satisfies the extensional function derived from
.
This gives a meaning to any statement about a class which can be made significantly about a function; and it will be found that technically it yields the results which are required in order to make a theory symbolically satisfactory.[41]
[41]See Principia Mathematica, vol. I. pp. 75-84 and * 20.
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. Now it is not difficult to show that the various functions which can take a given object
