We will, to begin with, ignore this last condition and the problems which it raises. The first two conditions may be [Pg 185] taken together. They state that there is to be one class, no more and no less, for each group of formally equivalent propositional functions; e.g. the class of men is to be the same as that of featherless bipeds or rational animals or Yahoos or whatever other characteristic may be preferred for defining a human being. Now, when we say that two formally equivalent propositional functions may be not identical, although they define the same class, we may prove the truth of the assertion by pointing out that a statement may be true of the one function and false of the other; e.g. "I believe that all men are mortal" may be true, while "I believe that all rational animals are mortal" may be false, since I may believe falsely that the Phoenix is an immortal rational animal. Thus we are led to consider statements about functions, or (more correctly) functions of functions. Some of the things that may be said about a function may be regarded as said about the class defined by the function, whereas others cannot. The statement "all men are mortal" involves the functions " is human" and " is mortal"; or, if we choose, we can say that it involves the classes men and mortals. We can interpret the statement in either way, because its truth-value is unchanged if we substitute for " is human" or for " is mortal" any formally equivalent function. But, as we have just seen, the statement "I believe that all men are mortal" cannot be regarded as being about the class determined by either function, because its truth-value may be changed by the substitution of a formally equivalent function (which leaves the class unchanged). We will call a statement involving a function an "extensional" function of the function , if it is like "all men are mortal," i.e. if its truth-value is unchanged by the substitution of any formally equivalent function; and when a function of a function is not extensional, we will call it "intensional," so that "I believe that all men are mortal" is an intensional function of " is human" or " is mortal." Thus extensional functions of a function may, for practical [Pg 186] purposes, be regarded as functions of the class determined by , while intensional functions cannot be so regarded.