ors, blondes,
brunettes, and dyspeptics is not to make a logical division. This is
seen more clearly in connexion with the second condition of a perfect
division.

II. In a perfect division, the subdivisions or species are
mutually exclusive.

Every object possessing the common character should be in one or other
of the groups, and none should be in more than one.

Confusion between classes, or overlapping, may arise from two
causes. It may be due (1) to faulty division, to want of unity in
the _fundamentum divisionis_; (2) to the indistinct character of the
objects to be defined.

(1) Unless the division is based upon a single ground, unless each
species is based upon some mode of the generic character, confusion is
almost certain to arise. Suppose the field to be divided, the objects
to be classified, are three-sided rectilineal plane figures, each
group must be based upon some modification of the three sides. Divide
them into equilateral, isosceles, and scalene according as the three
sides are all of equal length, or two of equal length, or each of
different length, and you have a perfect division. Similarly you can
divide them perfectly according to the character of the angles into
acute-angled, right-angled and obtuse-angled. But if you do not keep
to a single basis, as in dividing them into equilateral, isosceles,
scalene, and right-angled, you have a cross-division. The same
triangle might be both right-angled and isosceles.

(2) Overlapping, however, may be unavoidable in practice owing to
the nature of the objects. There may be objects in which the dividing
characters are not distinctly marked, objects that possess the
differentiae of more than one group in a greater or less degree. Things
are not always marked off from one another by hard and fast lines.
They shade into one another by imperceptible gradations. A clear
separation of them may be impossible. In that case you must allow a
certain indeterminate margin between your classes, and sometimes it
may be necessary to put an object into more than one class.

To insist that there is no essential difference unless a clear
demarcation can be made is a fallacy. A sophistical trick called the
_Sorites_ or Heap from the classical example of it was based upon this
difficulty of drawing sharp lines of definition. Assuming that it is
possible to say how many stones constitute a heap, you begin by asking
whether three stones form a heap. If your respo