Dichotomy

A dichotomy /daɪˈkɒtəmi/ is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be

• jointly exhaustive: everything must belong to one part or the other, and
• mutually exclusive: nothing can belong simultaneously to both parts.

This article is about dichotomy in logic and related topics. For usage of "dichotomous" in botany, see Glossary of botanical terms § dichotomous. For the album by Becoming the Archetype, see Dichotomy (album).

In this image, the universal set U (the entire rectangle) is dichotomized into the two sets A (in pink) and its complement A^c (in grey).

Such a partition is also frequently called a bipartition.

The two parts thus formed are complements. In logic, the partitions are opposites if there exists a proposition such that it holds over one and not the other.

Treating continuous variables or multicategorical variables as binary variables is called dichotomization. The discretization error inherent in dichotomization is temporarily ignored for modeling purposes.

Etymology

The term dichotomy is from the Greek language Greek: διχοτομία dichotomía "dividing in two" from δίχα dícha "in two, asunder" and τομή tomḗ "a cutting, incision".

Usage and examples

• The above applies directly when the term is used in mathematics, philosophy, literature, or linguistics. For example, if there is a concept A, and it is split into parts B and not-B, then the parts form a dichotomy: they are mutually exclusive, since no part of B is contained in not-B and vice versa, and they are jointly exhaustive, since they cover all of A, and together again give A.
• In set theory, a dichotomous relation R is such that either aRb, bRa, but not both.[1]
• In statistics, dichotomous data may only exist at first two levels of measurement, namely at the nominal level of measurement (such as "British" vs "American" when measuring nationality) and at the ordinal level of measurement (such as "tall" vs "short", when measuring height). A variable measured dichotomously is called a dummy variable.
• In the classification of mental disorders in psychiatry or clinical psychology, dichotomous classification or categorization refers to the use of cut-offs intended to separate disorder from non-disorder at some level of abnormality, severity or disability.
• A false dichotomy is an informal fallacy consisting of a supposed dichotomy which fails one or both of the conditions: it is not jointly exhaustive and/or not mutually exclusive. In its most common form, two entities are presented as if they are exhaustive, when in fact other alternatives are possible. In some cases, they may be presented as if they are mutually exclusive although there is a broad middle ground (see also undistributed middle).
• The divine dichotomy is mentioned in the Conversations With God series of books by religious author Neale Donald Walsch.
• In economics, the classical dichotomy is the division between the real side of the economy and the monetary side. According to the classical dichotomy, changes in monetary variables do not affect real values such as output, employment, and the real interest rate. Money is therefore neutral in the sense that its quantity cannot affect these real variables.
• In biology, a dichotomy is a division of organisms into two groups, typically based on a characteristic present in one group and absent in the other. Such dichotomies are used as part of the process of identifying species, as part of a dichotomous key, which asks a series of questions, each of which narrows down the set of organisms. A well known dichotomy is the question "does it have a backbone?" used to divide species into vertebrates and invertebrates.
• In botany, a dichotomy is a mode of branching by repeated bifurcation – thus a focus on branching rather than on division.
• In computer science, more specifically in programming-language engineering, dichotomies are fundamental dualities in a language's design. For instance, C++ has a dichotomy in its memory model (heap versus stack), whereas Java has a dichotomy in its type system (references versus primitive data types).
• In the anthropological field of theology and in philosophy, dichotomy is the belief that humans consist of a soul and a body. (See Mind-body dichotomy.) This stands in contrast to trichotomy.
• Perceived dichotomies are common in Western thought. C. P. Snow believes that Western society has become an argument culture (The Two Cultures). In The Argument Culture (1998), Deborah Tannen suggests that the dialogue of Western culture is characterized by a warlike atmosphere in which the winning side has truth (like a trophy). Such a dialogue virtually ignores the middle alternatives.
• In sociology and semiotics, dichotomies (also sometimes called 'binaries' and/or 'binarisms') are the subject of attention because they may form the basis to divisions and inequality. For example, the domestic–public dichotomy divides men's and women's roles in a society; the East-West dichotomy contrasts the Orient and the Occident. Some social scientists attempt to deconstruct dichotomies in order to address the divisions and inequalities they create: for instance Judith Butler's deconstruction of the gender-dichotomy (or gender binary) and Val Plumwood's deconstruction of the human-environment dichotomy.
• The I Ching and taijitu represent the yin yang theories of traditional Chinese culture. However, these do not represent a true dichotomy as the symbol incorporates a portion of each in the other, representing a dialectic.
• In dialectical behavioral therapy, a treatment shown to have some success in treating some clients with Borderline Personality Disorder, an essential tool used is based on the idea of dichotomy. Dichotomy, in this case, is a self-defeating behavior using "all-or-nothing" or "black-and-white" thinking. The therapy teaches the patient how to change the dichotomy to a more "dialectical" (or "seeing the middle ground") way of thinking.
• One type of dichotomy is dichotomous classification – classifying objects by recursively splitting them into two groups until all are separated and in their own unique category.
• Astronomy defines a dichotomy as "the phase of the moon or an inferior planet in which half its disk appears illuminated".

See also

• Binary opposition
• Bipartite
• Borderline personality disorder
• Class (set theory)
• Dialectical behavior therapy
• Dichotomy paradox
• Dilemma
• Law of excluded middle, which in logic asserts the existence of a dichotomy
• Mind–body dualism
• Polychotomy
• Trichotomy (disambiguation)
• Yin and yang
• Taxonomy (disambiguation)

Notes and references

1. Komjath, Peter; Totik, Vilmos (2006). Problems and Theorems in Classical Set Theory. Springer Science & Business Media. p. 497. ISBN 978-0-387-30293-5.

External links

• The dictionary definition of dichotomy at Wiktionary
• Media related to Dichotomies at Wikimedia Commons