# The Principles of Mathematics (1903)

### § 23

This brings me to the notion of such that. The values of x which render a propositional function ϕx true are like the roots of an equation—indeed the latter are a particular case of the former—and we may consider all the values of x which are such that ϕx is true. In general, these values form a class, and in fact a class may be defined as all the terms satisfying some propositional function. There is, however, some limitation required in this statement, though I have not been able to discover precisely what the limitation is. This results from a certain contradiction which I shall discuss at length at a later stage (Chap. X). The reasons for defining class in this way are, that we require to provide for the null-class, which prevents our defining a class as a term to which some other has the relation ∈, and that we wish to be able to define classes by relations, i.e. all the terms which have to other terms the relation R are to form a class, and such cases require somewhat complicated propositional functions.(§ 23 ¶ 1)