Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark holders
Every use of "(ɿx)(φx),” where it apparently occurs as a
constituent of a proposition in the place of an object, is defined in terms of
the primitive ideas already on hand. An example of this definition in use is
given by the proposition "E ! (ɿx)(φx)"
which is considered immediately. The whole subject is treated more fully ahead in
Chapter III.
The symbol should be compared and contrasted with
"x^(φx)" which in use can always be read
as "the x's which satisfy φx."
Both symbols are incomplete symbols defined only in use, and as such are
discussed in Chapter III. The symbol "
x^(φx)" always has an application,
namely to the class determined by φx;
but "(ɿx)(φx)" only has an application when φx^ is only satisfied by one value of x, neither more nor less. It should also
be observed that the meaning given to the symbol by the definition, given
immediately below, of E ! (ɿx)(φx). does not
presuppose that we know the meaning of "one." This is also
characteristic of the definition of any other use of (ɿx)(φx).
We now proceed to define "E! (ɿx)(φx)” so that it can
be read "the x satisfying φx
exists." (It will be observed that this is a different meaning of
existence from that which we express by “∃.") Its definition is
E ! ((ɿx) (φx)
. = : (∃c) : φx . ºx . x
=c Df,
i.e. "the x satisfying φx^
exists" is to mean "there is an object c such that φx is true when x is c but not
otherwise."
The following are equivalent forms:
┠ :. E! (ɿx) (φx) . º : (∃c) : φc :
φx. É . x = c,
┠ :. E! (ɿx) (φx) . º : (∃c) . φc
: φx φy Éxy . x = y,
┠ :. E! (ɿx) (φx) . º : (∃c) : φc :
x ≠ c
. Éx . ~ φx.
The last of these states that "the x
satisfying φ^ exists" is equivalent to
"there is an object c satisfying
φ^, and every object other than c does not satisfy φ^."
The kind of existence just defined covers a
great many cases. Thus for example "the most perfect Being exists"
will mean:
(∃c) : x is most perfect. . ºx . x
= c,
which, taking the last of the above
equivalences, is equivalent to
(∃c): c is most
perfect : x ≠ c . Éx . x is not most perfect.
A proposition such as "Apollo
exists" is really of the same logical form, although it does not
explicitly contain the word the. For
"Apollo" means really "the object having such-and-such
properties," say "the object having the properties enumerated in the
Classical Dictionary*. "If these properties make up the propositional
function φx, then "Apollo" means "(ɿx)(φx)," and
"Apollo exists" means "E! "(ɿx)(φx)." To take
another illustration, "the author of Waverley" means "the man
who (or rather, the object which) wrote Waverley." Thus "Scott is the
author of Waverley" is
Scott = (ɿx)(x wrote
Waverley).
Here (as we observed before) the importance
of identity in connection with descriptions plainly appears.
The notation “(ɿx) (φx),” which is long
and inconvenient, is seldom used, being chiefly required to lead up to another
notation, namely "Rʻy," meaning "the object having the relation R to y." That is, we put
Rʻy = (ɿx) (xRy) Df.
The inverted comma may be read
"of." Thus "Rʻy" is read "the R of y."
Thus if R is the relation of father
to son, " Rʻy" means "the father of y"; if R is the
relation of son to father, "Rʻy" means "the son of y," which will only "exist"
if y has one son and no more. R'y is a function of y, but not a propositional
function; we shall call it a descriptive
function. All the ordinary functions of mathematics are of this kind, as will
appear more fully in the sequel. Thus in our notation, " sin y" would
be written " sin ʻy," and "sin" would stand for the
relation which sin ʻy has to y. Instead
of a variable descriptive function ƒy, we put Rʻy, where the variable relation R takes the place of the variable
function ƒ. A descriptive function will in general
exist while y belongs to a certain
domain, but not outside that domain; thus if we are dealing with positive
rationals, √y will be
significant if y is a perfect square,
but not otherwise; if we are dealing with real numbers, and agree that "√y"
is to mean the positive square root
(or, is to mean the negative square root), √y will be significant provided y is
positive, but not otherwise; and so on.
Thus every descriptive function has what we may call a
"domain of definition" or a "domain of existence," which
may be thus defined: If the function in question is Rʻy,
its domain of definition or of existence will be the class of those arguments y for which we have E! Rʻy, i.e. for which E!(ɿx)(xRy), i.e. for which there is one x, and no more, having the relation R to y.
If R is any
relation, we will speak of Rʻy as the
"associated descriptive function." A great many of the constant
relations which we shall have occasion to introduce are only or chiefly
important on account of their associated descriptive functions. In such cases,
it is easier (though less correct) to begin by assigning the meaning of the
descriptive function, and to deduce the meaning of the relation from that of
the descriptive function. This will be explored in the explanations of notation
in the next post.
* The same principle applies to many uses
of the proper names of existent objects, e.g. to all uses of proper names for
objects known to the speaker only by report, and not by personal acquaintance (i.e. the Holy Grail, the North Pole, the elephants graveyard).
Script: Richard Hughes Pencils and inks: Chic Stone
©
Respective copyright/trademark holders.
The above part begins to address the process of constructing
symbols that can serve to represent thoughts, expressions, objects, relations,
etc. with which we may construct our propositional equations. Aspects of this
system appear akin to binary language in nature. In fact we will, in future
posts examine the construction of similar even more compact equations by means
of ‘the stroke,’ that are again a step towards even more simplification.
Clearly this system may be translated into binary code and function on a
computer in some fashion; but it would have no useful function. The purpose of
this system is to bring order to thought, improve the processing of stored data
of the mind and integrate it with new perceptions, in real time. This is
accomplished by understanding the nature of this data and systems of combining
it in increasingly more efficient ways, with avoiding error as the main
priority.
With regard to my thoughts in the last previous post here,
on Evident Reality as a logical tool to add equational ‘proof’ to Einstein’s
Theories of Special and General Relativity, applying physical and cultural data
derived from our shared and collected human experience; by means of Whitehead
and Russell’s system of logical philosophy. Below is an initial framing of the
equation I laid out this afternoon over three cups of coffee, using mainly
elements we have already encountered in previous posts. As we see below, with
our system, great complexity begins with a relatively simple set of divisions in
which great complexity may be divined.
I summed my scribbling up
for my waiter with the quote at the top of the sheet paraphrasing the authors: ‘analysis
applied to process equals division. ’
An example of this
blogger’s regrettable handwriting.
Scripted (perhaps by): Al Capp Pencils and Inks: Frank Frazetta © Respective copyright/trademark holders.
Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark holders
No comments:
Post a Comment