The Principles of Mathematics (1903)

§ 25

Most of the propositions of the class-calculus are easily deduced from those of the propositional calculus. The logical product or common part of two classes a nad b is the class of x’s such that the logical product of x is an a and x is a b is true. Similiarly we define the logical sum of two classes (a or b), and the negation of a class (not-a). A new idea is introduced by the logical product and sum of a class of classes. If k is a class of classes, its logical product is the class of terms belonging to each of the classes of k, i.e. the class of terms x such that u is a k implies x is a u for all values of u. The logical sum isthe class which is contained in every class of the class k is contained, i.e. the class of terms x such that, if u is a k implies u is contained in c for all values of u, then, for all values of c, x is a c. And we say that a class a is contained in class b when x is an a implies x is a b for all values of x. In like manner with the above we may define the product and sum of a class of propositions. Another very important notion is what is called the existence of a class—a word hwich must not be supposed to mean what existence means in philosophy. A class is said to exist when it has at least one term. A formal definition is as follows: a is an existent class when and only when any proposition is true provided x is an a always implies it whatevervalue we may give to x. It must be understood that the proposition implied must be a genuine proposition, not a propositional function of x. A class a exists when the logical sum of all propositions of the form x is an a is true, i.e. when not all such propositions are false.(§ 25 ¶ 1)

It is important to understand clearly the manner in which propositions in the class-calculus are obtained from those in the propositional calculus. Consider, for example, the syllogism. We have p implies q and q implies r imply p implies r. Now put x is an a, x is a b, x is a c for p, q, r, where x must have some definite value, but it is not necessary to decide what value. We then find that if, for the value of x in question, x is an a implies x is a b, and x is a b implies x is a c, then x is an a implies x is a c. Since the value of x is irrelevant, we may vary x, and thus we find that if a is contained in b, and b in c, then a is contained in c. This is the class-syllogism. But in applying this process it is necessary to employ the utmost caution if fallacies are to be successfully avoided. In this connection it will be instructive to examine a point upon which a dispute has arisen between Schröder and Mr McColl[19]. Schröder asserts that if p, q, r are propositions, pq implies r is equivalent to the disjunction p or q implies r. Mr McColl admits that the disjunction implies the other, but denies the converse implication. The reason for the divergence is, that Schröder is thinking of propositions and material implication, while Mr McColl is thinking of propositional functions and formal implication. As regards propositions, the truth of the principle may be easily made plain by the following considerations. If pq implies r, then, if either p or q be false, the one of them which is false implies r, because false propositions imply all propositions. But if both be true, pq is true, and therefore r is true, and therefore p implies r and q implies r, because true propositions are implied by every proposition. Thus in any case, one at least of the propositions p and q must imply r. (This is not a proof, but an elucidation.) But Mr McColl objects: Suppose p and q to be mutually contradictory, and r to be the null proposition, then pq implies r but neither p nor q implies r. Here we are dealing with propositional functions and formal implication. A propositional function is said to be null when it is false for all values of x; and the class of x’s satisfying the function is called the null-class, being in fact a class of no terms. Either the function or the class, following Peano, I shall denote by Λ. Now let our r be replaced by ϕx, and our q by not-ϕx, where ϕx is any propositional function. Then pq is false for all values of x, and therefore implies Λ. Thus the above formula can only be truly interpreted in the propositional calculus: in the class-calculus it is false. This may be easily rendered obvious by the following considerations: Let ϕx, ψx, χx be three propositional functions. Then ϕx . ψx implies χx implies, for all vlues of x, that either ϕx implies χx or ψx implies χx. But it does not imply that either ϕx implies χx for all values of x, or ψx implies χx for all values of x. The disjunction is what I shall call a variable disjunction, as opposed to a constant one: that is, in some cases one alternative is true, in others the other, whereas in a constant disjunction there is one of the alternatives (thought it is not stated which) that is always true. Wherever disjunctions occur in regard to propositional functions, they will only be transformable into statements in the class-calculus in cases where the disjunction is constant. This is a point which is both important in itself and instructive in its bearings. Another way of stating the matter is this: In the proposition: If ϕx . ψx implies χx, then either ϕx implies χx or ψx implies χx, the implication indicated by if and then is formal, while the subordinate implications are material; hence the subordinate implications do not lead to the inclusion of one class in another, which results only from formal implication.(§ 25 ¶ 2)

The formal laws of addition, multiplication, tautology and negation are the same as regards classes and propositions. The law of tautology states that no change is made when a class or proposition is added to or multiplied by itself. A new feature of the class-calculus is the null-class, or class having no terms. This may be defined as the class of terms that belong to every class, as the class which does not exist (in the sense defined above), as the class which is contained in every class, as the class Λ which is such that the propositional function x is a Λ is false for all values of x, or as the class of x’s satisfying any propositional function ϕx which is false for all values of x. All those definitions are easily shown to be equivalent.(§ 25 ¶ 3)

§ 25 n. 1. Schröder, Algebra der Logik, Vol. II, pp. 258-9; McColl, Calculus of Equivalent Statements, fifth paper, Proc. Lond. Math. Soc. Vol. XXVIII, p. 182.