INDUCTION-. 151 



But for the aggregate of alternatives we may now 

 substitute their equivalent as given in the first premise, 

 namely A, so that we get the required result 



A = AX. 



It may be remarked that we should have reached the 

 same final result if our original premise had been of the 

 form 



A = ABI-ACI I AQ. 



The difference of meaning is that all B's need not now 

 be A's, nor all C's, &c. But we should still have 

 A = ABX I ACX I I AQX = AX. 



We can always prove a proposition, if we find it more 

 convenient, by proving its equivalent. To assert that all 

 not-B's are not- A's, is exactly the same as to assert that all 

 A's are B's. Accordingly we may ascertain that A = AB 

 by first ascertaining that b = ab. If we observe, for in- 

 stance, that all substances which are not solids are also 

 not capable of double refraction, it follows necessarily 

 that all double refracting substances are solids. We may 

 convince ourselves that all electric substances are noncon- 

 ductors of electricity, by reflecting that all good conduc- 

 tors do not, and in fact cannot, retain electric excitation. 

 When we come to questions of probability it will be found 

 desirable to prove, as far as possible, both the original 

 proposition and its equivalent, as there is then an increased 

 area of observation. 



The number of alternatives which may arise in the 

 division of a class varies greatly, and may be any number 

 from two upwards. Thus it is probable that every sub- 

 stance is either magnetic or diamagnetic, and no substance 

 can be both at the same time. The division then must be 

 made in the form 



A = ABclA&C. 



If now we can prove that all magnetic substances 

 are capable of polarity, say B = BC, and also that all 



