Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/115

of classes is the class of all terms belonging to some one or other of the classes, and the sum of a class of relations is the relation consisting in the fact that some one relation of the class holds. The product of two propositions is their joint assertion, the product of two classes is their common part, the product of two relations is the relation consisting in the fact that both the relations hold. The product of a class of classes is the part common to all of them, and the product of a class of relations is the relation consisting in the fact that all relations of the class in question hold. The inclusion of one class in another consists in the fact that all members of the one are members of the other, while the inclusion of one relation in another consists in the fact that every pair of terms which has the one relation also has the other relation. It is then shown that the properties of negation, addition, multiplication and inclusion are exactly analogous for classes and relations, and are, with certain exceptions, analogous to the properties of negation, addition, multiplication and implication for propositions. (The exceptions arise chiefly from the fact that "$\scriptstyle {p}$ implies $\scriptstyle {q}$" is itself a proposition, and can therefore imply and be implied, while "$\scriptstyle {\alpha }$ is contained in $\scriptstyle {\beta }$," where $\scriptstyle {\alpha }$ and $\scriptstyle {\beta }$ are classes, is not a class, and can therefore neither contain nor be contained in another class $\scriptstyle {\gamma }$.) But classes have certain properties not possessed by propositions: these arise from the fact that classes have not a twofold division corresponding to the division of propositions into true and false, but a threefold division, namely into (1) the universal class, which contains the whole of a certain type, (2) the null-class, which has no members, (3) all other classes, which neither contain nothing nor contain everything of the appropriate type. The resulting properties of classes, which are not analogous to properties of propositions, are dealt with in *24. And just as classes have properties not analogous to any properties of propositions, so relations have properties not analogous to any properties of classes, though all the properties of classes have analogues among relations. The special properties of relations are much more numerous and important than the properties belonging to classes but not to propositions. These special properties of relations therefore occupy a whole section, namely section D.