Logic also treats of the relations of all sorts of things, but not as to their quantity.  It considers (i) that one thing may be like or unlike another in certain attributes, as that iron is in many ways like tin or lead, and in many ways unlike carbon or sulphur:  (ii) that attributes co-exist or 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 idiom to call the abstract ‘prior to,’ or ’higher than,’ the concrete (see Porphyry’s Tree, chap. xxii.  Sec. 8); and Logic is more abstract than Astronomy or Sociology.  Philosophy may thank that idiom for many a foolish notion.