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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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The first matrices that occur are those whose values
are of the forms
φx, y(x, y) c(x, y, z...)
i.e.
where the arguments, however many there may be, are all individuals. The
functions φ, y, c...,
since (by definition) they contain no apparent variables, and have no arguments
except individuals, do not presuppose any totality of functions. From the
functions y, c...
we may proceed to form other functions of x,
such as (y) . y (x, y), (∃y) . y(x, y), (y, z) . c(x, y, z), (y) : (∃z)
. c(x,
y, z), and so on. All these presuppose no totality except that of
individuals. We thus arrive at a certain collection of functions of x, characterized by the fact that they
involve no variables except individuals. Such functions we will call "first-order functions."
We
may now introduce a notation to express "any first-order function."
We will denote any first-order function by "φ!x^" and any value for such a
function by
"φ!x." Thus "φ!x" stands for any value for any
function which involves no variables except individuals. It will be seen that
"φ!x" is itself a function of two
variables, namely φ!z^ and x. Thus φ!x involves a variable which is not an
individual, namely φ!z^. Similarly " (x) .
φ!x" is a function of the variable φ!z^, and thus involves a variable other
than an individual. Again, if a
is a given individual,
"φ!x implies φ!a
with all possible values of φ"
is a function of x,
but it is not a function of the form φ!x, because it involves an (apparent)
variable φ which is not an individual. Let us give
the name "predicate" to any first-order function φ!x^. (This use of the word
"predicate" is only proposed for the purposes of the present
discussion.) Then the statement "φ!x implies φ!a with
all possible values of φ" may be read "all
the predicates of x are predicates of
a." This makes a statement about x, but does not attribute to x a predicate in the special sense just
defined.
Owing to the introduction of the variable first-order
function φ!z^,
we now have a new set of matrices. Thus "φ!x" is a function which contains no
apparent variables, but contains the two real variables φ!z^ and x. (It should be observed that when φ
is assigned, we may obtain a function whose values do involve individuals as
apparent variables, for example if φ!x is (y) . y (x, y). But so long as φ
is variable, φ!x
contains no apparent variables.) Again, if a
is a definite individual, φ!a is
a function of the one variable φ!z^. If a and b
are definite individuals, " φ!a^ implies y!b^" is a
function of the two variables φ!z^, y!z^, and so on. We are thus led to a
whole set of new matrices,
ƒ(φ!z^), g(φ!z^, y!z^), F(φ!z^, x),
and so on.
These matrices contain individuals and first-order
functions as arguments, but (like all matrices) they contain no apparent
variables. Any such matrix, if it contains more than one variable, gives rise
to new functions of one variable by turning all its arguments except one into
apparent variables. Thus we obtain the functions
(φ) . g (φ!z^, y!z^), which is a function of y!z^.
(x) . F(φ!z^, x),
which is a function of φ!z^.
(φ). F(φ!z^,
x), which is a function of x.
We will give the name of second-order matrices to such matrices as have first-order
functions among their arguments, and have no arguments except first-order
functions and individuals. (It is not necessary
that they should have individuals among their arguments.) We will give the name
of second-order functions to such as
either are second-order matrices or are derived from such matrices by turning
some of the arguments into apparent variables. It will be seen that either an
individual or a first-order function may appear as argument to a second-order function.
Second-order functions are such as contain variables which are first-order
functions, but contain no other variables except (possibly) individuals.
We now have various new classes of functions at our
command. In the first place, we have second-order functions which have one
argument which is a first-order function. We will denote a variable function of
this kind by the notation ƒ!(φ^!z^), and any value of such a function by ƒ!(φ!z^). Like φ!x, ƒ!(φ!z^) is a function
of two variables, namely ƒ!(φ^!z^) and φ!z^. Among possible values of ƒ!(φ!x^) will be φ!a (where
a is constant), (x) . φ!x, (∃x) . φ!x,
and so on. (These result from assigning a value to ƒ, leaving φ
to be assigned.) We will call such functions "predicative functions of
first-order functions."
In the second place, we have second-order functions of
two arguments, one of which is a first-order function while the other is an
individual. Let us denote undetermined values of such functions by the notation
ƒ!(φ!z^, x).
As soon as x
is assigned, we shall have a predicative function of φ!z^. If our function contains no
first-order function as apparent variable, we shall obtain a predicative
function of x if we assign a value to
φ!z^. Thus, to take the simplest possible case, if ƒ!(φ!z^, x) is φ!x,
the assignment of a value to φ gives us a predicative
function of x, in virtue of the
definition of "φ!x." But if ƒ!(φ!z^, x) contains a first-order function
as apparent variable, the assignment of a value to φ!z^ gives us a second-order function of
x.
In the third place, we have second-order functions of
individuals. These will all be derived from functions of the form ƒ!(φ!z^, x)
by turning φ into an apparent variable. We do not,
therefore, need a new notation for them.
We have also second-order functions of two first-order
functions, or of two such functions and an individual, and so on.
We may now proceed in exactly the same way to third-order
matrices, which will be functions containing second-order functions as
arguments, and containing no apparent variables, and no arguments except
individuals and first-order functions and second-order functions. Thence we
shall proceed, as before, to third-order functions; and so we can proceed
indefinitely. If the highest order of variable occurring in a function, whether
as argument or as apparent variable, is a function of the nth order, then the function in which it occurs is of the n + 1th
order. We do not arrive at functions of an infinite order, because the number
of arguments and of apparent variables in a function must be finite, and
therefore every function must be of a finite order. Since the orders of
functions are only defined step by step, there can be no process of
"proceeding to the limit," and functions of an infinite order cannot
occur.
We will define a function of one variable as
predicative when it is of the next order above that of its argument, i.e. of the lowest order compatible with
its having that argument. If a function has several arguments, and the highest
order of function occurring among the arguments is the nth, we call the function predicative if it is of the n + 1th
order, i.e. again, if it is of the
lowest order compatible with its having the arguments it has. A function of
several arguments is predicative if there is one of its arguments such that,
when the other arguments have values assigned to them, we obtain a predicative
function of the one undetermined argument.
It is important to observe that all possible functions
in the above hierarchy can be obtained by means of predicative functions and
apparent variables. Thus, as we saw, second-order functions of an individual x are of the form
(φ) . ƒ!(φ!z^, x)
or (∃φ) .
ƒ!(φ!z^, x)
or
(φ, y) . ƒ!(φ!z^, y!z^, x)
or etc.,
where ƒ is a second-order predicative function. And
speaking generally, a non-predicative function of the nth order is obtained from a predicative function of the nth order by turning all the arguments
of the n - 1th order into apparent variables. (Other arguments also may be
turned into apparent variables.) Thus we need not introduce as variables any
functions except predicative functions. Moreover, to obtain any function of one
variable x, we need not go beyond predicative functions of two variables. For the function (y) .
ƒ!(φ!z^, φ!z^, x), where ƒ is
given, is a function of φ!z^ and x, and is
predicative. Thus it is of the form F!(φ!z^, x), and therefore (φ, y) .
ƒ!(φ!z^, y!z^, x)
is of the form (φ) .
F!(φ!z^, x).
Thus speaking generally, by a succession of steps we find that, if φ!u^ is a predicative function of a
sufficiently high order, any assigned non-predicative function of x will be of one of the two forms
(φ) . F(φ!u^, x),
(∃φ) .
F!(φ!u^, x),
where F is a
predicative function of φ!u^ and x.
The nature of the above hierarchy of functions may be
restated as follows. A function, as we saw at an earlier stage, presupposes as
part of its meaning the totality of its values, or, what comes to the same
thing, the totality of its possible arguments. The arguments to a function may
be functions or propositions or individuals. (It will be remembered that
individuals were defined as whatever is neither a proposition nor a function.)
For the present we neglect the case in which the argument to a function is a
proposition. Consider a function whose argument is an individual. This function
presupposes the totality of individuals; but unless it contains functions as
apparent variables, it does not presuppose any totality of functions. If,
however, it does contain a function as apparent variable, then it cannot be
defined until some totality of functions has been defined. It follows that we
must first define the totality of those functions that have individuals as
arguments and contain no functions as apparent variables. These are the predicative functions of individuals.
Generally, a predicative function of a variable argument is one which involves
no totality except that of the possible values of the argument, and those that
are presupposed by any one of the possible arguments. Thus a predicative
function of a variable argument is any function which can be specified without
introducing new kinds of variables not necessarily presupposed by the variable
which is the argument.
A closely analogous treatment can be developed for
propositions. Propositions which contain no functions and no apparent variables
may be called elementary propositions.
Propositions which are not elementary, which contain no functions, and no
apparent variables except individuals, may be called first-order propositions. (It should be observed that no variables
except apparent variables can occur
in a proposition, since whatever contains a real
variable is a function, not a proposition.) Thus elementary and first-order
propositions will be values of first-order functions. (It should be remembered
that a function is not a constituent in one of its values: thus for example the
function "x^ is human" is
not a constituent of the proposition "Socrates is human.") Elementary
and first-order propositions presuppose no totality except (at most) the
totality of individuals. They are of one or other of the three forms
φ!x;
(x) . φ!x; (∃x) . φ!x,
where φ!x is a predicative function of an individual. It follows that, if p represents a variable elementary
proposition or a variable first-order proposition, a function ƒp is either ƒ(φ!x) or ƒ{(x) . φ!x} or {(∃x). φ!x}. Thus a function of an elementary or
a first-order proposition may always be reduced to a function of a first-order
function. It follows that a proposition involving the totality of first-order
propositions may be reduced to one involving the totality of first-order
functions; and this obviously applies equally to higher orders. The
propositional hierarchy can, therefore, be derived from the functional
hierarchy, and we may define a proposition of the nth order as one which involves an apparent variable of the n - 1th
order in the functional hierarchy. The propositional hierarchy is never
required in practice, and is only relevant for the solution of paradoxes; hence
it is unnecessary to go into further detail as to the types of propositions.
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Dan DeCarlo The
greatest of all time comic book artist!
Archie Andrews
Gang © Respective copyright/trademark
holders.
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“Well it ain’t no use to sit and wonder why babe.If yuh don’t know by now.”-Bob Dylan
The matrix is the womb, the rock where precious minerals are extracted;
and here basis for the construction of the Hierarchy of Functions and
Propositions. This very long post text above is deceiving and intimidating, in
that what meaning it conveys is dwarfed here by the authors’ explanation of
that meaning’s expression and notation. Or how it may be discussed and written.
These distinctions are well understood by all shamans and used exclusively
among themselves and our spirit helpers. Such use of language for us is merely
intensional, and rarely spoken aloud, only in oblique indication for points of
clarity. In our system the authors had no thought or intention to address or
define native American Indian thought or perception, this is clear by the
systems structure, and the use that they chose to demonstrate its practical
extensional use; the principle text of the work Principia Mathematica.
The authors' intention was to create a metaphysical end run past language
to a notational symbolic system that can convey extreme complexity in a clear,
compact manner and without error. Ironically once the system is assimilated
clearly in a reader's mind, her perception alters to a state where language
regains useful function. Unfortunately it works only one way, just as one
understands everything involving language, expression, intension and aspiration
in all cultures, she can only respond obliquely and mysteriously; or by means
of artistic expression to be understood. This should become clearer in posts to
follow.
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Pencils: Wallace Wood © Respective
copyright/trademark holders.
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark
holders.
|
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