# Dichotomy

Last updated

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

## Contents

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 : διχοτομίαdichotomía "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 Mandaeism, people, spirits, and places are often considered to have both earthly and heavenly counterparts. Also, a successful masiqta (death mass ritual) merges the incarnate soul (Classical Mandaic : ࡍࡉࡔࡉࡌࡕࡀ nišimta ) and spirit (Classical Mandaic : ࡓࡅࡄࡀ ruha ) from the Earth (Tibil) into a new merged entity in the World of Light called the ʿuṣṭuna. The ʿuṣṭuna can then reunite with its heavenly, non-incarnate counterpart (or spiritual image), the dmuta , in the World of Light, where it will reside in the world of ideal counterparts ( Mšunia Kušṭa ). [2]
• 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". [3]

## Related Research Articles

In logic, the law of non-contradiction (LNC) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "p is the case" and "p is not the case" are mutually exclusive. Formally this is expressed as the tautology ¬(p ∧ ¬p). The law is not to be confused with the law of excluded middle which states that at least one, "p is the case" or "p is not the case" holds.

A false dilemma, also referred to as false dichotomy, is an informal fallacy based on a premise that erroneously limits what options are available. The source of the fallacy lies not in an invalid form of inference but in a false premise. This premise has the form of a disjunctive claim: it asserts that one among a number of alternatives must be true. This disjunction is problematic because it oversimplifies the choice by excluding viable alternatives. For example, a false dilemma is committed when it is claimed that, "Stacey spoke out against capitalism; therefore, she must be a communist". One of the options excluded is that Stacey may be neither communist nor capitalist. False dilemmas often have the form of treating two contraries, which may both be false, as contradictories, of which one is necessarily true. Various inferential schemes are associated with false dilemmas, for example, the constructive dilemma, the destructive dilemma or the disjunctive syllogism. False dilemmas are usually discussed in terms of deductive arguments. But they can also occur as defeasible arguments. Our liability to commit false dilemmas may be due to the tendency to simplify reality by ordering it through either-or-statements, which is to some extent already built into our language. This may also be connected to the tendency to insist on clear distinction while denying the vagueness of many common expressions.

In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."

In logic and probability theory, two events are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both.

Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement originated in psychology and is widely criticized by scholars in other disciplines. Other classifications include those by Mosteller and Tukey, and by Chrisman.

In the analysis of multivariate observations designed to assess subjects with respect to an attribute, a Guttman scale is a single (unidimensional) ordinal scale for the assessment of the attribute, from which the original observations may be reproduced. The discovery of a Guttman scale in data depends on their multivariate distribution's conforming to a particular structure. Hence, a Guttman scale is a hypothesis about the structure of the data, formulated with respect to a specified attribute and a specified population and cannot be constructed for any given set of observations. Contrary to a widespread belief, a Guttman scale is not limited to dichotomous variables and does not necessarily determine an order among the variables. But if variables are all dichotomous, the variables are indeed ordered by their sensitivity in recording the assessed attribute, as illustrated by Example 1.

A binary opposition is a pair of related terms or concepts that are opposite in meaning. Binary opposition is the system of language and/or thought by which two theoretical opposites are strictly defined and set off against one another. It is the contrast between two mutually exclusive terms, such as on and off, up and down, left and right. Binary opposition is an important concept of structuralism, which sees such distinctions as fundamental to all language and thought. In structuralism, a binary opposition is seen as a fundamental organizer of human philosophy, culture, and language.

Chris Matthew Sciabarra is an American political theorist based in Brooklyn, New York. He is the author of three scholarly books—Marx, Hayek, and Utopia; Ayn Rand: The Russian Radical; and Total Freedom: Toward a Dialectical Libertarianism—as well as several shorter works. He is also the co-editor, with Mimi Reisel Gladstein, of Feminist Interpretations of Ayn Rand and co-editor with Roger E. Bissell and Edward W. Younkins of The Dialectics of Liberty: Exploring the Context of Human Freedom. His work has focused on topics including Objectivism, libertarianism, and dialectics.

In quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–Kochen–Specker theorem, is a "no-go" theorem proved by John S. Bell in 1966 and by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the permissible types of hidden-variable theories, which try to explain the predictions of quantum mechanics in a context-independent way. The version of the theorem proved by Kochen and Specker also gave an explicit example for this constraint in terms of a finite number of state vectors.

The complexity of constraint satisfaction is the application of computational complexity theory on constraint satisfaction. It has mainly been studied for discriminating between tractable and intractable classes of constraint satisfaction problems on finite domains.

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

Relational dialectics is an interpersonal communication theory about close personal ties and relationships that highlights the tensions, struggles and interplay between contrary tendencies. The theory, proposed respectively by Leslie Baxter and Barbara Montgomery in 1988, defines communication patterns between relationship partners as the result of endemic dialectical tensions. Dialectics are described as the tensions an individual feels when experiencing paradoxical desires that we need and/ or want. The theory contains four assumptions, one of them being that relationships are not one dimensional, rather, they consist of highs and lows, without moving in only one direction. The second assumption claims that change is a key element in relational life, in other words, as our lives change, our relationships change with it. Third, is the assumption that, “contradictions or tensions between opposites never go away and never cease to provide tension,” which means, we will always experience the feelings of pressure that come with our contradictory desires. The fourth assumption is that communication is essential when it comes to working through these opposing feelings. Relationships are made in dialogue and they can be complicated and dialogue with similarities and differences are necessary. Relational communication theories allow for opposing views or forces to come together in a reasonable way. When making decisions, desires and viewpoints that often contradict one another are mentioned and lead to dialectical tensions. Leslie A. Baxter and Barbara M. Montgomery exemplify these contradictory statements that arise from individuals experience dialectal tensions using common proverbs such as "opposites attract", but "birds of a feather flock together"; as well as, "two's company; three's a crowd" but "the more the merrier". This does not mean these opposing tensions are fundamentally troublesome for the relationship; on the contrary, they simply bring forward a discussion of the connection between two parties.

Facet theory is a metatheory for the multivariate behavioral sciences that posits that scientific theories and measurements can be advanced by discovering relationships between conceptual classifications of research variables and empirical partitions of data-representation spaces. For this purpose, facet theory proposes procedures for (1) Constructing or selecting variables for observation, using the mapping sentence technique, and (2) Analyzing multivariate data, using data representation spaces, notably those depicting similarity measures, or partially ordered sets, derived from the data.

Transference focused psychotherapy (TFP) is a highly structured, twice-weekly modified psychodynamic treatment based on Otto F. Kernberg's object relations model of borderline personality disorder. It views the individual with borderline personality organization (BPO) as holding unreconciled and contradictory internalized representations of self and significant others that are affectively charged. The defense against these contradictory internalized object relations leads to disturbed relationships with others and with self. The distorted perceptions of self, others, and associated affects are the focus of treatment as they emerge in the relationship with the therapist (transference). The treatment focuses on the integration of split off parts of self and object representations, and the consistent interpretation of these distorted perceptions is considered the mechanism of change.

Item tree analysis (ITA) is a data analytical method which allows constructing a hierarchical structure on the items of a questionnaire or test from observed response patterns.
Assume that we have a questionnaire with m items and that subjects can answer positive (1) or negative (0) to each of these items, i.e. the items are dichotomous. If n subjects answer the items this results in a binary data matrix D with m columns and n rows. Typical examples of this data format are test items which can be solved (1) or failed (0) by subjects. Other typical examples are questionnaires where the items are statements to which subjects can agree (1) or disagree (0).
Depending on the content of the items it is possible that the response of a subject to an item j determines her or his responses to other items. It is, for example, possible that each subject who agrees to item j will also agree to item i. In this case we say that item j implies item i. The goal of an ITA is to uncover such deterministic implications from the data set D.

Continuum theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, categorical theories or models explain variation using qualitatively different states.

Psychometric software is software that is used for psychometric analysis of data from tests, questionnaires, or inventories reflecting latent psychoeducational variables. While some psychometric analyses can be performed with standard statistical software like SPSS, most analyses require specialized tools.

Ayn Rand: The Russian Radical is a 1995 book by Chris Matthew Sciabarra tracing the intellectual roots of 20th-century Russian-American novelist and philosopher Ayn Rand and the philosophy she developed, Objectivism.

In science and research, an attribute is a quality of an object. Attributes are closely related to variables. A variable is a logical set of attributes. Variables can "vary" – for example, be high or low. How high, or how low, is determined by the value of the attribute. (For example see: Binary option)

Anton K. Formann was an Austrian research psychologist, statistician, and psychometrician. He is renowned for his contributions to item response theory, latent class analysis, the measurement of change, mixture models, categorical data analysis, and quantitative methods for research synthesis (meta-analysis).

## 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.
2. Buckley, Jorunn Jacobsen (2002). The Mandaeans: ancient texts and modern people. New York: Oxford University Press. ISBN   0-19-515385-5. OCLC   65198443.
3. "dichotomy". Academic Dictionaries and Encyclopedias.