Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/121

 PRINCIPLES OF EXTENSIVE ABSTRACTION 107 with the same formative condition σ or in the two cases, are K- equal. -3 Let σ be any assigned formative condition, let σp be the condition of * being a σ-prime/ and let σa be the condition of * being a σ-antiprime/ Thus an abstractive class, which satisfies the condition σp, (i) satisfies the condition σ, and (ii) is covered by every other abstractive class satisfying the same condi- tion σ. Hence any two abstractive classes which satisfy the condition σp cover each other. Hence every class which satisfies the condition σp is covered by every other class which satisfies the same condition σp. That is to say, every such class is a σp -prime. Analogously, it is a σp-antiprime. Similarly the σ-antiprimes are the σa -primes and σa -antiprimes. A formative condition σ will be called regular for primes when (i) there are σ-primes and (ii) the set of abstractive classes K-equal to any one assigned σ-prime is identical with the complete set of σ-primes; and σ will be called * regular for antiprimes when (i) there are σ-antiprimes and (ii) the set of abstractive classes K-equal to any one assigned σ-antiprime is identical with the complete set of σ-antiprimes. Thus if σ be a formative condition regular for primes, the set of σ-primes is the same as the set of abstractive classes K-equal to σ-primes; and if σ be a formative condition regular for antiprimes, the set of σ-antiprimes is the same as the set of abstractive classes K-equal to σ-antiprimes. 1 -4 Errors arise unless we remember the existence of some exceptional abstractive classes. Since we