Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/99

 since, if $$\scriptstyle{A}$$ asserts this concerning $\scriptstyle{\phi!\hat z}$, he certainly cannot assert it concerning all predicative functions that are equivalent to $\scriptstyle{\phi!\hat z}$, because life is too short. Again, consider the proposition "two white men claim to have reached the North Pole." This proposition states "two arguments satisfy the function '$\scriptstyle{\hat x}$ is a white man who claims to have reached the North Pole.'" The truth or falsehood of this proposition is unaffected if we substitute for "$\scriptstyle{\hat x}$ is a white man who claims to have reached the North Pole" any other statement which holds of the same arguments, and of no others. Hence it is an extensional function. But the proposition "it is a strange coincidence that two white men should claim to have reached the North Pole," which states "it is a strange coincidence that two arguments should satisfy the function '$\scriptstyle{\hat x}$ is a white man who claims to have reached the North Pole,'" is not equivalent to "it is a strange coincidence that two arguments should satisfy the function '$\scriptstyle{\hat x}$ is Dr Cook or Commander Peary.'" Thus "it is a strange coincidence that $$\scriptstyle{\phi!\hat x}$$ should be satisfied by two arguments" is an intensional function of $\scriptstyle{\phi!\hat x}$.

The above instances illustrate the fact that the functions of functions with which mathematics is specially concerned are extensional, and that intensional functions of functions only occur where non-mathematical ideas are introduced, such as what somebody believes or affirms, or the emotions aroused by some fact. Hence it is natural, in a mathematical logic, to lay special stress on extensional functions of functions.

When two functions are formally equivalent, we may say that they have the same extension. In this definition, we are in close agreement with usage. We do not assume that there is such a thing as an extension: we merely define the whole phrase "having the same extension." We may now say that an extensional function of a function is one whose truth or falsehood depends only upon the extension of its argument. In such a case, it is convenient to regard the statement concerned as being about the extension. Since extensional functions are many and important, it is natural to regard the extension as an object, called a class, which is supposed to be the subject of all the equivalent statements about various formally equivalent functions. Thus e.g. if we say "there were twelve Apostles," it is natural to regard this statement as attributing the property of being twelve to a certain col1ection of men, namely those who were Apostles, rather than as attributing the property of being satisfied by twelve arguments to the function "$\scriptstyle{x}$ was an Apostle." This view is encouraged by the feeling that there is something which is identical in the case of two functions which "have the same extension." And if we take such simple problems as "how many combinations can be made of $$\scriptstyle{n}$$ things?" it seems at first sight necessary that each "combination" should be a single object which can be counted as one. This, however, is certainly not necessary technically, and we see no reason to suppose that it is true