Apparently there are but two numbers which obey this law; for it is certain that
x × x = x
is true only in the two cases when x = 1, or x = 0.
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