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Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark
holders.
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Plural
descriptive functions. The class of terms x which have the relation R
to some member of a class a
is denoted by Rʻʻa
or Rєʻa.
The definition is
Rʻʻa =
x {(∃y).
y є a .
xRy} Df.
Thus for example let R be the relation of inhabiting, and a
the class of towns; then Rʻʻa =
inhabitants of towns. Let R be the
relation "less than" among rationals, and a
the class of those rationals which are of the form 1 - 2-n, for
integral values of n; then Rʻʻa
will be all rationals less than some member of a, i.e. all rationals less than 1. If P is the generating relation of a
series, and a is any class of members
of the series, Pʻʻa
will be predecessors of a's, i.e. the
segment defined by a. If P is a relation such that Pʻy always exists when
y є a, P"a
will be the class of all terms of the form Pʻy for values of y which are members of a; i.e.
Pʻʻa =
x^{(∃y) . y є a .
x = Pʻy}.
Thus a member of the class
"fathers of great men" will be the father of y, where y is some great
man. In other cases, this will not hold; for instance, let P be the relation of a number to any number of which it is a
factor; then Pʻʻ
(even numbers) = factors of even numbers, but this class is not composed of
terms of the form "the factor of x,"
where x is an even number, because
numbers do not have only one factor apiece.
Unit
classes. The class whose only member is x might be thought to be identical with x, but Peano and Frege have shown that this is not the case. (The
reasons why this is not the case will be explained in a preliminary way in
Chapter II of the Introduction.) We denote by "
ɿʻx "the
class whose only member is x: thus
ɿʻx = y^(y=x) Df,
i.e.
"
ɿʻx "
means "the class of objects which are identical with x."
The class consisting of x and y will be ɿʻx ◡ ɿʻx; the class got by
adding x to a class a
will be a ◡ ɿʻx; the class got by
taking away x from a class a
will be a- ɿʻx. (We write a -
b as an abbreviation for a ◠ -b.)
It will be observed that
unit classes have been defined without reference to the number 1; in fact, we
use unit classes to define the number 1. This number is defined as the class of
unit classes,
1 = a^
{(∃x). a =
ɿʻx } Df.
This leads to
┠ :.
a є 1 . ≡ : (∃x): y є a .
≡y . y = x.
From this it appears
further that
┠ :
a є 1 . ≡ . E! (ɿʻx) (x є a),
whence ┠ :
z^(φz) є 1 . ≡. E! (ɿʻx) (φx),
i.e.
"z^(φz) is a unit
class" is equivalent to "the x
satisfying φx^
exists."
If a є
1, ɿ⏑ʻa
is the only member of a, for the only
member of a is the only term to which
a has the relation ɿ.
Thus "ɿʻa" takes the
place of “(ɿʻx)
(φx)," if a
stands for z^(φz). In practice, "ɿʻa"
is a more convenient notation than “(ɿʻx) (φx)," and is
generally used instead of “(ɿʻx) (φx).”
The above account has
explained most of the logical notation employed in the present work. In the
applications to various parts of mathematics, other definitions are introduced;
but the objects defined by these later definitions belong, for the most part,
rather to mathematics than to logic. The reader who has mastered the symbols
explained above will find that any later formulae can be deciphered by the help
of comparatively few additional definitions.
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| Script: Stan Lee Pencils: & Inks: Steranko © Respective copyright/trademark holders.
Throughout the previous 24 posts we have seen so many
symbols, equations and definitions, but in all there was not a single number
used in any notation until finally the number 1 is elaborately defined here at
the end of chapter one of the introduction of Principia Mathematica. I can’t
help reading a bit of tongue-in-cheek in the author’s putting this definition
of 1 at the end of this chapter.
With the tools we have seen
so far we have the bare bones of the process, and may be able to begin thinking
of and ordering our outlines and propositional equations that are immediately more
useful and quantifiable than before. In the posts ahead we shall see many of
layers of fine distinction added to our present study.
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Script: Stan Lee Pencils: Jack Kirby Inks: Frank Giacoia
Captain America © Respective
copyright/trademark holders.
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark
holders.
|
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