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Harold R. (Hal)
Foster’s Prince Valiant
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d_
and
∫b
dx a
are incomplete symbols: something has to be supplied
before we have anything significant. Such symbols have what may be called a
"definition in use." Thus if we put
∇2 =
¶2 +
¶2 +
¶2
¶x2 ¶y2 ¶z2 Df,
we define the use of ∇2,
but ∇2
by itself remains without meaning. This distinguishes such symbols from what
(in a generalized sense) we may call proper names: "Socrates," for
example, stands for a certain man, and therefore has a meaning by itself,
without the need of any context. If we supply a context, as in "Socrates
is mortal," these words express a fact of which Socrates himself is a
constituent: there is a certain object, namely Socrates, which does have the
property of mortality, and this object is a constituent of the complex fact
which we assert when we say "Socrates is mortal." But in other cases,
this simple analysis fails us. Suppose we say: "The round square does not
exist." It seems plain that this is a true proposition, yet we cannot
regard it as denying the existence of a certain object called "the round
square." For if there were such an object, it would exist: we cannot first
assume that there is a certain object, and then proceed to deny that there is
such an object. Whenever the grammatical subject of a proposition can be
supposed not to exist without rendering the proposition meaningless, it is
plain that the grammatical subject is not a proper name, i.e. not a name directly representing some object. Thus in all such
cases, the proposition must be capable of being so analyzed that what was the
grammatical subject shall have disappeared. Thus when we say "the round
square 'does not exist," we may, as a first attempt at such analysis,
substitute " it is false that there is an object x which is both round and
square." Generally, when "the so-and-so" is said not to exist,
we have a proposition of the form*1
"~E!(⍳x)(φx),"
i.e. ~{(∃C):
φx .
≡x . x = c},
or some equivalent. Here the apparent grammatical
subject (⍳x)(φx)
has completely disappeared; thus in
"~E!(⍳x)(φx),”
(⍳x)(φx)
is an incomplete symbol.
By an extension of the above argument, it can easily be
shown that (⍳x)(φx)
is always an incomplete symbol. Take,
for example, the following proposition: "Scott is the author of
Waverley." [Here "the author of Waverley" is (⍳x)(x wrote
Waverley).] This proposition expresses an identity; thus if "the author of
Waverley" could be taken as a proper name, and supposed to stand for some
object c, the proposition would be
"Scott is c." But if c is anyone except Scott, this
proposition is false; while if c is
Scott, the proposition is "Scott is Scott," which is trivial, and
plainly different from "Scott is the author of Waverley."
Generalizing, we see that the proposition
a =
(⍳x)(φx)
is one which may be true or may be false, but is never
merely trivial, like a = a; whereas, if (⍳x)(φx)
were a proper name, a = (⍳x)(φx)
would necessarily be either false or the same as the trivial proposition a = a. We may
express this by saying that a = (⍳x)(φx)
is not a value of the propositional function a = y, from which it follows that (⍳x)(φx)
is not a value of y. But since y may be anything, it follows that (⍳x)(φx)
is nothing. Hence, since in use it has meaning, it must be an incomplete
symbol.
It might be suggested that "Scott is the author of
Waverley" asserts that "Scott" and "the author of
Waverley" are two names for the same object. But a little reflection will
show that this would be a mistake. For if that were the meaning of "Scott
is the author of Waverley,"what would be required for its truth would be
that Scott should have been called the author of Waverley: if he had been so
called, the proposition would be true, even if someone else had written
Waverley; while if no one called him so, the proposition would be false, even
if he had written Waverley. But in fact he was the author of Waverley at a time
when no one called him so, and he would not have been the author if everyone
had called him so but someone else had written Waverley. Thus the proposition
"Scott is the author of Waverley "is not a proposition about names,
like "Napoleon is Bonaparte"; and this illustrates the sense in which
"the author of Waverley "differs from a true proper name.
Thus all
phrases (other than propositions) containing the word the (in the singular) are incomplete symbols: they have a meaning
in use, but not in isolation. For "the author of Waverley" cannot
mean the same as "Scott," or "Scott is the author of
Waverley" would mean the same as "Scott is Scott," which it
plainly does not; nor can "the author of Waverley" mean anything
other than "Scott," or "Scott is the author of Waverley" would
be false. Hence "the author of Waverley" means nothing.
It follows from the above that we must not attempt to
define "(⍳x)(φx)," but must
define the uses of this symbol, i.e. the propositions in whose symbolic
expression it occurs. Now in seeking to define the uses of this symbol, it is
important to observe the import of propositions in which it occurs. Take as an
illustration: "The author of Waverley was a poet." This implies (1)
that Waverley was written, (2) that it was written by one man, and not in
collaboration, (3) that the one man who wrote it was a poet. If any one of
these fails, the proposition is false. Thus "the author of 'Slawkenburgius
on Noses' was a poet" is false, because no such book was ever written;
"the author of 'The Maid's Tragedy' was a poet" is false, because
this play was written by Beaumont and Fletcher jointly. These two possibilities
of falsehood do not arise if we say "Scott was a poet." Thus our
interpretation of the uses of (⍳x)(φx) must be such as
to allow for them. Now taking φx
to replace "x wrote
Waverley," it is plain that any statement apparently about (⍳x)(φx) requires (1) (∃x).(φx) and (2) φx . φ .⊃x,y .
x = y; here (1) states that at
least one object satisfies φx,
while (2) states that at most one
object satisfies φx.
The two together are equivalent to
(∃c) :
φx . ≡x . x = c,
which we defined as E! (⍳x)(φx).
Thus "E! (⍳x)(φx)" must be
part of what is affirmed by any proposition about (⍳x)(φx). If our
proposition is ƒ{(⍳x)(φx)}, what is further
affirmed is ƒc, if φx . ≡x . x = c. Thus we have
ƒ{(⍳x)(φx) . = : (∃C) :
φx . ≡x . x = c : ƒc Df,
i.e.
"the x satisfying φx satisfies ƒx" is to mean: "There is an
object c such that φx is true when, and
only when, x is c, and ƒc is true, "or,
more exactly: " There is a c
such that 'φx'
is always equivalent to 'x is c,' and ƒc." In this, " (⍳x)(φx)" has
completely disappeared; thus " (⍳x)(φx) " is merely
symbolic, and does not directly represent an object, as single small Latin
letters are assumed to do*2.
*1 see post 22.
*2 We shall generally write "ƒ(⍳x)(φx)" rather than "ƒ{(⍳x)(φx)}" in future.
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Sheldon Mayer’s Sugar and SpikeTM © Respective copyright/trademark holders.
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Script: Stan Lee Pencils and Inks: Bill Everett © Respective
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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