Principia mathematica, vol. 1 (of 3)

In this section, we consider first the classes contained in a given class and the relations contained in a given relation. If is any class, the classes contained in are the members of ; these are also called the sub-classes of , or (sometimes) the "parts" of . In this last usage, they are called "proper parts" when they are not coextensive with , this phrase being formed on the analogy of "proper fractions." The sub-classes of are all the classes that can be formed from members of ; they are the same thing as the "combinations" of members of taken any number at a time. If is the number of members of , is the number of sub-classes of , whether be finite or infinite. The number of sub-classes of is always greater than the number of members of . On account of these and other propositions, the class of sub-classes of a given class is an important function of the class. If the class is , we denote the class of its sub-classes by "." This is a descriptive function, derived from the relation "," defined as follows:

The sub-relations of a given relation are all the relations contained in the given relation, i.e. all relations which imply the given relation for all possible arguments. That is, if is the given relation, is a sub-relation of if . Thus denoting the class of sub-relations of by "," we are to have hence we take as the definition of "" the following: Sub-relations have properties analogous to those of sub-classes, but they are of somewhat less importance. It should, however, be observed that when one series is contained in another, i.e. is obtained by selecting some of the terms of the other series without changing their order, then the generating relation of the one series is a sub-relation of the generating relation of the other series. (It is not the case that a sub-relation of the generating relation of a series must generate a contained series, for its field may fall apart into detached portions, or otherwise fail of being serial.)

We shall also consider in this section (*62) the relation of membership of a class, i.e. the relation which has to when . This relation bears the same relation to "" as "" bears to "." Strictly speaking, we ought to introduce a new notation for it, putting (say) But as , unlike "," is a letter, and capable of being conveniently used alone, it seems more desirable, from the point of view of avoiding unnecessary duplication of symbols, to put Strictly speaking, this definition is faulty, since it gives two different meanings to "." But practically this does not matter, since the above definition gives where the first has the meaning just defined, while the second has the old meaning. Thus all that is really required of the above definition, namely to give a meaning to formulae in which occurs without referent or relatum, is effected without the danger of any confusion that could lead to errors.

The chief importance of as a relation arises from the fact that relations contained in play a very important part in arithmetic. Take, for example, the problem of selecting one term out of each member of a class of classes: in this case we require a selecting relation which is such that whenever , is a member of , i.e. such that . (This condition is only part of the definition of a selecting relation; the complete definition is given in *80.)

Three numbers in this section (*63, *64, *65) are devoted to the discussion of relative types. Given a variable , we often want to define the relative types of other variables, or of ambiguous symbols, occurring in the same context; that is, we wish to express the types of these other symbols in terms of that of . We use "" for the type of , "" for the type in which is contained. Then , , and . Also we introduce a notation (*65) for giving typical definiteness, relatively to , to typically ambiguous symbols. This notation is very useful in cardinal and ordinal arithmetic, since numbers are typically ambiguous, and the failure to take account of this fact has led to the contradictions concerning the greatest cardinal and the greatest ordinal.