Circular definition
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A circular definition is a type of
definition A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...
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 Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds ...
. Circular definitions are related to
circular reasoning Circular reasoning (, "circle in proving"; also known as circular logic) is a fallacy, logical fallacy in which the reasoner begins with what they are trying to end with. Circular reasoning is not a formal logical fallacy, but a pragmatic defect ...
in that they both involve a
self-referential Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields. In natural language, natural or formal languages, ...
approach. Circular definitions may be unhelpful if the audience must either already know the meaning of the key term, or if the term to be defined is used in the definition itself. In linguistics, a circular definition is a description of the meaning of a
lexeme A lexeme () is a unit of lexical meaning that underlies a set of words that are related through inflection. It is a basic abstract unit of meaning, a unit of morphological analysis in linguistics that roughly corresponds to a set of forms ta ...
that is constructed using one or more synonymous lexemes that are all defined in terms of each other.


Approaches to characterizing circular definitions


Classical

In
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
's ''
Meno ''Meno'' (; , ''Ménōn'') is a Socratic dialogue written by Plato around 385 BC., but set at an earlier date around 402 BC. Meno begins the dialogue by asking Socrates whether virtue (in , '' aretē'') can be taught, acquired by practice, o ...
'', several definitions of
virtue A virtue () is a trait of excellence, including traits that may be morality, moral, social, or intellectual. The cultivation and refinement of virtue is held to be the "good of humanity" and thus is Value (ethics), valued as an Telos, end purpos ...
are offered, the third of which states that virtue is the power of attaining good justly or with justice, where justice has already been defined as a form or aspect of virtue.
Socrates Socrates (; ; – 399 BC) was a Ancient Greek philosophy, Greek philosopher from Classical Athens, Athens who is credited as the founder of Western philosophy and as among the first moral philosophers of the Ethics, ethical tradition ...
, in response to Meno, states that the process of finding a definition will have to be started again because the wording offered defines virtue as a part of virtue.


Pragmatic

From a pragmatic point of view, circular definitions may be characterised in terms of new, useful or helpful information: a definition is deficient if the audience must either already know the meaning of the key term, or if the term to be defined is used in the definition itself. Such definitions lead to a need for additional information that motivated someone to look at the definition in the first place and, thus, violate the principle of providing new or useful information. Here are some examples: * Suppose we define "
oak An oak is a hardwood tree or shrub in the genus ''Quercus'' of the beech family. They have spirally arranged leaves, often with lobed edges, and a nut called an acorn, borne within a cup. The genus is widely distributed in the Northern Hemisp ...
" as a
tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only ...
which has
catkin A catkin or ament is a slim, cylindrical flower cluster (a spike), with inconspicuous or no petals, usually wind- pollinated ( anemophilous) but sometimes insect-pollinated (as in '' Salix''). It contains many, usually unisexual flowers, arra ...
s and grows from an acorn, and then define "acorn" as the nut produced by an oak tree. To someone who does not know which trees are oaks, nor which nuts are acorns, the definition is inadequate. * If someone wants to know what a
cellular phone A mobile phone or cell phone is a portable telephone that allows users to make and receive calls over a radio frequency link while moving within a designated telephone service area, unlike fixed-location phones ( landline phones). This radi ...
is, telling them that it is a "phone that is cellular" will not be especially illuminating. Much more helpful would be to explain the concept of a cell in the context of telecommunications, or at least to make some reference to portability. * Similarly, defining
dialectical materialism Dialectical materialism is a materialist theory based upon the writings of Karl Marx and Friedrich Engels that has found widespread applications in a variety of philosophical disciplines ranging from philosophy of history to philosophy of scien ...
as "materialism that involves dialectic" is unhelpful. Consequently, when constructing systems of definitions, authors should use good practices that avoid producing viciously circular definitions. In many learner's dictionaries, circular definitions are greatly reduced by writing definitions using only the words in a constrained defining vocabulary.


Lexicographic

From a lexicographic point of view, the simplest form of circular definition in a dictionary is in terms of synonyms, and the number of steps for closing the definition chain into a circle is known as the ''depth'' of the circular definition: the circular definition "object: a thing" → "thing: an object" is a circular definition with a depth of two. The circular definition "object: a thing" → "thing: an entity" → "entity: an object" has a depth of three. The classic "genus-difference" dictionary definition is in terms of nearest kind (''genus proximum'') and specific differences (''differentia specifica''). This genus-difference description may be involved in producing circular definitions of part and kind relationships, for example: "rake: an implement with three or more tines" → "tine: a part of a rake". However, if more specific differences are added, then the effect of circularity may disappear: "rake: a gardening implement with a long handle with three or more tines arranged on crossbar at 90° to the handle and the tines at 90° to both crossbar and handle"; in this case, "tine" is most usefully defined with reference to "rake", but with additional differences providing points of comparison, e.g.: "tine: a sharp spike at the end of a rake". In practice, a pragmatic approach is often taken in considering the effects of circularity in dictionary definitions.


Circular lexicographic (dictionary) definitions

Dictionary entries are often given as examples of apparent circular definitions. Dictionary production, as a project in
lexicography Lexicography is the study of lexicons and the art of compiling dictionaries. It is divided into two separate academic disciplines: * Practical lexicography is the art or craft of compiling, writing and editing dictionaries. * Theoretical le ...
, should not be confused with a
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
or
logical Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arg ...
activity, where giving a definition for a word is similar to providing an explanans for an explanandum in a context where practitioners are expected to use a
deductive system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...
. While, from a linguistic prescriptivist perspective, any
dictionary A dictionary is a listing of lexemes from the lexicon of one or more specific languages, often arranged Alphabetical order, alphabetically (or by Semitic root, consonantal root for Semitic languages or radical-and-stroke sorting, radical an ...
might be believed to dictate correct usage, the linguistic descriptivist perspective recognizes that looking up words in dictionaries is not itself a rule-following practice independent of the give-and-take of using words in context.Philip Seargeant, "Lexicography as a Philosophy of Language". ''Language Sciences'', 33:1-10 (2011). Thus, the example of a definition of oak given above (something that has catkins and grows from acorns) is not completely useless, even if "acorn" and "catkin" are defined in terms of "oak", in that it supplies additional concepts (e.g., the concept of catkin) in the definition. While a dictionary might produce a "circle" among the terms, "oak", "catkin", and "acorn", each of these is used in different (e.g., those related to plants, trees, flowers, and seeds) that generate ever-branching networks of usages. In another case it might produce a true circle. Taken as a whole, dictionaries are circular because each and every word is defined in terms of words that are also contained within the dictionary. () Definitions in lexicography can be broadly or narrowly circular. Narrowly circular definitions simply define one word in terms of another. A broadly circular definition has a larger circle of words. For example, the definition of the primary word is defined using two other words, which are defined with two other words, etc., creating a definitional chain. This can continue until the primary word is used to define one of the words used in the chain, closing the wide circle of terms. If all definitions rely on the definitions of other words in a very large, but finite chain, then all text-based definitions are ultimately circular.
Extension (semantics) 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 — the extension of a concept, idea, or sign consists of the things to which it ap ...
to the actual things that referring terms like nouns stand for, provided that agreement on reference is accomplished, is one method of breaking this circularity, but this is outside the capacity of a text-based definition.


Examples of narrowly circular definitions in dictionaries

The 2007
Merriam-Webster Merriam-Webster, Incorporated is an list of companies of the United States by state, American company that publishes reference work, reference books and is mostly known for Webster's Dictionary, its dictionaries. It is the oldest dictionary pub ...
dictionary defines a "hill" and a "mountain" this way: : hill - "1: a usually rounded natural elevation of land lower than a mountain" : mountain - "1a: a landmass that projects conspicuously above its surroundings and is higher than a hill" Merriam-Webster's online dictionary provides another example of a circular definition with the words "condescending" and "patronizing:" : Main Entry: condescending : Function: adjective : 1 : showing or characterized by condescension: patronizing From "condescension": : Main Entry: condescension : Function: noun : 1 : voluntary descent from one's rank or dignity in relations with an inferior : 2 : patronizing attitude or behavior To "patronize": : Main Entry: patronize : Function: transitive verb : 1 : to act as patron of: provide aid or support for : 2 : to adopt an air of condescension toward: treat haughtily or coolly From the Oxford Dictionary of English: * Punishment – "a ''penalty'' inflicted as ''retribution'' for an offence". * ''Penalty'' – "a punishment imposed for breaking a law, rule, or contract". * ''Retribution'' – "punishment inflicted on someone as vengeance for a wrong or criminal act". * Vengeance – "punishment inflicted or retribution exacted for an injury or wrong". Therefore, a punishment means "a ''punishment imposed for breaking a law, rule, or contract'' inflicted as ''punishment inflicted on someone as'' punishment inflicted or retribution exacted for an injury or wrong ''for a wrong or criminal act'' for an offence". Obviously, this is not the final result of substitution as this would result in an endlessly long sentence.


Mathematical theory

Formal approaches to characterizing circular definitions are found in
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
,
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and in
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
. A branch of mathematics called
non-well-founded set theory Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axio ...
allows for the construction of circular sets. Circular sets are good for modelling cycles, and, despite the field's name, this area of mathematics is well founded.
Computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
allows for procedures to be defined by using
recursion Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in m ...
. Such definitions are not circular as long as they terminate.


Linguistics

Linguistically, a circular definition is a description of the meaning of a
lexeme A lexeme () is a unit of lexical meaning that underlies a set of words that are related through inflection. It is a basic abstract unit of meaning, a unit of morphological analysis in linguistics that roughly corresponds to a set of forms ta ...
that is constructed using one or more synonymous lexemes that are all defined in terms of each other.


See also

*
Begging the question In classical rhetoric and logic, begging the question or assuming the conclusion (Latin: ) is an informal fallacy that occurs when an argument's premises assume the truth of the conclusion. Historically, begging the question refers to a fault i ...
*
Circular reasoning Circular reasoning (, "circle in proving"; also known as circular logic) is a fallacy, logical fallacy in which the reasoner begins with what they are trying to end with. Circular reasoning is not a formal logical fallacy, but a pragmatic defect ...
* Fallacies of definition * Genus–differentia definition *
Infinite regress Infinite regress is a philosophical concept to describe a series of entities. Each entity in the series depends on its predecessor, following a recursive principle. For example, the epistemic regress is a series of beliefs in which the justi ...
*
Lexicography Lexicography is the study of lexicons and the art of compiling dictionaries. It is divided into two separate academic disciplines: * Practical lexicography is the art or craft of compiling, writing and editing dictionaries. * Theoretical le ...
*
Lexical definition The lexical definition of a term, also known as the dictionary definition, is the definition closely matching the meaning of the term in common usage. As its other name implies, this is the sort of definition one is likely to find in the dictiona ...
* Meta-circular evaluator *
Self-reference Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields. In natural or formal languages, self-reference ...
*
Self-refuting idea A self-refuting idea or self-defeating idea is an idea or statement whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are called self-refuting by their detractors, and such accusations are ther ...
*
Recursive definition In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively definable objects include fact ...
* Revision theory * Tautology *
Vicious circle principle The vicious circle principle is a principle that was endorsed by many predicativist mathematicians in the early 20th century to prevent contradictions. The principle states that no object or property may be introduced by a definition that depen ...


References

{{DEFAULTSORT:Circular Definition Definition Philosophical logic