It may be objected that (1) and (2) ought not in a strict sense to be described as rules, but rather as constituting between them a precise statement of what is implied when we speak of a logical division. They become rules, however, in the sense that a professed logical division which fails to satisfy either of them implies relations between the members of the division which do not as a matter of fact hold good. Rules (3) and (4) are of a different character. They are rules in the sense that they must be complied with if a division is to have practical utility.
Rule (3) is not intended to condemn the processes of sub-division and co-division. Having made a division upon one principle, we may proceed to subdivide the classes thus arrived at in accordance with another principle, and so on indefinitely. A scientific classification will always consist of a hierarchy of classes thus obtained. There is again no reason why the same class should not for different purposes be divided in accordance with two or more different 444 principles, so long as these are kept distinct from one another, and the members of the different resulting divisions not confused together.
It has been said that a breach of rule (1) necessarily involves a breach of rule (3), since there cannot be any overlapping of classes so long as a division proceeds correctly upon a single principle. This does not, however, always hold good unless we interpret the word “correctly” as implying that precautions are taken to avoid any overlapping, which of course begs the question. Thus, if we divide triangles into those which have (a) a right angle, (b) an obtuse angle, (c) an acute angle, we may be said to proceed upon one principle, and yet the resulting classes are not mutually exclusive. It may, again, be argued that the classes equilateral triangle, isosceles triangle, scalene triangle (which result from a division based upon a single principle) are not mutually exclusive, since all equilateral triangles are isosceles.
This argument can only be met by saying that, in the first case, we are not proceeding upon any clear principle unless we make our division into triangles whose largest angle is an obtuse angle, a right angle, or an acute angle, respectively; nor unless, in the second case, our principle is the maximum number of sides that are equal to one another, so that an isosceles triangle is defined as a triangle that has two and only two sides equal. Any overlapping of classes is then in each case provided against; but only, it may be argued, because special precautions have been taken to attain this end. By the adoption of similar precautions, a division which proceeds “correctly” upon a single principle will also be exhaustive.
Looking at the question from the other side we may note that a division which satisfies both rule (1) and rule (2) may nevertheless be a cross-division; for it may happen that two different principles of division yield coincident results. For example, an isosceles triangle being defined as a triangle that has two and only two sides equal, there is a cross-division, but no overlapping of classes, or omission of any class contained in the totum divisum, if we divide triangles into scalene, isosceles, and equiangular; or if we divide plants into acotyledons, monocotyledons, and exogens.
As regards rule (4), it is to be observed that a division which proceeds per saltum will usually be much less effective than one in which the intermediate steps are filled in. The worst violation of this rule occurs when the division is disparate, that is, when “one of the classes into which we divide is an immediate and proximate 445 class, while others are mediate and remote” (Clarke, Logic, p. 242); as, for example, if we divide animals into invertebrates, fishes, amphibians, reptiles, birds, elephants, horses, dogs, &c.
Another rule of division is sometimes added, namely, that “none of the dividing members must be equal in extent to the divided whole” (Clarke, Logic, p. 236). When this rule is broken, the division is said to become null and void, because one of the sub-divisions contains no members. From the formal point of view, however, the observance of this rule can hardly be insisted upon. We need not regard a division as necessarily implying the actual occurrence of all its members in the universe of discourse; and the rule in question would deprive the logician of the right to employ the powerful method of division by contradictories. It may be a different matter when we are considering scientific classification from the material standpoint.
412. Division by Dichotomy.—Division by dichotomy or, as it is sometimes called more distinctively, dichotomy by contradiction is the division of a class simply with reference to the presence or absence of a given attribute or set of attributes; as, for example, when X is divided into XA and Xa (where a = not-A). An illustration is afforded by the Tree of Porphyry or Ramean Tree, in which Substances are first divided into Corporeal Substances (Bodies) and Incorporeal Substances, Bodies being then divided into Animate Bodies (Living Beings) and Inanimate Bodies, Living Beings being next divided into Sensitive Living Beings (Animals) and Insensitive Living Beings, and Animals being in their turn divided into Rational Animals (Men) and Irrational Animals. At each step in this scheme we proceed by taking contradictories. It was in praise of dichotomal division that Jeremy Bentham, who is here quoted with approval by Jevons (Principles of Science, 30, § 12), spoke of “the matchless beauty of the Ramean Tree.” When this method is employed we ensure formally that the members of our division shall be mutually exclusive and collectively exhaustive. For, by the law of contradiction, No X is both A and a ; and, by the law of excluded middle, Every X is either A or a.
It is pointed out by Spalding (Logic, p. 146) and by Jevons (Principles of Science, 30, § 9) that all logically perfect division is ultimately reducible to dichotomy, usually with the implication that some of the sub-classes which are à priori possible are not as a matter of fact to be found in the universe of discourse. Thus, 446 if we take the class X and divide it into XA and XB we imply that in the class X, A and B are never found either both present or both absent. Hence the division is equivalent to the following dichotomal division:—
