The Principles of Mathematics (1903)

§ 35

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 x is a not-a is equivalent to x is not an a. But we require an axiom to the effect that not-a is a class, and another to the effect that not-not-a is a. Peano gives also a third axiom, namely: If a, b, c be classes, and ab is contained in c, and x is an a but not a c, then x is not a b. This is simpler in the form: If p, q, r be propositions, and p, q together imply r, and q is true while r is false, then q is false. This would be still further improved by being put in the form: If q, r are propositions, and q implies r, then not-r implies not-q; a form which Peano obtains as a deduction. By dealing with propositions before classes or propositional functions, it is possible, as we saw, to avoid treating negation as a primitive idea, and to replace all axioms respecting negation by the principle of reduction.(§ 35 ¶ 1)

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, a or b is defined as the negation of the logical product of not-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: a or b consists of all terms which belong to any class which contains 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)