Principia mathematica, vol. 1 (of 3)

A double descriptive function is a non-propositional function of two arguments, such as , , , , , , , . The propositions of the present number apply to all such functions, assuming the notation to be (as in the above instances) a functional sign placed between the two arguments. In order to deal with all analogous cases at once, we shall in this number adopt the notation where "" stands for any such sign as , , , , , , , , or any functional sign to be hereafter defined and satisfying the condition The derived relations and classes with which we shall be concerned may be illustrated by taking the case of . The relation of to will be written , and the relation of to will be written . Thus we shall have The utility of this notation is chiefly due to the possibility of such notations as and . For example, take such a phrase as "the foreign members of English Clubs." Then if we put = foreigners, = English Clubs, we have Or again, let be a conic, and a pencil of lines; then In this case, since , we have . But when the function concerned is not commutative, this does not hold. Thus for example we do not have .

The notations of this number will be frequently applied hereafter to . In accordance with what was said above, we write for the relation of to , and for the relation of to . Hence we have Hence will be the class of relations obtained by taking members of and relatively multiplying them by . Thus if were the class of relations first cousin, second cousin, etc., and were the relation of parent to child, would be the class of relations first cousin once removed, second cousin once removed, etc., taken in the sense which goes from the older to the younger generation.

It is often convenient to be able to exhibit and kindred expressions as descriptive functions of the first argument instead of the second. For this purpose we put with similar notations for other descriptive double functions. We then have, just as in the case of , This enables us to form the class . This class is chiefly useful because the members of its members (i.e. , as we shall define it in *40) constitute the class of all products that can be formed of a member of and a member of .

Thus we are led to three general definitions for descriptive double functions, namely (if be any such function) Since is again a descriptive double function, the first two of the above definitions can be applied to it. The third definition, for typographical reasons, cannot be applied conveniently, though theoretically it is of course applicable. The relations and represent the general idea contained in some of the uses in mathematics of the term 'operation,' e.g. +1 is the operation of adding 1.

The uses of the notations introduced in the present number occur chiefly in arithmetic (Parts III and IV). Few propositions can be given at this stage, since most of the important uses of the notation here introduced depend upon the substitution of some special function for the general function "" here used. In the present number, the propositions given are all immediate consequences of the definitions.