This is a law which distinguishes mathematical terms from logical terms, because it obviously does not apply to the former. I propose to call it the Law of Unity, because it must really be involved in any definition of a mathematical unit. This law is closely analogous to the Law of Simplicity, AA = A; and the nature of the connection is worthy of
Few or no logicians except De Morgan have adequately noticed the close relation between combined and disjunctive terms, namely, that every disjunctive term is the negative of a corresponding combined term, and vice versâ. Consider the term
How shall we describe the class of things which are not malleable-dense-metals? Whatever is included under that term must have all the qualities of malleability, denseness, and metallicity. Wherever any one or more of the qualities is wanting, the combined term will not apply. Hence the negative of the whole term is
In the above the conjunction or must clearly be interpreted as unexclusive; for there may readily be objects which are both not-malleable, and not-dense, and perhaps not-metallic at the same time. If in fact we were required to use or in a strictly exclusive manner, it would be requisite to specify seven distinct alternatives in order to describe the negative of a combination of three terms. The negatives of four or five terms would consist of fifteen or thirty-one alternatives. This consideration alone is sufficient to prove that the meaning of or cannot be always exclusive in common language.
Expressed symbolically, we may say that the negative of
| ABC | |
|---|---|
| is | not-A or not-B or not-C; |
| that is, | a ꖌ b ꖌ c . |
Reciprocally the negative of