Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders. |
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^.
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.
Dan DeCarlo The
greatest of all time comic book artist!
Archie Andrews
Gang © Respective copyright/trademark
holders. |
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.
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders. |