Circular definition
Encyclopedia
A circular definition is one that uses the term(s) being defined as a part of the definition or assumes a prior understanding of the term being defined. Either the audience must already know the meaning of the key term(s), or the definition is deficient in including the term(s) to be defined 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. If someone wants to know what a cellular phone 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
as "materialism that involves dialectic" is unhelpful. For another example, we can define "oak
" as a tree
which has catkin
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. Consequently, many systems of definitions are constructed according to the vicious circle principle
in such a way that authors do not produce viciously circular definitions.
A circular definition occurred in an early definition of the kilogram. The kilogram was originally defined as the mass
of one liter
of water
at standard pressure and the temperature at which it is densest (which is about 4 °C). The unit of pressure
is the newton per square meter, where a newton is the force that accelerates one kilogram one meter per second squared. Thus the kilogram was defined in terms of itself. Since water is nearly incompressible, this circularity is of no consequence — with each iteration of the "circle," the resulting measure of a kilogram rapidly converges. Even so, to clear up any confusion, the kilogram was later defined as the mass of a certain piece of metal in Sèvres
.
A circular definition also crept into the classic definition of death that was once "the permanent cessation of the flow of vital bodily fluids", which raised the question "what makes a fluid vital?"
A branch of mathematics called non-well-founded set theory
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
allows for procedures to be defined by using recursion
. Such definitions are not circular as long as they terminate.
Dictionaries are sometimes used erroneously as sources for examples of circular definition. Dictionary production, as a project in lexicography
, should not be confused with a mathematical or logical 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
.. While, from a linguistic prescriptivist perspective, any dictionary
might be believed to dictate correct usage, linguists recognize that looking up words in dictionaries is not itself a rule-following practice independent of the give-and-take of using words in context.. 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 are used in contexts (e.g., those related to plants, trees, flowers, and seeds) that generate an ever-branching network of usages.
Definitions 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)
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.
Merriam-Webster's online dictionary provides another example of a circular definition with the words "condescending" and "patronizing:"
This definition alone is close to suffering from circular definition, but following the definition train:
Looking up the word "patronizing" then gives us:
In short: the two words define each other.
This is a parody on circular references in dictionaries, which are sometimes understood to be explanatory, rather than descriptive. Jokes often have an element of wisdom: In some cases, dictionary descriptions lead to apparent circular definitions among related words. However, jokes also can have an element of misunderstanding: This parody is the shortest possible example of an erroneous recursive
definition of an object, the error being the absence of the termination condition (or lack of the initial state, if looked at from an opposite point of view). A purer example of circular reference would be "Circle: See 'Circle'". The parody is also an erroneous example of the activity of giving a definition in a dictionary, where the more general error it makes is in mistaking dictionaries to involve procedures that are found in logical or mathematical contexts. Dictionaries are not self-contained texts, nor is their use expected to be so contained. If modeled on the practice of using dictionaries, there would be no circle in an illustration of the activity of looking up a word whose entry provides a definition in terms of that word (or in terms of another word defined in terms of this word). For this joke to be an example of giving a definition as modeled on dictionary usage, the practice of using dictionaries would have to be self-contained, would have to involve a function—say, a "look-up" procedure—that a computer can perform. If dictionaries were logico-mathematical texts, then so-called circular definition would amount to infinite regress
, where one of the steps involved in running the procedure is to run the procedure; and, in the context of explanation (as opposed to description in the form of dictionary definition), this would be a vicious infinite regress. Newcomers to recursion are often bewildered by its apparent circularity, until they learn to appreciate that a termination condition is key.
Dialectical materialism
Dialectical materialism is a strand of Marxism synthesizing Hegel's dialectics. The idea was originally invented by Moses Hess and it was later developed by Karl Marx and Friedrich Engels...
as "materialism that involves dialectic" is unhelpful. For another example, we can define "oak
Oak
An oak is a tree or shrub in the genus Quercus , of which about 600 species exist. "Oak" may also appear in the names of species in related genera, notably Lithocarpus...
" as a tree
Tree
A tree is a perennial woody plant. It is most often defined as a woody plant that has many secondary branches supported clear of the ground on a single main stem or trunk with clear apical dominance. A minimum height specification at maturity is cited by some authors, varying from 3 m to...
which has catkin
Catkin
A catkin or ament is a slim, cylindrical flower cluster, with inconspicuous or no petals, usually wind-pollinated but sometimes insect pollinated . They contain many, usually unisexual flowers, arranged closely along a central stem which is often drooping...
s and grows from an acorn, and then define "acorn" as the nut
Nut (fruit)
A nut is a hard-shelled fruit of some plants having an indehiscent seed. While a wide variety of dried seeds and fruits are called nuts in English, only a certain number of them are considered by biologists to be true nuts...
produced by an oak tree. To someone who does not know which trees are oaks, nor which nuts are acorns, the definition
Definition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...
is inadequate. Consequently, many systems of definitions are constructed according to the vicious circle principle
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 depends on that object or property itself...
in such a way that authors do not produce viciously circular definitions.
A circular definition occurred in an early definition of the kilogram. The kilogram was originally defined as the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of one liter
Litér
- External links :*...
of water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
at standard pressure and the temperature at which it is densest (which is about 4 °C). The unit of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
is the newton per square meter, where a newton is the force that accelerates one kilogram one meter per second squared. Thus the kilogram was defined in terms of itself. Since water is nearly incompressible, this circularity is of no consequence — with each iteration of the "circle," the resulting measure of a kilogram rapidly converges. Even so, to clear up any confusion, the kilogram was later defined as the mass of a certain piece of metal in Sèvres
Sèvres
Sèvres is a commune in the southwestern suburbs of Paris, France. It is located from the centre of Paris.The town is known for its porcelain manufacture, the Manufacture nationale de Sèvres, making the famous Sèvres porcelain, as well as being the location of the International Bureau of Weights...
.
A circular definition also crept into the classic definition of death that was once "the permanent cessation of the flow of vital bodily fluids", which raised the question "what makes a fluid vital?"
A branch of mathematics called non-well-founded set theory
Non-well-founded set theory
Non-well-founded set theories are variants of axiomatic set theory which allow sets to contain themselves and otherwise violate the rule of well-foundedness...
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
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
allows for procedures to be defined by using recursion
Recursion (computer science)
Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem. The approach can be applied to many types of problems, and is one of the central ideas of computer science....
. Such definitions are not circular as long as they terminate.
Dictionaries are sometimes used erroneously as sources for examples of circular definition. Dictionary production, as a project in lexicography
Lexicography
Lexicography is divided into two related disciplines:*Practical lexicography is the art or craft of compiling, writing and editing dictionaries....
, should not be confused with a mathematical or logical activity, where giving a definition for a word is similar to providing an explanans for an explanandum
Explanandum
An explanandum is a phenomenon that needs to be explained and its explanans is the explanation of that phenomenon. For example, one person may pose an explanandum by asking "Why is there smoke?", and another may provide an explanans by responding "Because there is a fire"...
in a context where practitioners are expected to use a deductive system
Deductive system
A deductive system consists of the axioms and rules of inference that can be used to derive the theorems of the system....
.. While, from a linguistic prescriptivist perspective, any dictionary
Dictionary
A dictionary is a collection of words in one or more specific languages, often listed alphabetically, with usage information, definitions, etymologies, phonetics, pronunciations, and other information; or a book of words in one language with their equivalents in another, also known as a lexicon...
might be believed to dictate correct usage, linguists recognize that looking up words in dictionaries is not itself a rule-following practice independent of the give-and-take of using words in context.. 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 are used in contexts (e.g., those related to plants, trees, flowers, and seeds) that generate an ever-branching network of usages.
Definitions 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)
Extension (semantics)
In any of several studies that treat the use of signs - for example, in linguistics, logic, mathematics, semantics, and semiotics - 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...
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
The 2007 Merriam-Webster dictionary defines a "hill" and a "mountain" this way:- Hill - "1: a usually rounded natural elevation of land lower than a mountain" http://www.m-w.com/dictionary/hill
- Mountain - "1a: a landmass that projects conspicuously above its surroundings and is higher than a hill" http://www.m-w.com/dictionary/mountain
Merriam-Webster's online dictionary provides another example of a circular definition with the words "condescending" and "patronizing:"
- Main Entry: condescending http://www.merriam-webster.com/dictionary/condescending
- Function: adjective
- 1 : showing or characterized by condescension : patronizing
This definition alone is close to suffering from circular definition, but following the definition train:
- Main Entry: condescension http://www.merriam-webster.com/dictionary/condescension
- Function: noun
- 1 : voluntary descent from one's rank or dignity in relations with an inferior
- 2 : patronizing attitude or behavior
Looking up the word "patronizing" then gives us:
- Main Entry: patronize http://www.merriam-webster.com/dictionary/patronizing
- 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
In short: the two words define each other.
Humorous Recursive Definition, so-called
A common joke is the following "definition" of recursion.- Recursion
- See "Recursion".
This is a parody on circular references in dictionaries, which are sometimes understood to be explanatory, rather than descriptive. Jokes often have an element of wisdom: In some cases, dictionary descriptions lead to apparent circular definitions among related words. However, jokes also can have an element of misunderstanding: This parody is the shortest possible example of an erroneous recursive
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
definition of an object, the error being the absence of the termination condition (or lack of the initial state, if looked at from an opposite point of view). A purer example of circular reference would be "Circle: See 'Circle'". The parody is also an erroneous example of the activity of giving a definition in a dictionary, where the more general error it makes is in mistaking dictionaries to involve procedures that are found in logical or mathematical contexts. Dictionaries are not self-contained texts, nor is their use expected to be so contained. If modeled on the practice of using dictionaries, there would be no circle in an illustration of the activity of looking up a word whose entry provides a definition in terms of that word (or in terms of another word defined in terms of this word). For this joke to be an example of giving a definition as modeled on dictionary usage, the practice of using dictionaries would have to be self-contained, would have to involve a function—say, a "look-up" procedure—that a computer can perform. If dictionaries were logico-mathematical texts, then so-called circular definition would amount to infinite regress
Infinite regress
An infinite regress in a series of propositions arises if the truth of proposition P1 requires the support of proposition P2, the truth of proposition P2 requires the support of proposition P3, .....
, where one of the steps involved in running the procedure is to run the procedure; and, in the context of explanation (as opposed to description in the form of dictionary definition), this would be a vicious infinite regress. Newcomers to recursion are often bewildered by its apparent circularity, until they learn to appreciate that a termination condition is key.
See also
- Fallacies of definitionFallacies of definitionFallacies of definition refer to the various ways in which definitions can fail to have merit. The term is used to suggest analogy with the logical fallacies...
- Begging the questionBegging the questionBegging the question is a type of logical fallacy in which the proposition to be proven is assumed implicitly or explicitly in the premise....
- TautologyTautology (logic)In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
- Self-referenceSelf-referenceSelf-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
- Meta-circular evaluatorMeta-circular evaluatorA meta-circular evaluator is a special case of a self-interpreter in which the existing facilities of the parent interpreter are directly applied to the source code being interpreted, without any need for additional implementation...
- Infinite regressInfinite regressAn infinite regress in a series of propositions arises if the truth of proposition P1 requires the support of proposition P2, the truth of proposition P2 requires the support of proposition P3, .....