User:Marshallsumter/Theory of definition

A theory of definition generally refers to those analytical tools for definitions so as to make predictions with definitions about definitions. A number of definitions included in the article uses definitions from wiktionary users or wikipedia users. Replace these with citational source definitions.

Notation
A theory usually begins with the introduction of notation or symbols for words or phrases in a commonly used language such as English, although for any needed words or phrases outside English that are difficult to express therein words or phrases from outside can be used.

Notation: Let the abbreviation Def. stand for the word definition; that is, as an indicator a definition of a term, word, phrase, or symbol is about to follow.

A few logical symbols may come in handy:

Definition of definition
“[D]efinitions are always of symbols, for only symbols have meanings for definitions to explain.” A term can be one or more of a set of symbols such as words, phrases, letter designations, or any already used symbol or new symbol.

In the theory of definition, “the symbol being defined is called the definiendum, and the symbol or set of symbols used to explain the meaning of the definiendum is called the definiens.” “The definiens is not the meaning of the definiendum, but another symbol or group of symbols which, according to the definition, has the same meaning as the definiendum.”

Def. "[t]he term - word or phrase - defined in a definition" is called a definiendum.

Def. "[t]he word or phrase that defines the definiendum in a definition" is called a definiens.

Def. a definiendum defined by a definiens is called a definition.

Russell's paradox for definitions
For every (or any) property, there exists a definition whose definiens is just those entities having that property.

Consider a definition whose definiens is "all those entities which have the property of not being in the definiens for their own definition". This is Russell's paradox for definitions.

This suggests that there is at least one property for inclusion in it's own definition whose definiens does not include that property.

In axiomatic set theory, admitting the first sentence above grants too much; however, language can have the paradox without any restriction.

Example:

Def. a. (from apple on Wiktionary) "[a] common, round fruit produced by the tree Malus domestica, cultivated in temperate climates" is called an apple.

Def. b. "[a] common, round fruit produced by the tree" Pyrus communis, "cultivated in temperate climates" is called an apple.

With these two definitions, an apple is a fruit of the tree Malus domestics, not the pear tree Pyrus communis. Definition b is customarily considered to be in error (a fallacy). It is either a fallacy because of the definiens or the definendum. A small branch of an apple tree grafted onto a pear tree that succeeds in producing apples has produced apples (not pears). Traditionally to amend this fallacy, we call such an apple something modified like a "papple". While this may make some people more comfortable, we also don't have to make an amend.

The definition of parallel lines is usually the definition from Euclidean geometry; however, change the definiens and many non-Euclidean geometries result.

Arguably, another form of this type of definition is a persuasive definition.

Those reading ahead may have noticed that certain alphabetical letters are missing: B, F, H, J, K, N, Q, U, V, W, X, Y, and Z, with respect to alternate types of definitions. Some of these gaps may be fillable with accepted terms for definitions, yet others may be filled only with adjective preceding noun. For example, a "being definition" could fill in the gap for the letter B: "The most obvious and common feature is language that may point toward some "Supreme Being" definition of religion." While appeal to any "Supreme Being" may be okay, an ordinary "being" such as an author is okay too. Usually, an authored definition falls under one or more of the customary definitional types below.

Axiomatic definition
It has been stated that "the rigorous definition of distance" fulfills "the three axioms that define an Euclidean metric" so that a "generalized metric can be defined using as distance an appropriate function ... that fulfills the three axioms of an Euclidean metric". Having met these three axioms as a criteria of an Euclidean metric, the definition of the generalized metric is said to be a "rigorous definition of distance".

An axiomatic definition is a rigorous definition: "the definition must clearly state the rules that are considered as binding, and on the other hand give the implementor enough freedom to achieve efficiency by leaving certain less important aspects undefined." This rigorous definition is for "an axiomatic definition of the programming language PASCAL".

Circular definition
Def. a definition relying directly or indirectly on the term being defined is called a circular definition.

Conceptual definition
Def. "[a] definition in terms of concepts, such as the one found in a dictionary, instead of in terms of the results of measuring procedures" is called a conceptual definition.

Dictionary definition
Dictionary definition is a description "specifying one of the commonly used meanings of the ... term.

Def. "[a] descriptive definition specifying one of the commonly used meanings of the defined term" is called a dictionary definition.

Def. 2: a definition is "a word or phrase expressing the essential nature of a person or thing".

Def. 3a: a definition is "a statement of the meaning of a word or word group or a sign or symbol".

Def. 3b: a definition is "the action or process of stating such a meaning".

Def. 4a: a definition is "the action or the power of making definite and clear".

Def. 4b: a definition is "clarity, distinctness".

Enumerative definition
Def. "[a] definition that exhaustively lists all the objects that fall under the defined term" is called an enumerative definition.

Extensional definition
Def. "[a] definition of a term that specifies its extension, that is, every object that falls under the definition" is called an extensional definition.

Genus–differentia definition
From the Wikipedia article genus-differentia definition: "[a] genus-differentia definition is a type of intentional definition ... composed by two parts:
 * 1) a genus (or family): An existing definition that serves as a portion of the new definition; ... [and]
 * 2) the differentia: The portion of the new definition that is not provided by the genera."

The rules for definition by genus and differentia are in the Wikipedia article definition.

Certain rules have traditionally been given for this particular type of definition.
 * 1) A definition must set out the essential attributes of the thing defined. That is "[a] definition should state the conventional connotation of the term being 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, a definition of a term must not be comprised 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. (See the definition of Free will, for instance).
 * 5) 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. We cannot define a point except as 'something with no parts', nor blindness except as 'the absence of sight in a creature that is normally sighted'.

Inductive definition
Def. a recursive definition is called an inductive definition.

Intensional definition
Def. "[a] definition that gives the meaning of a term by specifying all the properties of the things to which the term applies" is called an intensional definition.

Lexical definition
A lexical definition is usually a dictionary definition and "is either true or false."

Def. "[t]he meaning of a word in actual usage by speakers of a certain language" is called a lexical definition.

Metadefinition
Def. a metadefinition is "a set of attributes that address and satisfy a number of purposes or functions for [a particular] definition."

A definition serves five functions:


 * 1) a statement of identity,
 * 2) a way to define competitive and cooperative relationships with other terms,
 * 3) a way to end conceptual disputes and thus prepare the way for measurement - its preoperational or premeasurement function,
 * 4) a way to locate a term within a particular context - its orienting or contextual function, and
 * 5) a way to generate new ideas - its generative or revelatory function.

Operational definition
Def. "[a] showing of something - such as a variable, term, or object - in terms of the specific process or set of validation tests used to determine its presence and quantity" is an operational definition.

Ostensive definition
Def. "[a] process of binding the meaning to the defined term by pointing out examples and counterexamples" is called an ostensive definition.

Persuasive definition
From the Wikipedia article persuasive definition: "[a] persuasive definition is a form of 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 create or alter rights, duties or crimes. " I cannot confirm that this definition is in the references indicated.

"In his Ethics and Language, Stevenson defines the term 'persuasive definition' as follows: "In any 'persuasive definition' the term defined is a familiar one, whose meaning is both descriptive and strongly emotive. The purport of the definition is to alter the descriptive meaning of the term, usually by giving it greater precision within the boundary of its customary vagueness; but the definition does not make any substantial change in the term's emotive meaning. And the definition is used, consciously or unconsciously, in an effort to secure, by this interplay between emotive and descriptive meaning, a redirection of people's attitudes" (Stevenson, 1944)"

In argumentation the use of a stipulative definition is an example of the definist fallacy.

Precising definition
Def. 7a: a term is "a word or expression that has a precise meaning in some uses or is peculiar to a science, art, profession, or subject".

In the article precising definition, there is that "[a] precising definition is a definition that extends the lexical definition of a term for a specific purpose by including additional criteria that narrow down the set of things meeting the definition." The precising definition is usually aimed at the definiens.

Recursive definition
Def. a "definition of a term within which the term itself appears, and that is well-founded, avoiding an infinite regress" is called a recursive definition.

Rigorous definition
A rigorous definition of the term “stem cell” is as follows.

Definition: A cell is a stem cell if and only if it has the properties:
 * 1) unlimited self-renewal and
 * 2) within-tissue multipotentiality.

This definition has limited flexibility in that it “does not necessarily exclude cross-tissue plasticity.”

Semantic definition
Semantic definition is "[a] statement of the meaning of a word or word group or a sign or symbol. A semantic definition should be identical to a lexical definition.

Stipulative definition
A stipulative definition is a semantic "definition in which a new or currently-existing term is given a new meaning for the purposes of argument or discussion in a given context."

Symbolic definition
Def. 3: "A ⊆ B ↔ (∀x) (x ∈ A → x ∈ B)."

Synonymous definition
A synonymous definition is a definition “defining a single word [or symbol] by giving another single word [or symbol] which has the same meaning.” But, synonymous definitions have limitations:
 * 1) “some words have no exact synonyms” ,
 * 2) a synonymous definition “cannot be used in the construction of precising or theoretical definitions.”, and
 * 3) no synonym should appear in the definiens of a genus–differentia definition.

With relative synonymy instead of exact synonymy, relative synonyms may be useable in a genus–differentia definition.

Theoretical definition
Def. "the meaning of a word in terms of the theories of a specific discipline" is called a theoretical definition.

Def. an attempt "to formulate a theoretically adequate characterization of [an object] to which it is applied" is called a theoretical definition.

"The empirical and explanatory success of theories or theory-parts is a good indicator of their approximate truth. In turn, approximate truth is a good indicator of referential success."

There are "three dominant theories of reference, namely descriptivist, causal-historical and causal-descriptivist".

"Successor theories typically preserve all of the empirical and explanatory success of their predecessors as well as add to it. They are thus in general strictly more approximately true than their predecessors. Moreover, by preserving their predecessors’ approximately true parts they preserve any referential success the predecessors enjoy. This implies that successor theories that are more approximately true than their predecessors are typically also referentially continuous with them."

"[M]ost discussions of reference concern everyday language term reference and in particular the reference of proper names ... [here] primary focus will be on scientific term reference."

Here is an example of a causal-descriptivist definition:

"Def. 6: A term t refers to a(n) entity a (/property X) if and only if the dominant (causal) source of any descriptive content associated with t is a (/X).", where "a (/property X)" means an entity "a with property X", also as a "/X".

With full respect to a theoretical definition of a term that has continued to be used through several theories ("refcont" ), there is

“Def. 12: A scientific term t refers(MX) to a real entity a (/property X) if and only if (i) t is used to consistently identify a (/X) and (ii) all and only the theoretical and empirical descriptive claims associated with t are true of a (/X).”, where "refers(MX)" represents the maximum extreme "of the spectrum of noteworthy referential concepts."

And finally,

“Def. 13: Two successive scientific terms t and t' are refcont(MX)* if and only if (i) t' refers(MX) to a (/X), (ii) t nearly refers(MX) a (/X) and (iii) t' inherits all of the (non-trivial) theoretical and empirical descriptive claims true of a (/X) that are associated with t.”

By definition
By definition, or by means of a definition, indicates the use of a passage that explains the meaning of a term (a word, phrase or other set of symbols), or a type of thing, so as to demonstrate identicalness. “The phrase by definition has a precise meaning: the speaker [or writer] is asserting that a property can be assigned to an object that has been named, by virtue of the fact that the definition of the object requires it to have that property.”