In reality all numbers obey the law, for 2 × 2 = 2 is not really analogous to AA = A. According to the definition of a unit already given, each unit is discriminated from each other in the same problem, so that in 2′ × 2″, the first *two* involves a different discrimination from the second *two*. I get four kinds of things, for instance, if I first discriminate “heavy and light” and then “cubical and spherical,” for we now have the following classes-- heavy, cubical. light, cubical. heavy, spherical. light, spherical. But suppose that my two classes are in both cases discriminated by the same difference of light and heavy, then we have heavy heavy = heavy, heavy light = 0, light heavy = 0, light light = light. Thus, (heavy or light) × (heavy or light) = (heavy or light). In short, *twice two is two* unless we take care that the second two has a different meaning from the first. But under similar circumstances logical terms give the like result, and it is not true that A′A″ = A′, when A″ is different in meaning from A′. In a similar manner it may be shown that the Law of Unity A ꖌ A = A. holds true alike of logical and mathematical terms. It is absurd indeed to say that *x* + *x* = *x* except in the one case when *x* = absolute zero. But this contradiction *x* + *x* = *x* arises from the fact that we have already defined the units in one x as differing from those in the other. Under such circumstances the Law of Unity does not apply. For if in A′ ꖌ A″ = A′ we mean that A″ is in any way different from A′ the assertion of identity is evidently false. The contrast then which seems to exist between logical and mathematical symbols is only apparent. It is because the Laws of Simplicity and Unity must always be observed in the operation of counting that those laws seem no further to apply. This is the understood condition under which we use all numerical symbols. Whenever I write the symbol 5 I really mean 1 + 1 + 1 + 1 + 1, and it is perfectly understood that each of these units is distinct from each other. If requisite I might mark them thus 1′+ 1″ + 1‴ + 1″″ + 1″‴. Were this not the case and were the units really 1′ + 1″ + 1″ + 1‴ + 1″″, the Law of Unity would, as before remarked, apply, and 1″ + 1″ = 1″. Mathematical symbols then obey all the laws of logical symbols, but two of these laws seem to be inapplicable simply because they are presupposed in the definition of the mathematical unit. Logic thus lays down the conditions of number, and the science of arithmetic developed as it is into all the wondrous branches of mathematical calculus is but an outgrowth of logical discrimination. *Principle of Mathematical Inference.*