Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/120

 io6 III. METHOD OF EXTENSIVE ABSTRACTION subsequent rectangles. Let h n tend to zero as n increases in definitely. Then the set forms an abstractive class. Evidently the set of squares converges to a point, and the set of rectangles to a straight line. Similarly, using three dimen sions and volumes, we can thus diagrammatically find abstractive classes which converge to areas. If we suppose the centre of the set of squares to be the same as that of the set of rectangles, and place the squares so that their sides are parallel to the sides of the rectangles, then the set of rectangles covers the set of squares, but the set of squares does not cover the set of rect angles. Again, consider a set of concentric circles with their common centre at the centre of the squares, and let each circle be in scribed in one of the squares, and let each square have one of the circles inscribed in it. Then the circles form an abstractive class converging to their common centre. The set of squares covers the set of circles and the set of circles covers the set of squares. Accordingly the two sets are K-equal.]

. Primes and Antiprimes. 31-1 An abstractive class is called * prime in respect to the formative con- dition σ [whatever condition σ may be] when (i) it satisfies the condition σ, and (ii) it is covered by every other abstractive class satisfying the same condition σ. For brevity an abstractive class which is prime in respect to a formative condition σ is called a σ-prime. Evidently two σ-primes, with the same formative con- dition σ in the two cases, are .K-equal. respect to the formative condition σ [whatever con- dition σ may be] when (i) it satisfies the condition σ, and (ii) it covers every other abstractive class satisfying the same condition σ. For brevity an abstractive class which is antiprime in respect to a formative condition σ is called a σ-antiprime. Evidently two σ-antiprimes,
 * 2 An abstractive class is called antiprime in