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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 catkins 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 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.
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"
- 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
This definition alone is close to suffering from circular definition, but following the definition train:
- Main Entry: 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
- 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.
See also 
- Fallacies of definition
- Begging the question
- Meta-circular evaluator
- Infinite regress
- Michael Silverstein, "Old Wine, New Ethnographic Lexicography". Annual Review of Anthropology, 35:486-7.
- Philip Seargeant, "Lexicography as a Philosophy of Language". Language Sciences, 33:1-10 (2011).
- "Hill". Merriam-Webster. Retrieved January 17, 2013.
- "Mountain". Merriam-Webster. Retrieved January 17, 2013.
- "condescending". Merriam-Webster. Retrieved January 17, 2013.
- "condescension". Merriam-Webster. Retrieved January 17, 2013.
- "patronizing". Merriam-Webster. Retrieved January 17, 2013.