The Principles of Mathematics (1903)

§ 24

With regard to these three fundamental notions, we require two primitive propositions. The first asserts that if x belongs to the class of terms satisfying a propositional function ϕx, then ϕx is true. The second asserts that if ϕx and ψx are equivalent propositions for all values of x, then the class of x’s such that ϕx is true is identical with the class of x’s such that ψx is true. Identity, which occurs here, is defined as follows: x is identical with y if y belongs to every class to which x belongs, in other words, if x is a u implies y is a u for all values of u. With regard to the primitive proposition itself, it is to be observed that it decides in favour of an extensional view of classes. Two class concepts need not be identical when their extensions are so: man and featherless biped are by no means identical, and no more are even prime and integer betwen 1 and 3. These are class-concepts, and if our axiom is to hold, it must not be of these that we are to speak in dealing with classes. We must be concerned with the actual assemblage of terms, not with any kind of concept denoting that assemblage. For mathematical purposes, this is quite essential. Consider, for example, the problem as to how many combinations can be formed of a given set of terms taken any number at a time, i.e. as to how many classes are contained in a given class. If distinct classes may have the same extension this problem becomes utterly indeterminate. And certainly common usage would regard a class as determined when all its terms are given. The extensional view of classes, in some form, is thus essential to Symbolic Logic and to mathematics, and its necessity is expressed in the above axiom. But the axiom itself is not employed, until we come to Arithmetic; at least it need not be employed, if we choose to disginsuish the equality of clases, which is defined as mutual inclusion, from the identity of individuals. Formally the two are totally distinct: identity is defined as above, equality of a and b is defined by the equivalence of x is an a and x is a b for all values of x.(§ 24 ¶ 1)