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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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As was explained above, it is convenient to regard an
extensional function of a function as having for its argument not the function,
but the class determined by the function. Now we have seen that our derived
function is always extensional. Hence if our original function was ƒ(y!z^), we write the derived function ƒ{z^(φz^)},
where "z^(φz)" may be read
"the class of arguments which satisfy φz,"
or more simply "the class determined by φ^." Thus ƒ{z^(φz)},
will mean: "There is a predicative function y!z^
which is formally equivalent to φz^
and is such that ƒ(yz^),
is true." This is in reality a function of φz^, but we treat it
symbolically as if it had an argument z^(φz). By the help of
the axiom of reducibility, we find that the usual properties of classes result.
For example, two formally equivalent functions determine the same class, and
conversely, two functions which determine the same class are formally
equivalent. Also to say that x is a
member of z^(φz), i.e. of the class determined by φz^, is true when φx is true, and false
when φx is false. Thus all
the mathematical purposes for which classes might seem to be required are
fulfilled by the purely symbolic objects z^(φz), provided we
assume the axiom of reducibility.
In virtue of the axiom of reducibility, if φz is any function,
there is a formally equivalent predicative function y!z^; then the class z^(φz)
is identical with the class z^(y!z),
so that every class can be defined by a predicative
function. Hence the totality of the classes
to which a given term can be significantly said to belong or not to belong is a
legitimate totality, although the totality of functions which a given term can be significantly said to satisfy
or not to satisfy is not a legitimate totality. The classes to which a given
term a belongs or does not belong are the classes
defined by a-functions; they are also
the classes defined by predicative a-functions.
Let us call them a-classes. Then "a-classes"
form a legitimate totality, derived from that of predicative a-functions.
Hence many kinds of general statements become possible which would otherwise
involve vicious-circle paradoxes. These general statements are none of them
such as lead to contradictions, and many of them such as it is very hard to
suppose illegitimate. The fact that they are rendered possible by the axiom of
reducibility, and that they would otherwise be excluded by the vicious-circle
principle, is to be regarded as an argument in favor of the axiom of
reducibility.
The above definition of "the class defined by the
function φz^ or
rather, of any proposition in which this phrase occurs, is, in symbols, as
follows:
ƒ{z^(φz)}
. = : (∃y) :
φx . y!x :
ƒ{y!z^} Df.
In order to recommend this definition, we shall
enumerate five requisites which a definition of classes must satisfy, and we
shall then show that the above definition satisfies these five requisites.
We require of classes, if they are to serve the
purposes for which they are commonly employed, that they shall have certain
properties, which may be enumerated as follows. (1) Every propositional
function must determine a class, which may be regarded as the collection of all
the arguments satisfying the function in question. This principle must hold
when the function is satisfied by an infinite number of arguments as well as when
it is satisfied by a finite number. It must hold also when no arguments satisfy
the function; i.e. the "null-class"
must be just as good a class as any other. (2) Two propositional functions
which are formally equivalent, i.e.
such that any argument which satisfies either satisfies the other, must
determine the same class; that is to say, a class must be something wholly
determined by its membership, so that e.g.
the class "featherless bipeds" is identical with the class
"men," and the class "even primes" is identical with the
class "numbers identical with 2."
(3) Conversely, two propositional functions which determine the same class must
be formally equivalent; in other words, when the class is given, the membership
is determinate: two different sets of objects cannot yield the same class. (4)
In the same sense in which there are classes (whatever this sense may be), or
in some closely analogous sense, there must also be classes of classes. Thus
for example "the combinations of n
things m at a time," where the n things form a given class, is a class
of classes; each combination of m
things is a class, and each such class is a member of the specified set of
combinations, which set is therefore a class whose members are classes. Again,
the class of unit classes, or of couples, is absolutely indispensable; the
former is the number 1, the latter
the number 2. Thus without classes of
classes, arithmetic becomes impossible. (5) It must under all circumstances be
meaningless to suppose a class identical with one of its own members. For if
such a supposition had any meaning, "a Î a"
would be a significant propositional function*1, and so
would "a ~Î a"
Hence, by (1) and (4), there would be a class of all classes satisfying the
function "a ~Î a" If we call this class κ, we shall have
a Î κ
. ≡a a ~Î a
Since, by our hypothesis, "κ Î κ"
is supposed significant, the above equivalence, which holds with all possible
values of a, holds with the value κ, i.e.
κ Î κ
. ≡ . κ ~Î κ.
But this is a contradiction*2. Hence "a Î a"
and " a ~Î a"
must always be meaningless. In general, there is nothing surprising about this
conclusion, but it has two consequences which deserve special notice. In the
first place, a class consisting of only one member must not be identical with that
one member, i.e. we must not have ɿʻx = x. For we have x Î ɿʻx, and therefore, if
x Î ɿʻx, we have ɿʻx Î ɿʻx, which, we saw,
must be meaningless. It follows that "x
Î ɿʻx" must be
absolutely meaningless, not simply false. In the second place, it might appear
as if the class of all classes were a class, i.e. as if (writing "Cls"
for "class ") "Cls Î Cls" were a true proposition. But
this combination of symbols must be meaningless; unless, indeed, an ambiguity
exists in the meaning of "Cls,"
so that, in "Cls Î Cls," the first "Cls" can be supposed to have a
different meaning from the second.
As regards the above requisites, it is plain, to begin
with, that, in accordance with our definition, every propositional function φz^ determines a
class z^ (φz). Assuming the
axiom of reducibility, there must always be true propositions about z^(φz), i.e. true propositions of the form ƒ{z^(φz)}.
For suppose φz^ is formally equivalent to y!z^, and suppose y!z^ satisfies some function ƒ. Then z^(φz) also satisfies ƒ.
Hence, given any function φz,
there are true propositions of the form ƒ{z^(φz)},
i.e. true propositions in which
"the class determined by φz^"
is grammatically the subject. This shows that our definition fulfills the first
of our five requisites.
The second and third requisites together demand that
the classes z^(φz) and z^(yz)
should be identical when, and only when, their defining functions are formally
equivalent, i.e. that we should have
z^(φz) = z^(yz)
. ≡ : φx .
≡x . yx.
Here the meaning of "z^(φz)
= z^(yz)"
is to be derived, by means of a twofold application of the definition of ƒ(z^(φz)}, from the
definition of "χ!z^ = θ!z^,"
which is
χ!z^ = θ!z^ . = :(ƒ): ƒ!χ!z^. É .
ƒ!θ!z^ Df
by the general definition of identity.
In interpreting "z^(φz) = z^(yz),"
we will adopt the convention which we adopted in regard to (ix)(φx)
and (ix)(yx),
namely that the incomplete symbol which occurs first is to have the larger
scope. Thus z^(φz)
= z^(yz) becomes, by our definition,
(∃χ) : φx .≡x . χ! x : χ!z^ = z^(yz),
which, by eliminating z^(yz),
becomes
(∃χ):.
φx . ≡x . χ!x :. (∃θ) :
yx . ≡x
. θ!x
: χ!z^ = θ!z^,
which is equivalent to
(∃χ, θ) :
φx . ≡x . χ!x :
yx . ≡x
. χ!x,
which, again, is equivalent to
(∃x) : φx . ≡x
. χ!x : yx . ≡x . χ!x,
which, in virtue of the axiom of reducibility, is
equivalent to
φx .
≡x . yx.
Thus our definition of the use of z^(φz)
is such as to satisfy the conditions (2) and (3) which we laid down for
classes, i.e. we have
Before considering classes of classes, it will be well
to define membership of a class, i.e.
to define the symbol "x Î (φz),"
which may be read "x is a member
of the class determined by φz."
Since this is a function of the form ƒ{z^(φz^)}, it must be
derived, by means of our general definition of such functions, from the
corresponding function ƒ{y!z^}. We therefore put
x Î y!z^ . = . y!x
Df.
This definition is only needed in order to give a
meaning to " x Î y!z^"; the meaning it gives is, in
virtue of the definition of ƒ{z^(φz)},
(∃y)
: φy . ≡y . y!y
: y!x.
It thus appears that "x Î z^(φz)"
implies φx, since it implies y!x,
and y!x
is equivalent to φx;
also, in virtue of the axiom of reducibility, φx implies "x Î z^(φz),"
since there is a predicative function y
formally equivalent to φ, and x must satisfy y,
since x (ex hypothesi) satisfies φ.
Thus in virtue of the axiom of reducibility we have
┣ : x Î z^(φz)
. ≡ . φx,
i.e. x is a member of the class z^(φz) when,
and only when, x satisfies the
function φ which defines the class.
We have next to consider how to interpret a class of
classes. As we have defined ƒ{z^(φz)}, we shall
naturally regard a class of classes as consisting of those values of z^(φz)
which satisfy ƒ{z^(φz)}. Let us write ⍺
for z^(φz); then we may
write a^(ƒa)
for the class of values of a
which satisfy ƒa*3.
We shall apply the same definition, and put
F{a^(ƒa)}
. = : (∃g) : ƒβ .
≡β .
g!β : F {g!a} Df,
where "β"
stands for any expression of the form z^(yz^).
Let us take "τ Î a (ƒa)"
as an instance of F{a^(ƒa)}.
Then
┣ :. τ Î a(ƒa)
. ≡ : (∃g) : ƒβ .
≡β .
g!β :
τ Î g!a^.
Just as we put x ∈ ψ!z^ . = . ψ!x Df,
so we put
τ ∈ g!τ
. = . g!τ Df.
Thus we find
┣ :. τ ∈ ⍺^(ƒa) . ≡ : (∃g) : ƒβ . ≡β . g!β : g!τ
If we now extend the axiom of reducibility so as to
apply to functions of functions, i.e.
if we assume
(∃g) : ƒ(y!z^) . ≡ψ . g!{ψ!z),
we easily deduce
i.e.
┣ :
(∃g): ƒβ . ≡β .
g!β.
Thus
┣: τ
∈ ⍺^(ƒ⍺) .
≡. ƒτ.
Thus every function which can take classes as
arguments, i.e. every function of
functions, determines a class of classes, whose members are those classes which
satisfy the determining function. Thus the theory of classes of classes offers
no difficulty.
We have next to consider our fifth requisite, namely
that "z^(φz) ∈ z^(φz)"
is to be meaningless. Applying our definition of ƒ{z^(φz)}, we find that if
this collection of symbols had a meaning, it would mean
(∃ψ)
: φx . ≡x
. ψ!x : ψ!z^ ∈ ψ!z^,
in virtue of the definition
it would mean
But here the symbol "ψ!(ψ!z^)" occurs, which assigns a
function as argument to itself. Such a symbol is always meaningless, for the
reasons explained in posts 26 & 27. Hence "z^(φz) ∈ z^(φz)" is
meaningless, and our fifth and last requisite is fulfilled.
*1
As explained in post 17, "x ∈ ⍺"
means "x is a member of the class ⍺,"
or, more shortly, "x is an ⍺."
The definition of this expression in terms of our theory of classes will be
given shortly.
*2
This is the second of the contradictions discussed in post 37.
*3
The use of a single letter, such as ⍺
or A, to represent a variable class,
will be further explained shortly.
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Script : D. Alverez Pencils and Inks:
R. Garcia © Respective copyright/trademark holders.
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This post describing incomplete symbols clarifies where aspects of logical
philosophy diverge and are distinct from mathematical logic. Also in terms of
philosophy we learn techniques of thinking about, and also phrasing functions,
relations and classes in new ways, so that they do not lead to vicious-circle
fallacies as they might if carelessly thought of or expressed.
Remember that this system helps us to perceive and understand terrific
complexities and combine them clearly and without error, and most usefully with
brevity. As time is a true barrier in expressing complexity, especially in
common language, this system can be valuable to anyone with these goals. If
interested readers have even a fair grasp of the system presented in this blog
by now, this is a true mental achievement. Virtually everything presented is
abstract form that has no real meaning itself; but great value as a system of
organizing and expressing any subject we choose.
This sort of system is only made necessary by the complexity that has
evolved in Western language as a result of technology, science, philosophy,
theology, etc.;
North American Indian peoples do not think in the terms of our system. Instead,
our system allows us to think in similar
ways as those Indian peoples.
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Script: Stan Lee Pencils Pencils: Jack Kirby Inks: Paul
Reinman
The Uncanny X-Men, Angel, Marvel Girl Ô © Respective
copyright/trademark holders.
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
|
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