We shall come much nearer to a satisfactory theory if we try to identify classes with propositional functions. Every class, as we explained in Chapter II., is defined by some propositional function which is true of the members of the class and false of other things. But if a class can be defined by one propositional function, it can equally well be defined by any other which is true whenever the first is true and false whenever the first is false. For this reason the class cannot be identified with any one such propositional function rather than with any other—and given a propositional function, there are always many others which are true when it is true and false when it is false. We say that two propositional functions are "formally equivalent" when this happens. Two propositions are "equivalent" [Pg 183] when both are true or both false; two propositional functions , are "formally equivalent" when is always equivalent to . It is the fact that there are other functions formally equivalent to a given function that makes it impossible to identify a class with a function; for we wish classes to be such that no two distinct classes have exactly the same members, and therefore two formally equivalent functions will have to determine the same class. When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And if we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions. This is an example of Occam's razor, namely, "entities are not to be multiplied without necessity." But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them: like Laplace, we can say, "je n'ai pas besoin de cette hypothèse." Let us set forth the conditions that a symbol must fulfil if it is to serve as a class. I think the following conditions will be found necessary and sufficient:—