# Definitions/Theory

For the theory of definition, as a learning resource, the topic of definitions probably arises during the secondary level as students find themselves more and more consulting a dictionary to understand a term, its meaning, and it use. This learning resource proceeds from the secondary level into the tertiary, or university, level, and to a limited extent into the research and exploration level.
A theory generally refers to analytical tools for understanding, describing, or explaining a subject so as to make predictions within that subject. A theory of definition generally refers to those analytical tools for definitions so as to make predictions with definitions about definitions. A theory whose subject matter is a theory of definition is the metatheory of definition. A theory of a theory of a theory, recursively, is still a metatheory.
As a type of resource, it is closest to an article.

## Notations

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.
A few logical symbols may come in handy:
Logical notation or symbols :
| Sentential | symbol | English | example | meaning |
| negation | - | not, or not the case | -apple | not an apple |
| conjunction | & | and | an apple & an orange | an apple and an orange |
| exclusive disjunction | ⋁ | one or the other but not both | an apple ⋁ an orange | either an apple or an orange but not both |
| inclusive disjunction | ⋃ | one or the other or both | an apple ⋃ an orange | an apple, an orange, or both |
| implication | → | if ... then ... | an apple is a fruit → not all fruit are apples | if an apple is a fruit then not all fruit are apples |
| equivalence | ↔ | if and only if | an apple eater eats fruit ↔ fruit are only apples | an apple eater eats fruit if and only if the fruit are only apples |
| identity | ≡ | identical | A ≡ A | A is identical to A |
| universal quantifier | ∀ | for all, for every | ∀apple | for every apple, for all apples |
| existence | ∃ | for some or one | ∃apple | at least one apple exists |
| unitary existence | ∃! | exactly one exists | ∃!apple | exactly one apple exists |
| such that | ∣ | such that, so that | ∣apple | such that an apple |

## Semantics

Def.
    1. "the study of meanings:"
    1.a: "the historical and psychological study and the classification of changes in the signification of words or forms viewed as factors in linguistic development",
    1.b: "a branch of semiotic dealing with the relations between signs and what they refer to and including theories of denotation, extension, naming, and truth"
    2.a: "the meaning or relationship of meanings of a sign or set of signs; [especially]: connotative meaning"
    2.b: "the exploitation of connotation and ambiguity (as in propaganda)"
is called semantics.
Def. "the branch of linguistics devoted to the investigation of linguistic meaning, the interpretation of expression in a language system." is called semantics.
Starting with universals and ontology, where the word "semantics" is in category 543. MEANING, a definition of semantics using concepts from categories lower in number than 543 may be as follows:
Def. the knowledge of the nature of ideas transferred among entities is called semantics.
Every word after Def. and before "is called" has its most popular category of usage less than 543.

## Definitions

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

## Dictionaries

Def. 1: "a reference book containing words usu. alphabetically arranged along with information about their forms, pronunciations, functions, etymologies, meanings, and syntactical and idiomatic uses" is called a dictionary.
Def. 2: "a reference book listing alphabetically terms or names important to a particular subject or activity along with discussion of their meanings and applications" is called a dictionary.
Def. 3: "a general comprehensive list, collection, or repository of information alphabetically arranged" is called a dictionary.

## Glossary

Def. "[a] list of terms in a particular domain of knowledge with the definitions for those terms" is called a glossary.

## 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. 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 for the types of definitions certain alphabetical letters are missing: B, F, H, J, K, N, Q, U, V, 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 definitions

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".

## By definitions

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.”

## Circular definitions

Def. a definition relying directly or indirectly on the term being defined is called a circular definition.
The image at the right is an example of a circular definition: a tomato facing left is used to define a tomato facing right.
Def. "a usually rounded natural elevation of land lower than a mountain" is called a hill.
Def. "a landmass that projects conspicuously above its surroundings and is higher than a hill" is called a mountain.

## Conceptual definitions

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.
A conceptual description makes no reference "to any technical realization."

## Coordinative definitions

Def. "a postulate which assigns a partial meaning to the theoretical terms of a scientific theory by correlating the mathematical objects of the pure or formal/syntactical aspects of a theory with physical objects in the world" is called a coordinative definition.

## Dictionary definitions

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 word or phrase expressing the essential nature of a person or thing" is called a definition.
Def. 3a: "a statement of the meaning of a word or word group or a sign or symbol" is called a definition.
Def. 3b: "the action or process of stating such a meaning" is called a definition.
Def. 4a: "the action or the power of making definite and clear" is called a definition.
Def. 4b: "clarity, distinctness" is called a definition.
If every term of every definiens must itself be defined, "where at last should we stop?" This problem parallels the diallelus, but leads to scepticism about meaning rather than knowledge. A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity. 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. 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. In the game of Vish, players compete to find circularity in a dictionary.

## Enumerative definitions

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

## Extensional definitions

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

## Functional definitions

Def. a definition that is "intended to be used; practical rather than attractive" is called a functional definition.

## Genus differentia definitions

A genus-differentia definition is a type of intentional definition composed by two parts:
* a genus (or family): An existing definition that serves as a portion of the new definition; and
* the differentia: The portion of the new definition that is not provided by the genera.
There are rules for definition by genus and differentia.
Certain rules have traditionally been given for this particular type of definition.
* 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."
* 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.
* 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. (See the definition of Free will, for instance).
* 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 definitions

Def. a recursive definition is called an inductive definition.

## Intensional definitions

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.
An intensional definition, also called a coactive definition, specifies the necessary and sufficient conditions for a thing being 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.

## Lexical definitions

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.

## Metadefinitions

Def. a "[smallest] distinct unit of language [which has] with a particular meaning" [and can be expressed by itself]; the [smallest discrete, meaningful unit of language]" is called a word.
Def. a definiendum defined by a definiens is called a definition.
Def. "[a] statement of the meaning of a word or word group or a sign or symbol" is called a definition.
Many philosophers have chosen instead to leave some terms undefined. The scholastic philosophers claimed that the highest genera (the so-called ten generalissima) cannot be defined, since we cannot assign any higher genus under which they may fall. Thus we cannot define being, unity and similar concepts. John Locke supposes in An Essay Concerning Human Understanding that the names of simple concepts do not admit of any definition.
Here's an example of a metadefinition (Metadef.):
* Metadefinition of a clear liquid:
Metadef.
* A clear flowing substance
* keeping or retaining no shape or definite shape
       a. except that determined by the containing receptacle
    3. composed of molecules
       a. not tending to separate from one another like those of a gas
       b. readily changing relative position
is called a clear liquid.
A clear liquid diet consists of transparent liquid foods such as vegetable broth, bouillon, clear fruit juices, clear fruit ices, popsicles, clear gelatin desserts, and no carbonated drinks. Soda's carbonation expands the gastrointestinal tract.
Herein are many clear liquids that fit within the metadefinition.
* Definition of water:
Def. a clear liquid having the chemical formula H$_2$O, required by all forms of life on Earth is called water.
* Definition of water from the metadefinition of a clear liquid:
* A clear flowing substance
* keeping or retaining no shape or definite shape
       a. except that determined by the containing receptacle
    3. composed of H$_2$O molecules
       a. not tending to separate from one another like those of a gas
       b. readily changing relative position
is called water.
* Definition of ethanol from the metadefinition of a clear liquid:
* A clear flowing substance
* keeping or retaining no shape or definite shape
       a. except that determined by the containing receptacle
    3. composed of C$_2$H$_5$OH molecules
       a. not tending to separate from one another like those of a gas
       b. readily changing relative position
is called ethanol.

## Albert metadefinitions

Def. "a set of attributes that address and satisfy a number of purposes or functions for [a particular] definition" is called a metadefinition.
A definition serves five functions:
* a statement of identity,
* a way to define competitive and cooperative relationships with other terms,
* a way to end conceptual disputes and thus prepare the way for measurement - its preoperational or premeasurement function,
* a way to locate a term within a particular context - its orienting or contextual function, and
* a way to generate new ideas - its generative or revelatory function.

## Operational definitions

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.
"An operational definition, also known as an empirical definition, defines a concept on the basis of criteria that must be applied in order to determine whether and to what extent that concept exists, ie, an operational definition defines a concept in terms of observable data."

## Ostensive definitions

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

## Persuasive definitions

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.
Def. an "effort to influence attitudes by surreptitiously attaching emotive significance to the meaning of a term" is called a persuasive definition.
"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.
Here are some examples of persuasive definitions:
* Def. "someone who doesn't yet realize that God exists" is called an atheist.
* Def. "a leftist who desires to overtax the corporations and abolish freedom in the economic sphere" is called a Democrat.
* Def. "an old white man who feels threatened by change" is called a Republican.
* Def. "a tool to get people to do things they don't want to do" is called loyalty.
* Def. "a slogan used by ordinary common sense against educated reason" is called sophistry.

## Precising definitions

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
Def. "[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 definitions

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.
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:
* 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.
* All other things are excluded from the set.

## Rigorous definitions

A rigorous definition of the term “stem cell” is as follows.
Def. A cell is a stem cell if and only if it has the properties:
* unlimited self-renewal and
* within-tissue multipotentiality.
This definition has limited flexibility in that it “does not necessarily exclude cross-tissue plasticity.”

## Semantic definitions

From Wiktionary, definition (semantics):
Def. "[a] statement of the meaning of a word or word group or a sign or symbol" is called a definition (from semantics), a semantic definition. A semantic definition should be identical to a lexical (or dictionary) definition.

## Stipulative definitions

Def. 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 definitions

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

## Synonymous definitions

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:
* “some words have no exact synonyms”,
* a synonymous definition “cannot be used in the construction of precising or theoretical definitions.”, and
* 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 definitions

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."
Here's a theoretical definition:
Def. any natural object radiating through a portion or all of the Earth's or another natural, astronomical object's atmosphere is called a meteor.
These are usage notes:
* Such an object may be as small as an electron or much larger.
* Astronomical objects that are atoms, nuclei, or subatomic particles are part of cosmic-ray astronomy.
* Astronomical objects larger than atoms, nuclei, or subatomic particles that are fast-moving relative to perceived, almost motionless objects, radiating through another natural object's atmosphere or gaseous environment are also here referred to as meteors.
* These can be a high-velocity star moving through the interstellar medium or a larger object moving through an intergalactic medium.
* At the extreme a meteor can be a galaxy cluster moving relative to apparently stationary clusters in its neighborhood of the universe.
Here's another theoretical definition:
Def. an astronomy of meteors (as a form of radiation) is called meteor astronomy.
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.”

## Working definitions

A working definition is either chosen for an occasion and may not fully conform with established or authoritative definitions. Not knowing of established definitions would be grounds for selecting or devising a working definition. Or it refers to a definition being developed; a tentative definition that can be tailored to create an authoritative definition.

## Hypotheses

* A rigorous definition is not necessarily a functional definition.