Now, if we substitute for each alternative of the first premise its description as found among the succeeding premises, we obtain A = BX ꖌ CX ꖌ .... ꖌ PX ꖌ QX or A = (B ꖌ C ꖌ .... ꖌ Q)X 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. We should have reached the same result if the first premise had been of the form A = AB ꖌ AC ꖌ .... ꖌ AQ. 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 instance, 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 nonconductors of electricity, by reflecting that all good conductors 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 substance 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 = AB*c* ꖌ A*b*C. If now we can prove that all magnetic substances are capable of polarity, say B = BD, and also that all diamagnetic substances are capable of polarity, C = CD, it follows by substitution that all substances are capable of polarity, or A = AD. We commonly divide the class substance into the three subclasses, solid, liquid, and gas; and if we can show that in each of these forms it obeys Carnot’s thermodynamic law, it follows that all substances obey that law. Similarly we may show that all vertebrate animals possess red blood, if we can show separately that fish, reptiles, birds, marsupials, and mammals possess red blood, there being, as far as is known, only five principal subclasses of vertebrata.