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V. The Hierarchy of Functions and Propositions.
Just a few days ago I awoke from a dream and marveled at
it. In the dream I was talking to some people, and realized that I knew what I
was talking about. I thought it was just a joke with the punchline “I woke up
and in the dream I knew what I was talking about.” I saw the humor but could
not connect the meaning to the truth of the humor until the dream I had today.
I have been puzzling about what to write about today’s post. I knew it is important, but how do I express its importance in words? My dream today gave me clarity on this question, and it lies in the relation of variables and apparent variables. In the statement “The first part of each and every previous post that is quoted from Principia Mathematica is definitively without error and therefore correct.” This is a definite statement that obviously contains no ‘real’ variables. Still, one may infer two apparent variables in the previous sentence, 1) future posts may contain error in the first part of the post; 2) other parts of the previous posts are or may not be without error. Although this example seems obscure, in some form it arises in every use of language beyond simple declarative sentences. The element of time acts to degrade everything and language structure is not exempt from the effect. This points out the meaning of my dream: not that I knew what I was saying, but that others understood my meaning.
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
|
We are thus led to the conclusion, both from the
vicious-circle principle and from direct inspection, that the functions to
which a given object a can be an argument are incapable of being arguments to
each other, and that they have no term in common with the functions to which
they can be arguments. We are thus led to construct a hierarchy. Beginning with
a and the other terms which can be arguments
to the same functions to which a
can be argument*, we
come next to functions to which a
is a possible argument, and then to functions to which such functions are
possible arguments, and so on. But the hierarchy which has to be constructed is
not as simple as might at first appear. The functions which can take a
as argument form an illegitimate totality, and themselves require division into
a hierarchy of functions. This is easily seen as follows. Let ƒ(φz^, x) be a function of the two variables φz^ and x. Then if, keeping x fixed for the moment, we assert this with all possible values of φ,
we obtain a proposition:
(φ) . ƒ(φz,
x).
Here, if x is
variable, we have a function of x;
but as this function involves a totality of values of φz^*2, it cannot
itself be one of the values included in the totality, by the vicious-circle
principle. It follows that the totality of values of φz^ concerned in (φ) .
ƒ(φz^, x) is not the totality of all functions
in which x can occur as argument, and
that there is no such totality as that of all functions in which x can occur as argument.
It follows from the above that a function in which φz^ appears as
argument requires that " φz^"
should not stand for any function which is capable of a given argument, but
must be restricted in such a way that none of the functions which are possible
values of "φz^"
should involve any reference to the totality of such functions. Let us take as
an illustration the definition of identity. We might attempt to define "x
is identical with y" as meaning "whatever is true of x is true of y," i.e. "φx
always implies φy." But here, since
we are concerned to assert all values of "φx implies φy" regarded as
a function of φ, we shall be compelled to impose upon φ
some limitation which will prevent us from including among values of φ
values in which "all possible values of φ"
are referred to. Thus for example "x
is identical with a" is a
function of x; hence, if it is a
legitimate value of φ in "φx always implies φy,"
we shall be able to infer, by means of the above definition, that if x is identical with a,
and x is identical with y, then y is identical with a.
Although the conclusion is sound, the reasoning embodies a vicious-circle
fallacy, since we have taken "(φ) . φx implies φa"
as a possible value of φx,
which it cannot be. If, however, we impose any limitation upon φ,
it may happen, so far as appears at present, that with other values of φ we
might have φx
true and φy
false, so that our proposed definition of identity would plainly be wrong. This
difficulty is avoided by the "axiom of reducibility," to be explained
later. For the present, it is only mentioned in order to illustrate the
necessity and the relevance of the hierarchy of functions of a given argument.
Let us give the name "a-functions"
to functions that are significant for a given argument a.
Then suppose we take any selection of a-functions,
and consider the proposition "a
satisfies all the functions belonging to the selection in question." If we
here replace a by a variable, we obtain
an a-function; but by the vicious-circle principle this a-function
cannot be a member of our selection, since it refers to the whole of the selection.
Let the selection consist of all those functions which satisfy ƒ(φz^). Then our new
function is
(φ).
{ƒ(φz^) implies φx},
where x is
the argument. It thus appears that, whatever selection of a-functions
we may make, there will be other a-functions
that lie outside our selection. Such a-functions,
as the above instance illustrates, will always arise through taking a function
of two arguments, φz^
and x, and asserting all or some of
the values resulting from varying φ. What is
necessary, therefore, in order to avoid vicious-circle fallacies, is to divide
our a-functions into " types," each of
which contains no functions which refer to the whole of that type.
When something is asserted or denied about all possible
values or about some (undetermined) possible values of a variable, that
variable is called apparent, after
Peano. The presence of the words all
or some in a proposition indicates
the presence of an apparent variable; but often an apparent variable is really
present where language does not at once indicate its presence. Thus for example
"A is mortal" means "there is a time at which A will die."
Thus a variable time occurs as apparent variable.
The clearest instances of propositions not containing
apparent variables are such as express immediate judgments of perception, such
as "this is red" or " this is painful," where "this"
is something immediately given. In other judgments, even where at first sight
no variable appears to be present, it often happens that there really is one.
Take (say) "Socrates is human." To Socrates himself, the word
"Socrates" no doubt stood for an object of which he was immediately
aware, and the judgment "Socrates is human" contained no apparent
variable. But to us, who only know Socrates by description, the word
"Socrates" cannot mean what it meant to him; it means rather
"the person having such-and-such properties," (say) "the
Athenian philosopher who drank the hemlock." Now in all propositions about
"the so-and-so" there is an apparent variable, as will be shown in
Chapter III. Thus in what we have in
mind when we say "Socrates is human" there is an apparent variable,
though there was no apparent variable in the corresponding judgment as made by
Socrates, provided we assume that there is such a thing as immediate awareness
of oneself.
Whatever may be the instances of propositions not
containing apparent variables, it is obvious that propositional functions whose
values do not contain apparent variables are the source of propositions
containing apparent variables, in the sense in which the function φx^ is the source of
the proposition (x) . φx. For the values
for φx^ do not contain
the apparent variable x, which
appears in (x) . φx; if they contain
an apparent variable y, this can be
similarly eliminated, and so on. This process must come to an end, since no
proposition which we can apprehend can contain more than a finite number of
apparent variables, on the ground that whatever we can apprehend must be of
finite complexity. Thus we must arrive at last at a function of as many
variables as there have been stages in reaching it from our original
proposition, and this function will be such that its values contain no apparent
variables. We may call this function the matrix
of our original proposition and of any other propositions and functions to be
obtained by turning some of the arguments to the function into apparent
variables. Thus for example, if we have a matrix-function whose values are φ(x, y),
we shall derive from it
(y). φ(x, y),
which is a function of x,
(x). φ(x, y),
which is a function of y,
(x, y). φ(x, y),
meaning " φ(x, y) is true with all
possible values of x and y."
This last is a proposition
containing no real variable, i.e. no variable except apparent
variables.
It is thus plain that all possible propositions and
functions are obtainable from matrices by the process of turning the arguments
to the matrices into apparent variables. In order to divide our propositions
and functions into types, we shall, therefore, start from matrices, and
consider how they are to be divided with a view to the avoidance of
vicious-circle fallacies in the definitions of the functions concerned. For
this purpose, we will use such letters as a,
b, c, x, y, z,
w, to denote objects which are
neither propositions nor functions. Such objects we shall call individuals. Such objects will be constituents
of propositions or functions, and will be genuine
constituents, in the sense that they do not disappear on analysis, as (for
example) classes do, or phrases of the form "the so-and-so."
*
Cf. Chapter III.
*2When
we speak of "values of φz^"
it is φ,
not z, that is to be assigned. This
follows from the explanation in the note on post (link)27. When the function
itself is the variable, it is possible and simpler to write φ
rather than φz^,
except in positions where it is necessary to emphasize that an argument must be
supplied to secure significance.
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| Script: J. Mariscal & F. Trueba Art: various © Respective copyright/trademark
holders. |
I have been puzzling about what to write about today’s post. I knew it is important, but how do I express its importance in words? My dream today gave me clarity on this question, and it lies in the relation of variables and apparent variables. In the statement “The first part of each and every previous post that is quoted from Principia Mathematica is definitively without error and therefore correct.” This is a definite statement that obviously contains no ‘real’ variables. Still, one may infer two apparent variables in the previous sentence, 1) future posts may contain error in the first part of the post; 2) other parts of the previous posts are or may not be without error. Although this example seems obscure, in some form it arises in every use of language beyond simple declarative sentences. The element of time acts to degrade everything and language structure is not exempt from the effect. This points out the meaning of my dream: not that I knew what I was saying, but that others understood my meaning.
 |
|
Pencils and Inks: Carl Burgos Human Torch and Sub-mariner © Respective
copyright/trademark holders.
|
 |
|
Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
|
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