# The Principles of Mathematics (1903)

## § 486

The chief difficulty which arises in the above theory of classes is as to the kind of entity that a range is to be. The reason which led me, against my inclination, to adopt an extensional view of classes, was the necessity of discovering some entity determinate for a given propositional function, and the same for any equivalent propositional function. Thus x is a man is equivalent (we will suppose) to x is a featherless biped, and we wish to discover some one entity which is determined in the same way by both these propositional functions. The only single entity I have been able to discover is the class as one--except the derivative class (also as one) of propositional functions equivalent to either of the given propositional functions. This latter class is plainly a more complex notion, which will not enableus to dispense with the general notion of class; out of this more complex notion (so we agreed in § 73) must be substituted for the class of terms in the symbolic treatment, if there is to be any null-class and if the class whose only member is a given term is to be distinguished from that term. It would certainly be a very great simplification to admit, as Frege does, a range which is something other than the whole composed of the terms satisfying the propositional function in question; but for my part, inspection reveals to me no such entity. On this ground, and also on account of the contradiction, I feel compelled to adhere to the extensional theory of classes, though not quite as set forth in Chapter VI.(§ 486 ¶ 1)