Dichotomous Division

division of the extension of a concept (division of a class, set) into two subordinate (derivative) classes according to the principle of the excluded middle: “A or not-A.” In other words, only that division into two will be dichotomous in which derivative classes are defined by a pair of logically contradictory properties (terms), one of which is used as a basis for the division. Thus, the division of the set of all human beings into males and not-males (on the basis of the attribute of “being a male”) is dichotomous. But the division of such a set into the class of males and the class of females (on the basis of the attribute of sex) is not a dichotomous division. In such a case the bases for division are different, and the property “being a male” does not logically contradict the property “being a female.”;

The last type of division (as an analogy of “division into two”) is sometimes called pseudodichotomous. The results of both types of division may coincide. In this sense, the classification of a certain “division into two” as a dichotomous division (if it is not such “absolutely,” by definition) depends in a number of cases on which assumptions are accepted. Thus, within the framework of the principle of bivalence, pseudodichotomous division of statements into true and false (the basis for division being the truth value of a proposition) is equivalent to their dichotomous division into a class of true and a class of not-true propositions (the basis for division being the property of the proposition “to be true”). But if the principle of bivalence is not accepted, then it is evident that from the point of view of their results these two divisions are clearly different—among the not-true statements may also be those which we have no basis to consider false.

Any pseudodichotomous division may be converted into a dichotomous division, but the converse is not true. This results specifically from the fact that in dichotomous division one of the derivative classes—the complementary class— is always defined negatively (by means of a negative term), while in pseudodichotomous division both classes are defined positively. It is not always possible to replace a negative definition with a positive one. For example, since there is no positive definition of the concept “transcendental function,” there is no pseudodichotomous division corresponding to the dichotomous division of functions into algebraic and transcendental (nonalgebraic).