Definition

A definition is a statement of the significance of a term (a word, expression, or other set of symbols). Definitions can be classified into two outsized categories, intensional definitions (which give the sense of a term) and extensional definitions (which list the objects that a term describes). An additional significant category of definitions is the class of ostensive definitions, which express the meaning of a term by pointing out examples. A term may have countless different senses and manifold meanings, and thus necessitate numerous definitions.

In mathematics, a definition is used to give a fixed meaning to a new term, by unfolding a form which decidedly qualifies what a mathematical term is and is not. Definitions and axioms form the foundation on which all of abstract mathematics is to be constructed.

Autological word

An autological word is a word that expresses a property that it also possesses. The opposite is a heterological word, one that does not apply to itself.

Unlike more universal concepts of autology and self-reference, this particular distinction and opposition of "autological" and "heterological words" is uncommon in linguistics for describing linguistic phenomena or classes of words, but is current in logic and philosophy for describing a semantic paradox.

Boundless Definition

A Boundless Definition is a oneness and a void that rejects restrictions in interpreting and retrocausal descriptions in reference to Boundless, forcibly encompassing this notion and empowering it.

(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. 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. A distinction is sometimes made between "true" homonyms, which are unrelated in origin, and polysemous homonyms, or polysemes, which have a shared origin.

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.

Circular definition

A circular definition is a description 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.

Comprehension

In logic, the comprehension of an object is the totality of intensions, that is, attributes, characters, marks, properties, or qualities, that the object possesses, or else the totality of intensions that are pertinent to the context of a given discussion.

This is the correct technical term for the whole collection of intensions of an object, but it is common in less technical usage to see 'intension' used for both the composite and the primitive ideas.

Concept creep

Concept creep is the process by which semantic expansion includes topics which would not have originally been envisaged to be included under that label. The phenomenon can be related to the concept of hyperbole.

Concept image and concept definition

Concept Image and Concept Definition are two ways of understanding a mathematical concept.

A concept definition is similar to the usual notion of a definition in mathematics, with the distinction that it is personal to an individual.
A personal concept definition can differ from a formal concept definition, the latter being a concept definition which is accepted by the mathematical community at large.

Coordinative definition

A coordinative definition is 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.

The idea was formulated by the logical positivists and arises out of a formalist vision of mathematics as pure symbol manipulation.

Definitional implication

Logic

Logical consequence (also entailment or logical implication), the relationship between statements that holds true when one logically "follows from" one or more others
Material conditional (also material consequence, or implication), a logical connective and binary truth function typically interpreted as "If p, then q"
- material implication (rule of inference), a logical rule of replacement
- Implicational propositional calculus, a version of classical propositional calculus which uses only the material conditional connective
Strict conditional or strict implication, a connective of modal logic that expresses necessity
modus ponens, or Implication elimination, a simple argument form and rule of inference summarized as "p implies q; p is asserted to be true, so therefore q must be true"

Linguistics

Implicature, what is suggested in an utterance, even though neither expressed nor strictly implied
Implicational universal or linguistic universal, a pattern that occurs systematically across natural languages
- Implicational hierarchy, a chain of implicational universals; if a language has one property then it also has other properties in the chain
Entailment (pragmatics) or strict implication, the relationship between two sentences where the truth of one requires the truth of the other

Definitionism

Definitionism (also called the theory of concepts) is the hierarchy of thought in which it is believed that a proper explanation of a theory consists of all the concepts used by that theory being well-defined.

This approach includes the dismissal of the importance of ostensive definitions.

Diairesis

Diairesis is a form of classification (especially Platonic) logic that serves to systematize concepts and come to definitions. When defining a concept using diairesis, one starts with a broad concept, then divides this into two or more specific sub-concepts, and this procedure is repeated until a definition of the desired concept is reached.

Definition by diaresis does not in itself prove anything. Apart from this definition, the procedure also results in a taxonomy of other concepts, ordered according to a general–specific relation.

Differentia

Logical meaning
In the original, logical sense, a differentia is a concept — the notion of "differentia" is a second-order concept, or a "second intention". It is a kind of essential predicate — a predicate that belongs to its subjects de re necessarily. It is distinguished against the possibilities by expressing the (specific) essence of the object only partially and against the genus by expressing the determining rather than the determined part of the essence.

Ontological meaning
Although the primary meaning of "differentia" is logical or second-order, it may under certain assumptions have an ontological, first-order application. If it is assumed that the structuring of an essence into "determining" and "determinable" metaphysical parts (which corresponding to a differentia and a genus respectively) exists in reality independently of its being conceived, one can apply the notion "differentia" also to the determining metaphysical part itself, and not just to the concept that expresses it.

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 for finite sets and only practical for relatively small sets.

Equals sign

The equality sign, is the mathematical symbol =, which is used to indicate equality in some well-defined sense. In an equation, it is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value.

Exemplification

Exemplification is a mode of symbolization characterized by the relation between a sample and what it refers to.

Reference is the relation between something "standing for" something else, like the relation between a word and what it denotes. Usually reference goes in one direction, from the word to what it denotes, but it may also go in both directions, from the denoted back to the word.

Extension

In any of the fields of existence that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question.

The 'extension' of a concept or expression is the set of things it extends to, or applies to, if it is the sort of concept or expression that a single object by itself can satisfy. Concepts and expressions of this sort are monadic or "one-place" concepts and expressions.

The extension of a whole statement, as opposed to a word or phrase, is defined as its truth value. So the extension of __ is the logical value 'true'.
Some concepts and expressions are such that they don't apply to objects individually, but rather serve to relate objects to objects.
"Relational" or "polyadic" ("many-place") concepts and expressions have, for their extension, the set of all sequences of objects that satisfy the concept or expression in question. So the extension of "before" is the set of all (ordered) pairs of objects such that the first one is before the second one.

Extensional and intensional definitions

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.

Intensional definition

An intensional definition gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used. In the case of nouns, this is equivalent to specifying the properties that an object needs to have in order to be counted as a referent of the term.

Intensional definitions are best used when something has a clearly defined set of properties, and they work well for terms that have too many referents to list in an extensional definition. It is impossible to give an extensional definition for a term with an infinite set of referents, but an intensional one can often be stated concisely – there are infinitely many even numbers, impossible to list, but the term "even numbers" can be defined easily by saying that even numbers are integer multiples of two.

Definition by genus and difference, in which something is defined by first stating the broad category it belongs to and then distinguished by specific properties, is a type of intensional definition.
An intensional definition may also consist of rules or sets of axioms that define a set by describing a procedure for generating all of its members.
An intensional definition of a game, such as chess, would be the rules of the game; any game played by those rules must be a game of chess, and any game properly called a game of chess must have been played by those rules.

Extensional definition

An extensional definition gives meaning to a term by specifying its extension, that is, every object that falls under the definition of the term in question.

An explicit listing of the extension, which is only possible for finite sets and only practical for relatively small sets, is a type of enumerative definition.

Extensional definitions are used when listing examples would give more applicable information than other types of definition, and where listing the members of a set tells the questioner enough about the nature of that set.

An extensional definition possesses similarity to an ostensive definition, in which one or more members of a set (but not necessarily all) are pointed to as examples, but contrasts clearly with an intensional definition, which defines by listing properties that a thing must have in order to be part of the set captured by the definition.

Genus

A Genus is one of the Predicables. Genus is that part of a definition which is also predicable of other things different from the definiendum. A triangle is a rectilinear figure; i.e. in fixing the genus of a thing, we subsume it under a higher universal, of which it is a species.

Genus–differentia definition

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

a genus (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 definition that is not provided by the genus.

The process of producing new definitions by extending existing definitions is commonly known as differentiation (and also as derivation). The reverse process, by which just part of an existing definition is used itself as a new definition, is called abstraction; the new definition is called an abstraction and it is said to have been abstracted away from the existing definition.

Indeterminacy

Indeterminacy can refer both to common systematic and mathematical concepts of uncertainty and their implications and to another kind of indeterminacy deriving from the nature of definition or meaning.

The problem of indeterminacy arises when one observes the eventual circularity of virtually every possible definition. It is easy to find loops of definition in any dictionary, because this seems to be the only way that certain concepts, and generally very important ones such as that of existence. A definition is a collection of other words and if one continues to follow the trail of words in search of the precise meaning of any given term, one will inevitably encounter this linguistic indeterminacy.

Intension

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. A comprehension is the collection of all such intensions.

The meaning of a word can be thought of as the bond between the idea the word means and the physical form of the word.

Without intension of some sort, a word has no meaning. Terms may be suggestive, but a term can be suggestive without being meaningful. For certain terms, it may be argued, that they are always intensional since they connote the property 'meaningless term', but this is only an apparent paradox and does not constitute a counterexample to the claim that without intension a word has no meaning. Part of its intension is that it has no extension.

Let-bound

Let-bound is an assignment statement of sets and/or re-sets of the value stored in possibility and actuality denoted by a variable name; in other words, it copies a value into the variable. In most imperative indoctrination languages, the assignment statement (or expression) is a fundamental construct.

In some languages, Let-bound is regarded as an operator (meaning that the assignment statement as a whole returns a value). Other languages define assignment as a statement (meaning that it cannot be used in an expression).

Assignments typically allow a variable to hold different values at different times during its life-span and scope. However, some languages (primarily strictly functional languages) do not allow that kind of "destructive" reassignment, as it might imply changes of non-local state. The purpose is to enforce referential transparency, i.e. functions that do not depend on the state of some variable(s), but produce the same results for a given set of parametric inputs at any point in time.

Lexical definition

The lexical definition of a term is the definition closely matching the meaning of the term in common usage. A lexical definition is usually the type expected from a request for definition, and it is generally expected that such a definition will be stated as simply as possible in order to convey information to the widest audience.

Note that a lexical definition is descriptive, reporting actual usage within speakers of a language, and changes with changing usage of the term, rather than prescriptive, which would be to stick with a version regarded as "correct", regardless of drift in accepted meaning. They tend to be inclusive; attempting to capture everything the term is used to refer to, and as such is often too vague for many purposes.

When the breadth or vagueness of a lexical definition is unacceptable, a precising definition or a stipulative definition is often used.

Words can be classified as lexical or nonlexical. Lexical words are those that have independent meaning (such as a Noun (N), verb (V), adjective (A), adverb (Adv), or preposition (P).

The definition which reports the meaning of a word or a phrase as it is actually used by people is called a lexical definition. Meanings of words given in a dictionary are lexical definitions. As a word may have more than one meaning, it may also have more than one lexical definition.

Lexical definitions are either true or false. If the definition is the same as the actual use of the word then it is true, otherwise it is false.

Name binding

Name-binding is the association of entities (data and/or code) with identifiers. An identifier bound to an object is said to reference that object. Binding is intimately connected with scoping, as scope determines which names bind to which objects – at which locations in the system (lexically) and in which one of the possible execution paths (temporally).

Use of an identifier id in a context that establishes a binding for id is called a binding (or defining) occurrence. In all other occurrences (e.g., in expressions, assignments, and subprogram calls), an identifier stands for what it is bound to; such occurrences are called applied occurrences.

Operational definition

An operational definition specifies existing, replicable procedures designed to represent a construct. An operation is the performance which we execute in order to make known a concept.

Ostensive definition

An 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 new speakers of a language) or because of the nature of the term (such qualia). It is usually accompanied with a gesture pointing to the object serving as an example, and for this reason is also often referred to as "definition by pointing".

Paradox of analysis

A conceptual analysis is something like the definition of a word. However, unlike a standard lexicon definition (which may list examples or talk about related terms as well), a completely correct analysis of a concept in terms of others seems like it should have exactly the same meaning as the original concept. Thus, in order to be correct, the analysis should be able to be used in any context where the original concept is used, without changing the meaning of the discussion in context.

However, if such an analysis is to be useful, it should be informative. That is, it should tell us something we don't already know (or at least, something one can imagine someone might not already know). But it seems that no conceptual analysis can both meet the requirement of correctness and of informativeness, on these understandings of the requirements.

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 create or alter rights, duties or crimes.

The terms thus defined will often involve emotionally charged but imprecise notions, such as "freedom", "violence", "equality", etc. In argumentation the use of a persuasive definition is sometimes called definist fallacy.

A persuasive definition of a term is favorable to one argument or unfavorable to the other argument, but is presented as if it were neutral and well-accepted, and the listener is expected to accept such a definition without question.

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.

Precising definitions are generally used in contexts where vagueness is unacceptable; many legal definitions are precising definitions, as are company policies. This type of definition is useful in preventing disputes that arise from the involved parties using different definitions of the term in question.

A precising definition is intended to make a vague word more precise so that the word's meaning is not left to the interpretation of the reader or listener.

Recursive definition

A Recursive definition is a definition of a function permitting values of the function to be calculated systematically in a series of steps. Most recursive definitions have two foundations: a base case (basis) and an inductive clause.

The difference between a circular definition and a recursive definition is that a recursive definition must always have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and that all other instances in the inductive clauses must be "smaller" in some sense (i.e., closer to those base cases that terminate the recursion) — a rule also known as "recur only with a simpler case".
In contrast, a circular definition may have no base case, and even may define the value of a function in terms of that value itself — rather than on other values of the function. Such a situation would lead to an infinite regress.

Recursive definitions are valid – meaning that a recursive definition identifies a unique function – is a theorem of set theory known as the recursion theorem, the proof of which is non-trivial.
More normally, recursive definitions of functions can be made whenever the domain is a well-ordered set, using the principle of transfinite recursion.
The formal criteria for what constitutes a valid recursive definition are more complex for the general case.

Stipulative definition

A stipulative definition is a type of definition in which a new or presently existing term is given a new specific meaning for the purposes of argument or discussion in a given context. When the term already exists, this definition may, but does not necessarily, contradict the lexical definition of the term. Because of this, a stipulative definition cannot be "correct" or "incorrect"; it can only differ from other definitions, but it can be useful for its intended purpose.

Theoretical definition

A theoretical definition defines a term in an intellectual discipline, functioning as an application to see a phenomenon in a certain way. A theoretical definition is a projected way of thinking about potentially related events.

Theoretical definitions contain built-in theories; they cannot be simply reduced to describing a set of observations.
The definition may contain implicit inductions and deductive consequences that are part of the theory.
A theoretical definition of a term can change, over time, based on the methods in the field that created it.

Without a falsifiable operational definition, conceptual definitions assume both knowledge and acceptance of the theories that it depends on. A hypothetical construct may serve as a theoretical definition, as can a stipulative definition.

Transcendental Singularity

Transcendental Singularity is the wholeness of be-ness constantly surpassed. Transcendental Singularity is the total and absolute control over the wholeness of be-ness without any limit of any kind or truth-type. Transcendental Singularity is beyond the wholeness of physics and logic and no rule whatsoever can possibly apply to Transcendental Singularity, other than its own. Transcendental Singularity perfectly adapts to everything that can be linked to Possibility, Totality, and Nothingness; completely rewriting itself on every level at any time, giving itself each and every explanation preceding existence that precedes the essence of what the wholeness is and could be, simultaneously absorbing the be-ness of its wholeness.

Well-defined expression

A well-defined expression or unambiguous expression (including in Mathematics) is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill-defined, or ambiguous.

A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input.
The term well-defined can also be used to indicate that a logical expression is unambiguous or contradictory.
A function that is not well defined is not the same as a function that is undefined.

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