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 idiom to call the abstract 'prior to,' or 'higher
than,' the concrete (see Porphyry's Tree, chap. xxii. § 8); and Logic is
more abstract than Astronomy or Sociology. Philosophy may thank that
idiom for many a foolish notion.

(c) But, as we have seen, Logic does not investigate the truth,
trustworthiness, or validity of its own principles; nor does