What we have said just now as regards the definition of classes is sufficient to satisfy our first four conditions. The way in which it secures the third and fourth, namely, the possibility of classes of classes, and the impossibility of a class being or not being a member of itself, is somewhat technical; it is explained in Principia Mathematica, but may be taken for granted here. It results that, but for our fifth condition, we might regard our task as completed. But this condition—at once the most important and the most difficult—is not fulfilled in virtue of anything we have said as yet. The difficulty is connected with the theory of types, and must be briefly discussed.[42] [42]The reader who desires a fuller discussion should consult Principia Mathematica, Introduction, chap. II.; also * 12. We saw in Chapter XIII. that there is a hierarchy of logical types, and that it is a fallacy to allow an object belonging to one of these to be substituted for an object belonging to another. [Pg 188] Now it is not difficult to show that the various functions which can take a given object as argument are not all of one type. Let us call them all -functions. We may take first those among them which do not involve reference to any collection of functions; these we will call "predicative -functions." If we now proceed to functions involving reference to the totality of predicative -functions, we shall incur a fallacy if we regard these as of the same type as the predicative -functions. Take such an everyday statement as " is a typical Frenchman." How shall we define a "typical" Frenchman? We may define him as one "possessing all qualities that are possessed by most French men." But unless we confine "all qualities" to such as do not involve a reference to any totality of qualities, we shall have to observe that most Frenchmen are not typical in the above sense, and therefore the definition shows that to be not typical is essential to a typical Frenchman. This is not a logical contradiction, since there is no reason why there should be any typical Frenchmen; but it illustrates the need for separating off qualities that involve reference to a totality of qualities from those that do not.