There is only one fundamental kind of addition, namely the logical kind. All other kinds can be defined in terms of this and logical multiplication. In the present chapter the addition of integers is to be defined by its means. Logical addition, as was explained in Part I, is the same as disjunction; if `p` and `q` are propositions, their logical sum is the proposition

and if `p` or `q`,`u` and `v` are classes, their logical sum is the class

i.e. the class to which belongs every term which either belongs to `u` or `v`,`u` or belongs to `v`. The logical sum of two classes `u` and `v` may be defined in terms belonging to every class in which both `u` and `v` are contained^{[77]}. This definition is not essentially confined to two classes, but may be extended to a class of classes, whether finite or infinite. Thus if `k` be a class of classes, the logical sum of the classes composing `k` (called for short the sum of `k`) is the class of terms belonging to every class which contains every class which is a term of `k`. It is this notion which underlies arithmetical addition. If `k` be a class of classes no two of which have any common terms (called for short an exclusive class of classes), then the arithmetical sum of the numbers of the various classes of `k` is the number of terms in the logical sum of `k`. This definition is absolutely general, and applies equally whether `k` or any of its constituent classes be finite or infinite. In order to assure ourselves that the resulting number depends only upon the *numbers* of the various classes belonging to `k`, and not upon the particular class `k` that happens to be chosen, it is necessary to prove (as is easily done) that if `k′` be another exclusive class of classes, similar to `k`, and every member of `k` is similar to its correlate in `k′`, and vice versâ, then the number of terms in the sum of `k` is the same as the number in the sum of `k′`. Thus, for example, suppose `k` has only two terms, `u` and `v`, and suppose `u` and `v` have no common part. Then the number of terms in the logical sum of `u` and `v` is the sum of the number of terms in `u` and `v`; and if `u′` be similar to `u`, and `v′` to `v`, and `u′`, `v′` have no common part, then the sum of `u′` and `v′` is similar to the sum of `u` and `v`.(§ 113 ¶ 1)

§ 113 n. 1. F. 1901, § 2, Prop. 1 ·0. ↩

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.