coinhere (or do not) in the same subject, as metallic
lustre, hardness, a certain atomic weight and a certain specific gravity
coinhere in iron: and (iii) that one event follows another (or is the
effect of it), as that the placing of iron in water causes it to rust.
The relations of likeness and of coinherence are the ground of
Classification; for it is by resemblance of coinhering attributes that
things form classes: coinherence is the ground of judgments concerning
Substance and Attribute, as that iron is metallic; and the relation of
succession, in the mode of Causation, is the chief subject of the
department of Induction. It is usual to group together these relations
of attributes and of order in time, and call them qualitative, in order
to contrast them with the quantitative relations which belong to
Mathematics. And it is assumed that qualitative relations of things,
when they cannot be directly perceived, may be proved indirectly by
assuming the axiom of the Syllogism (chap. ix.) and the law of Causation
(chap. xiv.).

So far, then, Logic and Mathematics appear to be co-ordinate and
distinct sciences. But we shall see hereafter that the satisfactory
treatment of that special order of events in time which constitutes
Causation, requires a combination of Logic with Mathematics; and so does
the treatment of Probability. And, again, Logic may be said to be, in a
certain sense, 'prior to' or 'above' Mathematics as usually treated. For
the Mathematics assume that one magnitude must be either equal or
unequal to another, and that it cannot be both equal and unequal to it,
and thus take for granted the principles of Contradiction and Excluded
Middle; but the statement and elucidation of these Principles are left
to Logic (chap. vi.). The Mathematics also classify and define
magnitudes, as (in Geometry) triangles, squares, cubes, spheres; but the
principles of classification and definition remain for Logic to
discuss.

(b) As to the concrete Sciences, such as Astronomy, Chemistry, Zoology,
Sociology--Logic (as well as Mathematics) is implied in them all; for
all the propositions of which they consist involve causation,
co-existence, and class-likeness. Logic is therefore said to be prior to
them or above them: meaning by 'prior' not that it should be studied
earlier, for that is not a good plan; meaning by 'above' not in dignity,
for distinctions of dignity amongst liberal studies are absurd. But it
is a philosophical