2. The Sub-classes taken together should be equal to the Class to be divided: the sum of the Species constitutes the Genus. This provides that the division shall be exhaustive; which dichotomy always secures, according to the principle of Excluded Middle; because whatever is not in the positive class, must be in the negative: Red and Not-red include all colours.
3. The Sub-classes must be opposed or mutually exclusive: Species must not overlap. This again is secured by dichotomy, according to the principle of Contradiction, provided the division be made upon one attribute at a time. But, if we attempt to divide simultaneously upon two attributes, as 'Musicians' upon 'nationality' and 'method,' we get what is called a Cross-division, thus 'German Musicians.' 'Not-German,' 'Classical,' 'Not-Classical;' for these classes may overlap, the same men sometimes appearing in two groups—Bach in 'German' and 'Classical,' Pergolesi in 'Not-German' and 'Classical.' If, however, we divide Musicians upon these attributes successively, cross division will be avoided, thus:
Here no Musician will be found in two classes, unless he has written works in two styles, or unless there are works whose style is undecided. This "unless—or unless" may suggest caution in using dichotomy as a short cut to the classification of realities.
4. No Sub-class must include anything that is not comprised in the class to be divided: the Genus comprises all the Species. We must not divide Dogs into fox-terriers and dog-fish.
§ 6. The process of Inductive Classification may be represented thus:
Given any multitude of individuals to be classified:
(1) Place together in groups (or in thought) those things that have in common the most, the most widely diffused and the most important qualities.
(2) Connect those groups which have, as groups, the greater resemblance, and separate those that have the greater difference.
(3) Demarcate, as forming higher or more general classes, those groups of groups that have important characters in common; and, if possible, on the same principle, form these higher classes into classes higher still: that is to say, graduate the classification upwards.