From this point, we advance successfully until we require the idea
of *negation*. This is taken, in the edition of the Formulaire we are considering, as a new primitive idea,
and disjunction is defined by its means. By means of the negation of a
proposition, it is of course easy to define the negation of a class: for

is equivalent to `x` is a not-`a`

But we require an axiom to the effect that
`x` is
not an `a`.*not- a* is a class, and another to the effect that
not-not-

We come next to the definition of the disjunction or logical sum of two classes. On this subject Peano has many times changed his procedure. In the edition we are considering,

is defined as the negation of the logical product of not-`a` or `b``a` and not-`b`, i.e. as the class of terms which are not both not-`a` and not-`b`. In later editions (e.g. F. 1901, p. 19), we find a somewhat less artificial definition, namely:

consists of all terms which belong to any class which contains `a` or `b``a` and contains `b`. Either definition seems logically unobjectionable. It is to be observed that `a` and `b` are classes, and that it remains a question for philosophical logic whether there is not a quite different notion of the disjunction of individuals, as e.g. Brown or Jones.

I shall consider this question in Chapter V. It will be remembered that, when we begin by the calculus of propositions, disjunction is defined before negation; with the above definition (that of 1897), it is plainly necessary to take negation first.(§ 35 ¶ 2)

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