Page:The World and the Individual, Second Series (1901).djvu/104

Rh from either ɑ or b. The same result follows if you not merely neglect or abstract from the difference of the two classes, but positively know them to be identical classes. For in that case both ɑ and b become identical with m.

Generalizing from these cases, one may go quite beyond Mr. Kempe’s instances of the classes and the points and say, Let there be any system or collection of objects such that, if they are really different, these objects can be discriminated by an attention once properly directed. Let it be also possible for a given intelligence not to discriminate two objects belonging to that collection. Or again, let it be possible for this intelligence, although discriminating them, still to regard two of them at will as “equivalent,” that is, as such that their difference does not count for a given purpose. Then let an object m of the system in question be so related to ɑ and to b that if you, either by inattention, neglect, or deliberate choice, disregard their difference, so that in any way they blend or become equivalent, m thereupon of necessity blends with both, or becomes equivalent to both. In this case we shall say that, in the generalized sense, m is such a member of the system in question as to lie between ɑ and b. The mathematical way of symbolizing this relation would be briefer. It would take the form of merely saying: “m is such that, if ɑ = b, then m = ɑ = b. And if this is the case, m is regarded as between ɑ and b.”

Now the advantage of this formal generalization is the power that it here gives us of facing an important logical aspect of all discrimination, comparison, and differentiation. We usually say that the relation between ɑ and