Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/119

 PRINCIPLES OF EXTENSIVE ABSTRACTION 105 converging end of β subsequent to the first member of β which it extends over. -3 Two classes of events are called * K-equal when each covers the other. Evidently such classes cannot have a finite number of members, K-equality is a relation in which two abstractive classes can stand to each other. The relation is symmetrical and tran- sitive, and every abstractive class is K-equal to itself. [Note. Abstractive classes and the relation of * covering can be illustrated by spatial diagrams, with the same caution as to their possibly misleading character.

Consider a series of squares, concentric and similarly situated. Let the lengths of the sides of the successive squares, stated in order of diminishing size, be h1, h2, ... hn, ....

Then each square extends over all the subsequent squares of the set. Also let L n-> &prop; = O ; Fig. 5-

namely, let hn tend to zero as n increases indefinitely. Then the set forms an abstractive class.

Again, consider a series of rectangles, concentric and similarly situated. Let the lengths of the sides of the successive rectangles , stated in order of diminishing size, be (a, h1), (a, h2), ... (a, hn ),.... ]

Fig. 6. Thus one pair of opposite sides is of the same length through- out the whole series. Then each rectangle extends over all the