Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
The definitions which occur in the theory of classes,
by which the idea of a class (at least in use) is based on the other ideas
assumed as primitive, cannot be understood without a fuller discussion than can
be given now (cf. Chapter II of this Introduction and also *20). Accordingly,
in this preliminary survey, we proceed to state the more important simple
propositions which result from those definitions, leaving the reader to employ
in his mind the ordinary unanalyzed idea of a class of things. Our symbols in
their usage conform to the ordinary usage of this idea in language. It is to be
noticed that in the systematic exposition our treatment of classes and
relations requires no new primitive ideas and only two new primitive
propositions, namely the two forms of the "Axiom of Reducibility"
(cf. next Chapter) for one and two variables respectively. The propositional
function "x is a member of the
class a " will be expressed, following Peano,
by the notation
x є
a.
Here є is chosen as the initial of the word є’τσί (so). "x є a " may be read
" x is an a."
Thus "x є man" will mean
"x is a man," and so on.
For typographical convenience we shall put
x ~
є a . = . ~ (x є a) Df,
x,y є
a . =. x
є a . y
є a Df.
For "class" we shall write "Cls";
thus "a є Cls" means" a
is a class."
We have-:
├ : x є z^ (φz). º .
φx,
i.e. "'x is a member of the class determined by φz^' is equivalent to 'x
satisfies φz^,' or to 'φx is true.’"
A class is wholly determinate when its membership is
known, that is, there cannot be two different classes having the same
membership. Thus if φx, yx are formally
equivalent functions, they determine the same class; for in that case, if x is a member of the class determined by
φx^, and therefore satisfies φx, it also satisfies yx, and is therefore
a member of the class determined by yx^. Thus we have
├ :. z^ (φz) = z^(yz) . º :
φx . ºx .
yx.
The following propositions are obvious and important:
├ :. a =
z^(φz) . º : x є a .
ºx .
φx,
i.e. a
is identical with the class determined by φz^
when, and only when, "x is an a"
is formally equivalent to φx;
├ :. a =
b . º :
x є a .
ºx .
x є b,
i.e.
two classes a and b
are identical when, and only when, they have the same membership;
├ . x^(x є a)
= a,
i.e.
the class whose determining function is "x is an a" is a,
in other words, a is the class of
objects which are members of a;
├ . z^(φz) є Cls,
i.e.
the class determined by the function φz^
is a class.
It will be seen that, according to the above, any
function of one variable can be replaced by an equivalent function of the form
"x є a."
Hence any extensional function of functions which holds when its argument is a
function of the form "z^ є a,"
whatever possible value a may have, will
hold also when its argument is any function φz^. Thus variation of classes can replace variation of functions of
one variable in all the propositions of the sort with which we are concerned.
 |
| Dan DeCarlo The
greatest of all time comic book artist!Archie Andrews
Gang
© Respective copyright/trademark
holders.
|
Class is kind of a
dirty word today in North America, with its connotation of snooty judgments of
persons based on seemingly random criteria. In our system here class is one of the
keys to building propositions in equations that can be handled and crunched
like numbers.
Script: Gaylord Du Bois Pencils and Inks: Russ Manning © Respective
copyright/trademark holders.
Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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