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. 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. Such a partition is also frequently called a bipartition. The two parts thus formed are complements. In logic, the partitions are 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. The term dichotomy is from the Greek language διχοτομία dichotomía "dividing in two" from δίχα dícha "in two, asunder" and τομή tomḗ "a cutting, incision". In set theory, a dichotomous relation R is such that either aRb, bRa, but not both. 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). One type of dichotomy is dichotomous classification – classifying objects by recursively splitting them into two groups. As Lewis Carroll explains, "After dividing a Class, by the Process of Dichotomy, into two smaller Classes, we may sub-divide each of these into two still smaller Classes; and this Process may be repeated over and over again, the number of Classes being doubled at each repetition.