1.  Each Sub-class, or Species, should comprise less than the Class, or Genus, to be divided.  This provides that the division shall be a real one, and not based upon an attribute common to the whole class; that, therefore, the first rule for making a division shall have been adhered to.  But, as in Sec. 4, we are here met by a logical difficulty.  Suppose that the class to be divided is A, and that we attempt to divide upon the attribute B, into AB and Ab; is this a true division, if we do not know any A that is not B?  As far as our knowledge extends, we have not divided A at all.  On the other hand, our knowledge of concrete things is never exhaustive; so that, although we know of no A that is not B, it may yet exist, and we have seen that it is a logical caution not to assume what we do not know.  In a deductive classification, at least, it seems better to regard every attribute as a possible ground of division.  Hence, in the above division of ‘All Things,’—­’Not-phenomenal,’ ‘Extended-Not-resistant,’ ‘Resistant-Not-gravitating,’ appear as negative classes (that is, classes based on the negation of an attribute), although their real existence may be doubtful.  But, if this be justifiable, we must either rewrite the first test of a division thus:  ’Each sub-class should possibly comprise less than the class to be divided’; or else we must confine the test to (a) thoroughly empirical divisions, as in dividing Colour into Red and Not-red, where we know that both sub-classes are real; and (b) divisions under demonstrable conditions—­as in dividing the three kinds of triangles by the quality equilateral, we know that it is only applicable to acute-angled triangles, and do not attempt to divide the right-angled or obtuse-angled by it.

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: