Principia mathematica, vol. 1 (of 3)

When "" was defined, in *20, it was defined as a propositional function; and this mode of definition was necessary, because we had to treat of this function before treating of relations. But for many purposes it is desirable to regard as a relation, so that "" becomes an instance of the notation "." This requires, strictly speaking, a change in the meaning of "," but it is a change which does not falsify any of the previous propositions in which "" occurs; for if we call the new meaning "," i.e. if we put

Hence it is unnecessary in practice to have a new notation for the new meaning, and we put simply This definition, though strictly incorrect, is recommended by its convenience, and by the fact that it cannot lead to any harmful confusions. The new meaning of may be taken as replacing the old throughout the remainder of this work.

The uses of the propositions of the present number occur almost exclusively in the theory of selections from a class of classes (*83, *84, *85 and *88). Such selections are effected by means of selective relations, part of whose definition is that they are contained in . Hence the uses of the present number. If is the class of classes from which a selection is to be made, a selective relation will in fact be contained in ; hence the properties of become important. Some of these properties are given in *62·4 ff.

In the above proposition, the first has the newly-defined meaning, while the second has the old meaning. In virtue of the above proposition, the new meaning may be substituted for the old in all propositions hitherto proved concerning , and may take the place of the old meaning in all that follows.

Note that, since is not a homogeneous relation, i.e. not one in which referent and relatum belong to the same type, is strictly meaningless. For if we have , the two 's have different meanings, and do not therefore properly give . But it is convenient to allow , on the understanding that the ambiguity of is to be differently determined for the two factors in the product , namely the second must make both referent and relatum belong to the next type above that to which they respectively belong for the first .