42. The Distinction Between Intensional and Extensional Functions

Harold R. (Hal) Foster’s Prince Valiant

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A function of a function is called intensional when it is not extensional.

The nature and importance of the distinction between intensional and extensional functions will be made clearer by some illustrations. The proposition "'x is a man' always implies 'x is a mortal'" is an extensional function of the function "x^ is a man," because we may substitute, for "x is a man, "'x is a featherless biped," or any other statement which applies to the same objects to which "x is a man "applies, and to no others. But the proposition "A believes that 'x is a man' always implies 'x is a mortal"' is an intensional function of "x^ is a man," because A may never have considered the question whether featherless bipeds are mortal, or may believe wrongly that there are featherless bipeds which are not mortal. Thus even if "x is a featherless biped" is formally equivalent to "x is a man," it by no means follows that a person who believes that all men are mortal must believe that all featherless bipeds are mortal, since he may have never thought about featherless bipeds, or have supposed that featherless bipeds were not always men. Again, the proposition "the number of arguments that satisfy the function φ!z^ is n" is an extensional function of φ!^, because its truth or falsehood is unchanged if we substitute for φ!z^ any other function which is true whenever φ!z^ is true, and false whenever φ!z^ is false. But the proposition "A asserts that the number of arguments satisfying φ!z^ is n" is an intensional function of φ!z^, since, if A asserts this concerning φ!z, he certainly cannot assert it concerning all predicative functions that are equivalent to φ!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 “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 "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 '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 'x^ is Dr. Cook or Commander Peary."' Thus "it is a strange coincidence that φ!z^ should be satisfied by two arguments" is an intensional function of φ!z^.

The above instances illustrate  the fact that the functions of functions with which mathematics is especially 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 collection of men, namely those who were Apostles, rather than as attributing the property of being satisfied by twelve arguments to the function "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 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 philosophically. The technical procedure by which the apparent difficulty is overcome is as follows.

We have seen that an extensional function of a function may be regarded as a function of the class determined by the argument-function, but that an intensional function cannot be so regarded. In order to obviate the necessity of giving different treatment to intensional and extensional functions of functions, we construct an extensional function derived from any function of a predicative function y!z^, and having the property of being equivalent to the function from which it is derived, provided this function is extensional, as well as the property of being significant (by the help of the systematic ambiguity of equivalence) with any argument φ^ whose arguments are of the same type as those of y!z^. The derived function, written "ƒ{z^(φz)},"is defined as follows: Given a function ƒ(y!z^), our derived function is to be "there is a predicative function which is formally equivalent to φz^ and satisfies ƒ." If φz^ is a predicative function, our derived function will be true whenever ƒ(φz^) is true. If ƒ(φz^)  is an extensional function, and φz^ is a predicative function, our derived function will not be true unless ƒ(φz^) is true; thus in this case, our derived function is equivalent to ƒ(φz^). If ƒ(φz^) is not an extensional function, and if φz^ is a predicative function, our derived function may sometimes be true when the original function is false. But in any case the derived function is always extensional.

In order that the derived function should be significant for any function φz^, of whatever order, provided it takes arguments of the right type, it is necessary and sufficient that ƒ(y!z^) should be significant, where y!z^ is any predicative function. The reason of this is that we only require, concerning an argument φz^, the hypothesis that it is formally equivalent to some predicative function y!z^, and formal equivalence has the same kind of systematic ambiguity as to type that belongs to truth and falsehood, and can therefore hold between functions of any two different orders, provided the functions take arguments of the same type. Thus by means of our derived function we have not merely provided extensional functions everywhere in place of intensional functions, but we have practically removed the necessity for considering differences of type among functions whose arguments are of the same type. This effects the same kind of simplification in our hierarchy as would result from never considering any but predicative functions.

If ƒ(y!z^) can be built up by means of the primitive ideas of disjunction, negation, (x) . φx, and (∃x) . φx, as is the case with all the functions of functions that explicitly occur in the present work, it will be found that, in virtue of the systematic ambiguity of the above primitive ideas, any function φz^  whose arguments are of the same type as those of y!z^ can significantly be substituted for y!z^  in ƒ without any other symbolic change. Thus in such a case what is symbolically, though not really, the same function ƒ can receive as arguments functions of various different types. If, with a given argument φz, the function ƒ(φ!z^), so interpreted, is equivalent to ƒ(y!z^) whenever y!z^ is formally equivalent to φz^, then ƒ{z^(φz)} is equivalent to ƒ(φz^) provided there is any predicative function formally equivalent to φz^. At this point, we make use of the axiom of reducibility, according to which there always is a predicative function formally equivalent to φz^. 

Dan DeCarlo The greatest of all time comic book artist!

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 We tend to believe mathematical problems have reliable answers that are provable and constant, but that more human questions involving beliefs, emotions, speculations, etc., are a completely different matter. In fact, although the propositional equations we are learning to construct may seem a bit more complex than typical math, they are actually similar; and plainly behave in the same ways, and reveal any errors clearly.

 

Once again these aspects are only in small part about understanding the system, or thinking like an Indian does. These ending posts more directly go towards communicating complex thought clearly to others, and a systematic means of notation. As I have said above I feel that by the time most persons reach an understanding of the system we find that we no longer require the purpose we set out to satisfy. This is because if one understands the system well at every step, she changes in her perception of dominant culture, common use of language, the process of metaphysics, and all of life as it goes on around us. If one is very strongly attached to a stake in dominant culture, earning money as a means of living life, such as responsibility for supporting family or debt; I am not certain that she can learn this system at all, or would even consider investing the time it requires.


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Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.