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Harold R. (Hal)
Foster’s Prince Valiant
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VI. The Axiom of Reducibility.
It remains to consider the "axiom of
reducibility." It will be seen that, according to the above hierarchy, no
statement can be made significantly about "all a-functions,"
where a is some given object. Thus such a notion
as "all properties of a,"
meaning "all functions which are true with the argument a," will be
illegitimate. We shall have to distinguish the order of function concerned. We
can speak of "all predicative properties of a,"
"all second-order properties of a,"
and so on. (If a is not an
individual, but an object of order n,
"second-order properties of a"
will mean "functions of order n
+ 2 satisfied by a.")
But we cannot speak of "all properties of a."
In some cases, we can see that some statement will hold of "all nth-order properties of a,"
whatever value n may have. In such
cases, no practical harm results from regarding the statement as being about
"all properties of a," provided we
remember that it is really a number of statements, and not a single statement
which could be regarded as assigning another property to a,
over and above all properties. Such cases will always involve some systematic
ambiguity, such as that involved in the meaning of the word "truth,"
as explained above. Owing to this systematic ambiguity, it will be possible,
sometimes, to combine into a single verbal statement what are really a number
of different statements, corresponding to different orders in the hierarchy.
This is illustrated in the case of the liar, where the statement "all A's
statements are false" should be broken up into different statements
referring to his statements of various orders, and attributing to each the
appropriate kind of falsehood.
The axiom of reducibility is introduced in order to
legitimate a great mass of reasoning, in which, prima facie, we are concerned
with such notions as "all properties of a"
or "all a-functions," and in
which, nevertheless, it seems scarcely possible to suspect any substantial
error. In order to state the axiom, we must first define what is meant by
"formal equivalence." Two functions φx^, yx^ are said to be
"formally equivalent" when, with every possible argument x, φx is equivalent to yx,
i.e. φx and yx are either both
true or both false. Thus two functions are formally equivalent when they are
satisfied by the same set of arguments. The axiom of reducibility is the
assumption that, given any function φx^,
there is a formally equivalent predicative function, i.e. there is a predicative function which is true when φx is true and false
when φx is false. In
symbols, the axiom is:
┠ : (∃y) :
φx . ≡x . y!x.
For two variables, we require a similar axiom, namely:
Given any function φ(x^, y^), there is a
formally equivalent predicative function, i.e.
┠ : (∃y) :
φ(x, y). ≡x,y . y!x.
In order to explain the purposes of the axiom of
reducibility, and the nature of the grounds for supposing it true, we shall
first illustrate it by applying it to some particular cases.
If we call a predicate
of an object a predicative function which is true of that object, then the
predicates of an object are only some among its properties. Take for example
such a proposition as "Napoleon had all the qualities that make a great
general." We may interpret this as meaning "Napoleon had all the
predicates that make a great general. "Here there is a predicate which is an
apparent variable. If we put "ƒ(φ!z^)" for
"φ!z^ is a predicate required in a great
general," our proposition is
(φ): ƒ(φ!z^) implies φ!(Napoleon).
Since this refers to a totality of predicates, it is
not itself a predicate of Napoleon. It by no means follows, however, that there
is not some one predicate common and peculiar to great generals. In fact, it is
certain that there is such a predicate. For the number of great generals is
finite, and each of them certainly possessed some predicate not possessed by
any other human being-for example, the exact instant of his birth. The
disjunction of such predicates will constitute a predicate common and peculiar
to great generals*1.
If we call this predicate y!z^, the statement we made about Napoleon
was equivalent to y!(Napoleon).
And this equivalence holds equally if we substitute any other individual for
Napoleon. Thus we have arrived at a predicate which is always equivalent to the
property we ascribed to Napoleon, i.e.
it belongs to those objects which have this property, and to no others. The
axiom of reducibility states that such a predicate always exists, i.e. that any property of an object
belongs to the same collection of objects as those that possess some predicate.
We may next illustrate our principle by its application
to identity. In this connection, it
has a certain affinity with Leibniz's identity of indiscernibles. It is plain
that, if x and y are identical, and φx
is true, then φy
is true. Here it cannot matter what sort of function φx^ may be: the
statement must hold for any function.
But we cannot say, conversely: "If, with all values of φ, φx implies φy, then x and y are identical"; because "all values of φ"
is inadmissible. If we wish to speak of "all values of φ,"
we must confine ourselves to functions of one order. We may confine φ
to predicates, or to second-order functions, or to functions of any order we
please. But we must necessarily leave out functions of all but one order. Thus
we shall obtain, so to speak, a hierarchy of different degrees of identity. We
may say "all the predicates of x
belong to y," "all
second-order properties of x belong
to y," and so on. Each of these
statements implies all its predecessors: for example, if all second-order
properties of x belong to y, then all
predicates of x belong to y, for to have all the predicates of x is a second-order property, and this
property belongs to x. But we cannot,
without the help of an axiom, argue conversely that if all the predicates of x belong to y, all the second-order properties of x must also belong to y. Thus
we cannot, without the help of an axiom, be sure that x and y are identical if
they have the same predicates. Leibniz's identity of indiscernibles supplied
this axiom. It should be observed that by "indiscernibles" he cannot
have meant two objects which agree as to all
their properties, for one of the properties of x is to be identical with x, and therefore this property would
necessarily belong to y if x and y agreed in all their properties. Some limitation of the common
properties necessary to make things indiscernible is therefore implied by the
necessity of an axiom. For purposes of illustration (not of interpreting
Leibniz) we may suppose the common properties required for indiscernibility to
be limited to predicates. Then the identity of indiscernibles will state that
if x and y agree as to all their predicates, they are identical. This can be
proved if we assume the axiom of reducibility. For, in that case, every
property belongs to the same collection of objects as is defined by some
predicate. Hence there is some predicate common and peculiar to the objects
which are identical with x. This
predicate belongs to x, since x is identical with itself; hence it
belongs to y, since y has all the predicates of x; hence y is identical with x. It
follows that we may define x and y as identical when all the predicates
of x belong to y, i.e. when (φ):
φ!x . ⊃ .
φ!y. We therefore adopt the following
definition of identity*2:
x=y
. = : (φ) :
φ!x . ⊃ .
φ!y Df.
But apart from the axiom of reducibility, or some axiom
equivalent in this connection, we should be compelled to regard identity as
indefinable, and to admit (what seems impossible) that two objects may agree in
all their predicates without being identical.
The axiom of reducibility is even more essential in the
theory of classes. It should be observed, in the first place, that if we assume
the existence of classes, the axiom of reducibility can be proved. For in that
case, given any function φz^
of whatever order, there is a class a
consisting of just those objects which satisfy φz. Hence "φz" is equivalent to "x belongs to a."
But "x belongs to a"
is a statement containing no apparent variable, and is therefore a predicative
function of x. Hence if we assume the
existence of classes, the axiom of reducibility becomes unnecessary. The
assumption of the axiom of reducibility is therefore a smaller assumption than
the assumption that there are classes. This latter assumption has hitherto been
made unhesitatingly. However, both on the ground of the contradictions, which
require a more complicated treatment if classes are assumed, and on the ground
that it is always well to make the smallest assumption required for proving our
theorems, we prefer to assume the axiom of reducibility rather than the
existence of classes. But in order to explain the use of the axiom in dealing
with classes, it is necessary first to explain the theory of classes, which is
a topic belonging to Chapter III. We therefore postpone to that Chapter the explanation
of the use of our axiom in dealing with classes.
It is worth while to note that all the purposes served
by the axiom of reducibility are equally well served if we assume that there is
always a function of the nth order
(where n is fixed) which is formally
equivalent to φx^,
whatever may be the order of φx^.
Here we shall mean by "a function of the nth order" a function of the nth order relative to the arguments to φx^; thus if these arguments are absolutely of the mth order, we assume the existence of a
function formally equivalent to φx^
whose absolute order is the m + nth. The axiom of reducibility in the
form assumed above takes n = 1, but
this is not necessary to the use of the axiom. It is also unnecessary that n
should be the same for different values of m; what is necessary is that n should be constant so long as m is constant. What is needed is that,
where extensional functions of functions are concerned, we should be able to
deal with any a-function by means of some
formally equivalent function of a given type, so as to be able to obtain
results which would otherwise require the illegitimate notion of "all a-functions";
but it does not matter what the given type is. It does not appear, however,
that the axiom of reducibility is rendered appreciably more plausible by being
put in the above more general but more complicated form.
The axiom of reducibility is equivalent to the
assumption that "any combination or disjunction of predicates*3 is equivalent
to a single predicate," i.e. to
the assumption that, if we assert that x
has all the predicates that satisfy a function ƒ(φ!z^), there is some one predicate which x
will have whenever our assertion is true, and will not have whenever it is
false, and similarly if we assert that x
has some one of the predicates that satisfy a function ƒ(φ!z^). For by means of this assumption, the order of a non-predicative
function can be lowered by one; hence, after some finite number of steps, we
shall be able to get from any non-predicative function to a formally equivalent
predicative function. It does not seem probable that the above assumption could
be substituted for the axiom of reducibility in symbolic deductions, since its
use would require the explicit introduction of the further assumption that by a
finite number of downward steps we can pass from any function to a predicative
function, and this assumption could not well be made without developments that
are scarcely possible at an early stage. But on the above grounds it seems
plain that in fact, if the above alternative axiom is true, so is the axiom of
reducibility. The converse, which completes the proof of equivalence, is of
course evident.
*1
When a (finite) set of predicates is given by actual enumeration, their
disjunction is a predicate, because no predicate occurs as apparent variable in
the disjunction.
*2
Note that in this definition the second sign of equality is to be regarded as
combining with "Df" to form one symbol; what is defined is the sign
of equality not followed by the letters "Df."
*3
Here the combination or disjunction is supposed to be given intensionally. If
given extensionally (i.e. by enumeration), no assumption is required; but in
this case the number of predicates concerned must be finite.
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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