| Thread Links | Date Links | ||||
|---|---|---|---|---|---|
| Thread Prev | Thread Next | Thread Index | Date Prev | Date Next | Date Index |
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note D36
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Transformations of General Type
| 'Es ist passiert', "it just sort of happened", people said there
| when other people in other places thought heaven knows what had
| occurred. It was a peculiar phrase, not known in this sense to
| the Germans and with no equivalent in other languages, the very
| breath of it transforming facts and the bludgeonings of fate
| into something light as eiderdown, as thought itself.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 34]
Consider the situation illustrated in Figure 30, where the alphabets
!U! = {u, v} and !X! = {x, y, z} are used to label basic features in
two different logical universes, U% = [u, v] and X% = [x, y, z].
o-------------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------o---------------------------o
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
o-------------------------o o-------------------------o o-------------------------o
| U | | U | | U |
| o---o o---o | | o---o o---o | | o---o o---o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / / \ \ | | / / \ \ | | / / \ \ |
| o o o o | | o o o o | | o o o o |
| | u | | v | | | | u | | v | | | | u | | v | |
| o o o o | | o o o o | | o o o o |
| \ \ / / | | \ \ / / | | \ \ / / |
| \ o / | | \ o / | | \ o / |
| \ / \ / | | \ / \ / | | \ / \ / |
| o---o o---o | | o---o o---o | | o---o o---o |
| | | | | |
o-------------------------o o-------------------------o o-------------------------o
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ g | \ f / | h /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ o----------|-----------\-----/-----------|----------o /
\ | X | \ / | | /
\ | | \ / | | /
\ | | o-----o-----o | | /
\| | / \ | |/
\ | / \ | /
|\ | / \ | /|
| \ | / \ | / |
| \ | / \ | / |
| \ | o o | / |
| \ | | | | / |
| \ | | x | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \| | | |/ |
| o--o--------o o--------o--o |
| / \ \ / / \ |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o--o-----o--o o |
| | | | | |
| | | | | |
| | | | | |
| | y | | z | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------------------------------o
\ /
\ /
\ /
\ /
\ /
\ p , q /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
o
Figure 30. Generic Frame of a Logical Transformation
Enter the picture, as we usually do, in the middle of things, with features
like x, y, z that present themselves to be simple enough in their own right
and that form a satisfactory, if a temporary, foundation to provide a basis
for discussion. In this universe and on these terms we find expression for
various propositions and questions of principal interest to ourselves, as
indicated by the maps p, q : X -> B. Then we discover that the simple
features {x, y, z} are really more complex than we thought at first,
and it becomes useful to regard them as functions {f, g, h} of other
features {u, v}, that we place in a preface to our original discourse,
or suppose as topics of a preliminary universe of discourse U% = [u, v].
It may happen that these late-blooming but pre-ambling features are found
to lie closer, in a sense that may be our job to determine, to the central
nature of the situation of interest, in which case they earn our regard as
being more fundamental, but these functions and features are only required
to supply a critical stance on the universe of discourse or an alternate
perspective on the nature of things in order to be preserved as useful.
A particular transformation F : [u, v] -> [x, y, z] may be expressed
by a system of equations, as shown below. Here, F is defined by its
component maps F = <F_1, F_2, F_3> = <f, g, h>, where each component
map in {f, g, h} is a proposition of type B^n -> B^1.
o-------------------------------------------------o
| |
| x = f<u, v> |
| |
| y = g<u, v> |
| |
| z = h<u, v> |
| |
o-------------------------------------------------o
Regarded as a logical statement, this system of equations expresses a relation
between a collection of freely chosen propositions {f, g, h} in one universe
of discourse and the special collection of simple propositions {x, y, z} on
which are founded another universe of discourse. Growing familiarity with
a particular transformation of discourse, and the desire to achieve a ready
understanding of its implications, requires that we be able to convert this
information about generals and simples into information about all the main
subtypes of propositions, including the linear and singular propositions.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o