The Principles of Mathematics (1903)

§ 34

The remainder of the definitions preceding the primitive propositions are less important, and may be passed over. Of the primitive propositions, some appear to be merely concerned with the symbolism, and not to express any real properties of what is symbolized; others, on the contrary, are of high logical importance.(§ 34 ¶ 1)

(1) The first of Peano’s axioms is every class is contained in itself. This is equivalent to every proposition implies itself. There seems no way of evading this axiom, which is equivalent to the law of identity, except by the method adopting above, of using self-implication to define propositions. (2) Next we have the axiom that the product of two classes is a class. This ought to have been stated, as ought also the definition of the logical product, for a class of classes; for when stated for only two classes, it cannot be extended to the logical product of an infinite class of classes. If class is taken as indefinable, it is a genuine axiom, which is very necessary to reasoning. But it might perhaps be somewhat generalized by an axiom concerning the terms satisfying propositions of a given form: e.g. the terms having one or more given relations to one or more given terms form a class. In Section B, above, the axiom was wholly evaded by using a generalized form of the axiom as the definition of class. (3) We have next two axioms which are really only one, and appear distinct only because Peano defines the common part of two classes instead of the common part of a class of classes. The two axioms state that, if a, b be classes, their logical product, ab, is contained in a and is contained in b. These appear as different axioms, because, as far as the symbolism shows, ab might be different from ba. It is one of the defects of most symbolisms that they give an order to terms which intrinsically have none, or at least none that is relevant. So in this case: if K be a class of classes, the logical product of K consists of all terms belonging to every class that belongs to K. With this definition, it becomes at once evident that no order of the terms of K is involved. Hence if K has only two terms, a and b, it is indifferent whether we represent the logical product of K by ab or by ba, since the order exists only in the symbols, not in what is symbolized. It is to be observed that the corresponding axiom with regard to propositions is, that the simultaneous assertion of a class of propositions implies any proposition of the class; and this is perhaps the best form of the axiom. Nevertheless, though an axiom is not required, it is necessary, here as elsewhere, to have a means of connecting the case where we start from a class of classes or of propositions or of relations with the case where the class results from enumeration of its terms. Thus although no order is involved in the product of a class of propositions, there is an order in the product of two definite propositions p, q, and it is significant to assert that the products pq and qp are equivalent. But this can be proved by means of the axioms with which we began the calculus of propositions (§ 18). It is to be observed that this proof is prior to the proof that the class whose terms are p and q is identical with the class whose terms are q and p. (4) We have next two forms of syllogism, both primitive propositions. The first asserts that, if a, b, c be classes, and a is contained in b, and x is an a, then x is a b; the second asserts that if a, b, c be classes, and a is contained in b, b in c, then a is contained in c. It is one of the greatest of Peano’s merits to have clearly distinguished the relation of the individual to its class from the relation of inclusion between classes. The difference is exceedingly fundamental: the former relation is the simplest and most essential of all relations, the latter a complicated relation derived from logical implication. It results from the distinction that the syllogism in Barbara has two forms, usually confounded: the one the time-honoured assertion that Socrates is a man, and therefore mortal, the other the assertion that Greeks are men, and therefore mortal. Thetwo forms are stated in PEano’s axioms. It is to be observed that, in ivritue of the other definition of what is meant by one class being contained in another, the first form results from the axiom that, if p, q, r be propositions, and p implies that q implies r, then the product of p and q implies r. This axiom is now substituted by Peano for the first form of the syllogism[32]: it is more general and cannot be deduced from the said form. The second form of the syllogism, when applied to propositions instead of classes, asserts that implication is transitive. This principle is, of course, the very life of all chains of reasoning. (5) We have next a principle of reasoning which Peano calls composition: this asserts that if a is contained in b and also in c, then it is contained in the common part of both. Stating this principle with regard to propositions, it asserts that if a proposition implies each of two others, then it implies their joint assertion or logical product; and this is the principle which was called composition above.(§ 34 ¶ 2)

§ 34 n. 1. See e.g. F. 1901, Part I, § 1, Prop. 3. 3 (p. 10).