Circular definition

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. There are several kinds of circular definition, and several ways of characterising the term: pragmatic, lexicographic and linguistic.
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.
Approaches to characterizing circular definitions.
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:
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.
Linguistic.
From a linguistic point of view, some intuitively circular definitions in the derivation of words can easily be shown to be non-circular. For example, sometimes a definition like "musicality: the quality or state of being musical" is said to be circular. But strictly speaking, the condition "the term(s) being defined as a part of the definition or assumes a prior understanding of the term being defined" is false in this case. The definition chain "musicality: the quality or state of being musical" → "musical: associated with music" → "music: an acoustic art form" is a two-step derivation of the word "musicality" from the root "music", where the chain ends. A definition chain which ends is not circular. It may be objected that the term to be defined and one of the difference terms in the definition share the same root; the answer is that the objection requires prior analysis of the terms in order to identify identical parts, yet the terms themselves cannot be reduced to these parts: the meaning of "musicality" is composed of the meaning of "musical" and the meaning of "ity", the meaning of "musical" is composed of the meaning of "music" and "al". In each case, the terms to be defined and the terms in the definition are different.
Mathematical.
Formal approaches to characterizing circular definitions are found in logic, mathematics and in computer science. 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.
Circular lexicographic (dictionary) definitions.
Dictionary entries are often given as examples of apparent circular definitions. 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, 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. 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) 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 dictionary defines a "hill" and a "mountain" this way:
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:
The following definition of the adjective "inspiring" appears to be a circular definition, but in the phrase "effect of inspiring someone", the word "inspiring" is actually the inflected verb "inspire", which is defined without using the adjective "inspiring":