 |
Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark
holders.
|
II. The Nature of Propositional Functions.
By a "propositional function" we mean
something which contains a variable x,
and expresses a proposition as soon as a value is assigned to x. That is to say, it differs from a
proposition solely by the fact that it is ambiguous: it contains a variable of
which the value is unassigned. It agrees with the ordinary functions of mathematics
in the fact of containing an unassigned variable: where it differs is in the
fact that the values of the function are propositions. Thus e.g. "x is a man" or "sin x
= 1 " is a propositional function. We shall find that it is possible to
incur a vicious-circle fallacy at the very outset, by admitting as possible
arguments to a propositional function terms which presuppose the function. This
form of the fallacy is very instructive, and its avoidance leads, as we shall
see, to the hierarchy of types. The question as to the nature of a function†1 is by no means an easy
one. It would seem, however, that the essential characteristic of a function is
ambiguity. Take, for example, the law of identity in the form "A is
A," which is the form in which it is usually enunciated. It is plain that,
regarded psychologically, we have here a single judgment. But what are we to
say of the object of the judgment? We are not judging that Socrates is
Socrates, nor that Plato is Plato, nor any other of the definite judgments that
are instances of the law of identity. Yet each of these judgments is, in a
sense, within the scope of our judgment. We are in fact judging an ambiguous
instance of the propositional function " A is A." We appear to have a
single thought which does not have a definite object, but has as its object an
undetermined one of the values of the function "A is A." It is this
kind of ambiguity that constitutes the essence of a function. When we speak of
"φx," where x is not specified, we mean one value of
the function, but not a definite one. We may express this by saying that "φx" ambiguously
denotes φa, φb, φc, etc., where φa, φb, φc, etc., are the
various values of "φx."
When we say that "φx" ambiguously
denotes φa, φb, φc, etc., we mean that
“φx"
means one of the objects φa, φb, φc, etc., though not
a definite one, but an undetermined one. It follows that "φx" only has a
well-defined meaning (well-defined, that is to say, except in so far as it is
of its essence to be ambiguous) if the objects φa, φb, φc, etc., are
well-defined. That is to say, a function is not a well-defined function unless
all its values are already well defined. It follows from this that no function
can have among its values anything which presupposes the function, for if it had,
we could not regard the objects ambiguously denoted by the function as definite
until the function was definite, while conversely, as we have just seen, the
function cannot be definite until its values are definite. This is a particular
case, but perhaps the most fundamental case, of the vicious-circle principle. A
function is what ambiguously denotes some one of a certain totality, namely the
values of the function; hence this totality cannot contain any members which
involve the function, since, if it did, it would contain members involving the
totality, which, by the vicious-circle principle, no totality can do.
It will be seen that,
according to the above account, the values of a function are presupposed by the
function, not vice versa. It is sufficiently obvious, in any particular case,
that a value of a function does not presuppose the function. Thus for example
the proposition "Socrates is human" can be perfectly apprehended
without regarding it as a value of the function "x is human." It is true that, conversely, a function can be
apprehended without its being necessary to apprehend its values severally and
individually. If this were not the case, no function could be apprehended at
all, since the number of values (true and false) of a function is necessarily
infinite and there are necessarily possible arguments with which we are
unacquainted. What is necessary is not that the values should be given
individually and extensionally, but that the totality of the values should be
given intentionally, so that, concerning any assigned object, it is at least
theoretically determinate whether or not the said object is a value of the
function.
It is necessary
practically to distinguish the function itself from an undetermined value of
the function. We may regard the function itself as that which ambiguously
denotes, while an undetermined value of the function is that which is
ambiguously denoted. If the undetermined value is written "φx," we will
write the function itself "φx^."
(Any other letter may be used in place of x.)
Thus we should say " φx
is a proposition," but "φx^
is a propositional function." When we say "φx is a
proposition," we mean to state something which is true for every possible
value of x, though we do not decide what value x is to have. We are making an ambiguous statement about any value
of the function. But when we say "φx^
is a function," we are not making an ambiguous statement. It would be more
correct to say that we are making a statement about an ambiguity, taking the
view that a function is an ambiguity. The function itself, φx^ is the single thing
which ambiguously denotes its many values; while φx, where x is not specified, is one of the
denoted objects, with the ambiguity belonging to the manner of denoting.
We have seen that, in
accordance with the vicious-circle principle, the values of a function cannot
contain terms only definable in terms of the function. Now given a function φx^, the values for
the function†2
are all pro positions of the form φx.
It follows that there must be no propositions, of the form φx, in which x has a value which involves φx^. (If this were
the case, the values of the function would not all be determinate until the
function was determinate, whereas we found that the function is not determinate
unless its values are previously determinate.) Hence there must be no such
thing as the value for φx^
with the argument φx^,
or with any argument which involves φx.
That is to say, the symbol "φ(φx^)" must not
express a proposition, as " φa" does if φa
is a value for φx^. In
fact "φ(φx^)"
must be a symbol which does not express anything: we may therefore say that it
is not significant. Thus given any function φx^,
there are arguments with which the function has no value, as well as arguments
with which it has a value. We will call the arguments with which φx^ has a value
"possible values of x." We
will say that φx^ is
"significant with the argument x"
when φx^ has a value with
the argument x.
†1
When the word "function" is used in the sequel, "propositional
function" is always meant. Other functions will not be in question in the
present Chapter.
†2
We shall speak in this Chapter of "values for φx^" and of
"values of φx,"
meaning in each case the same thing, namely φa, φb, φc, etc. The
distinction of phraseology serves to avoid ambiguity where several variables
are concerned, especially when one of them is a function.
 |
| Legion of Superheroes™ © Respective copyright/trademark holders. |
This examination of propositional functions is so important
in understanding our system, but also the intent of the system. In our culture
we seek to define all of our surroundings; and then live within these
understandings, using them as a shield against the unknown. It is the way of Indian
peoples to exist comfortably in ambiguity; where a decision is made ad hoc,
only as it is called for, without pre-conceived supposition about every person
or situation encountered.
Like all metaphysical
systems, the details are of little use as memorized and repeated on a conscious
level. Their value is as absorbed into our being, and our understanding
intentional within the ambiguity that is all of life. Although all of dominant
culture seeks rules and laws, principles and moral judgments of what is right
and wrong in rigid structures that apply to all persons in all circumstances.
The problem with this is that these rigid structures are always only approximate
at best. Formulas of agreed upon equations that are compromises, because ‘truths’
are mostly impossible to agree upon or to pin down. Inevitably the intended results
are subverted to the forms invented to define them. It is always the case that
a definition is not the object defined, and vice versa.
 |
|
Dan DeCarlo The
greatest of all time comic book artist!
Archie Andrews
Gang © Respective copyright/trademark
holders.
|
 |
|
Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark
holders.
|
No comments:
Post a Comment