versality of such processes, we next need a generalization of the relation expressed by the word between.
Such a logical generalization has been suggested, although for a purpose decidedly different from that of my present inquiry, by Mr. A. B. Kempe, in a very remarkable paper on “The Relation between the Logical Theory of Classes and the Geometrical Theory of Points.”[1] If I venture to follow out the suggestion of Mr. Kempe’s work, it is in my own way, and his discussion must not be viewed as responsible either for the intent or for the outcome of my speculations. In Mr. Kempe’s research, what is most important for us at the moment is that a relation of a logically identical character is shown to exist in two apparently very different cases. When three points are on the same line, one of them is said to be between the two others. But when two logical classes of objects, ɑ and b, are so related to a third class, m, that this class includes all the objects which are common to both ɑ and b, and at the same time is included within the class of the objects which are either ɑ or b, then Mr. Kempe defines the class m as a class between ɑ and b. The interest of the identification of the relation between in the geometrical and in the logical realms, lies in the proof, given at length by Mr. Kempe, that the exactly definable properties of any complete system of logical classes, or “Universe of Discourse,” are, up to a certain limit, identical with the properties of
- ↑ Published in the Proceedings of the London Mathematical Society, Vol. XXI, for the year 1890. Another statement of his main results was printed by Mr. Kempe in Nature, Vol. XLIII, pp. 156-162. These papers have been far too much neglected by the students of exact logic to whom they were addressed. Their interest goes far beyond that of the special idea which I here borrow from them.
