The Principles of Mathematics (1903)

§ 32

For the sake of definiteness, let us now examine some one of Professor Peano’s expositions of the subject. In his later expositions[28] he has abandoned the attempt to distinguish clearly certain ideas and propositions as primitive, probably because of the realization that any such distinction is largely arbitrary. But the distinction appears useful, as introducing greater definiteness, and as showing that a certain set of primitive ideas and propositions are sufficient; so far from being abandoned, it ought rather to be made in every possible way. I shall, therefore, in what follows, expound one of his earlier expositions, that of 1897[29].(§ 32 ¶ 1)

The primitive notions with which Peano starts are the following: Class, the relation of an individual to a class of which it is a member, the notion of a term, implication where both propositions contain the same variables, i.e. formal implication, the simultaneous affirmation of two propositions, the notion of definition, and the negation of a proposition. From these notions, together with the division of a complex proposition into parts, Peano professes to deduce all symbolic logic by means of certain primitive propositions. Let us examine the deduction in outline.(§ 32 ¶ 2)

We may observe, to begin with, that the simultaneous affirmation of two propositions might seem, at first sight, not enough to take as a primitive idea. For although this can be extended, by successive steps, to the simultaneous affirmation of any finite number of propositions, yet this is not all that is wanted; we require to be able to affirm simultaneously all the propositions of any class, finite or infinite. But the simultaneous assertion of a class of propositions, oddly enough, is much easier to define than that of two propositions (see § 34, (3)). If k be a class of propositions, their simultaneous affirmation is the assertion that p is a k implies p. If this holds, all propositions of the class are true; if it fails, one at least must be false. We have seen that the logical product of two propositions can be defined in a highly artificial manner; but it might almost as well be taken as indefinable, since no further property can be proved by means of the definition. We may observe, also, that formal and material implication are combined by Peano into one primitive idea, whereas they ought to be kept separate.(§ 32 ¶ 3)

§ 32 n. 1. F. 1901 and R. d. M. Vol. VII, No. 1 (1900).

§ 32 n. 2. F. 1897, Part I.