Principia mathematica, vol. 1 (of 3)

The above proposition, besides being used in the next two, is used in the theory of couples (*54·6), in the theory of greater and less (*117·632), and in the chapter on the ordering of classes by the principle of first differences (*170·68).

In the remainder of this number we shall be concerned with the existence of classes. Many of the properties of the existence of classes follow from the fact that to say a class exists is equivalent to saying that the class is not equal to the null-class. This is proved in *24·54.

This proposition states that the class of all objects of the type in question is not null, but has at least one member. The assumption that there is something, which is equivalent to this proposition, is implicit in the proposition *10·1, that what is true always is true in any instance. This would not hold if there were no instances of anything; hence it implies the existence of something. It will be observed that the above proposition (*24·52) depends on *24·1, which depends on *23·351, which depends on *10·251, which depends on *10·24, which depends on *10·1 or on *9·1. The assumption that there is something is involved in the use of the real variable, which would otherwise be meaningless. This is made explicit in *9·1, and in the proof of *9·2, which is the same proposition as *10·1.