuals, who are real worlds objects, or events, and types, or classes, who are sets of real world objects. Class expressions or definitions gives the properties that the individuals must fulfill to be members of the class. Individuals that fulfill the property are called Instances.

In logic, extensional and intensional definitions are two key ways in which the objects, concepts, or referents a term refers to can be defined. They give meaning or denotation to a term.

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  12. Posterior Analytics Bk 2 c. 7
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  26. Copi 1982 pp 165–169
  27. 1 2 Joyce, Ch. X
  28. Joseph, Ch. V
  29. Macagno & Walton 2014, Ch. III
  30. Locke, Essay, Bk. III, Ch. iv, 5
  31. This problem parallels the diallelus, but leads to scepticism about meaning rather than knowledge.
  32. Generally lexicographers seek to avoid circularity wherever possible, but the definitions of words such as "the" and "a" use those words and are therefore circular. Lexicographer Sidney I. Landau's essay "Sexual Intercourse in American College Dictionaries" provides other examples of circularity in dictionary definitions. (McKean, p. 73–77)
  33. An exercise suggested by J. 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 and G. J. Warnock. Oxford: Oxford UP, 1961. 1979.)
  34. In the game of Vish, players compete to find circularity in a dictionary.
  35. Locke, Essay, Bk. III, Ch. iv
  36. See especially Philosophical Investigations Part 1 §48
  37. He continues: "Whereas an explanation may indeed rest on another one that has been given, but none stands in need of another – unless we 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
  38. This theory of meaning is one of the targets of the private language argument
  39. Locke, Essay, Bk. III, Ch. iii, 3
  40. Philosophical Investigations