ONT Re: Differential Logic
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DLOG. Note D26
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| And, despite the care which she took to look behind her at every moment,
| she failed to see a shadow which followed her like her own shadow, which
| stopped when she stopped, which started again when she did and which made
| no more noise than a well-conducted shadow should.
|
| Gaston Leroux, 'The Phantom of the Opera', [Ler, 126]
Foreshadowing Transformations: Extensions and Projections of Discourse
Many times in our discussion we have occasion to place one universe of discourse
in the context of a larger universe of discourse. An embedding of the general
type [!X!] -> [!Y!] is implied any time that we make use of one alphabet !X!
that happens to be included in another alphabet !Y!. When we are discussing
differential issues we usually have in mind that the extended alphabet !Y!
has a special construction or a specific lexical relation with respect to
the initial alphabet !X!, one that is marked by characteristic types of
accents, indices, or inflected forms.
Extension from 1 to 2 Dimensions
Figure 18-a lays out the "angular form" of venn diagram for universes
of 1 and 2 dimensions, indicating the embedding map of type B^1 -> B^2
and detailing the coordinates that are associated with individual cells.
Because all points, cells, or logical interpretations are represented
as connected geometric areas, we can say that these pictures provide
us with an "areal view" of each universe of discourse.
o-----------------------------------------------------------o
| |
| o o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / o o 1 1 o |
| / / \ / \ / \ |
| / / \ / \ / \ |
| / 1 / \ / \ / \ |
| / / \ !e! / \ / \ |
| o / o ----> o 1 0 o 0 1 o |
| |\ / / |\ / \ /| |
| | \ / 0 / | \ / \ / | |
| | \ / / | \ / \ / | |
| |x_1\ / / |x_1\ / \ /x_2| |
| o----o / o----o 0 0 o----o |
| \ / \ / |
| \ / \ / |
| \ / \ / |
| \ / \ / |
| o o |
| |
o-----------------------------------------------------------o
Figure 18-a. Extension from 1 to 2 Dimensions: Areal
Figure 18-b shows the differential extension from X% = [x] to EX% = [x, dx] in
a "bundle of boxes" form of venn diagram. As awkward as it may seem at first,
this type of picture is often the most natural and the most easily available
representation when we want to conceptualize the localized information or
momentary knowledge of an intelligent dynamic system. It gives a ready
picture of a "proposition at a point", in the present instance, of a
proposition about changing states which is itself associated with a
particular dynamic state of a system. It is easy to see how this
application might be extended to conceive of more general types
of instantaneous knowledge that are possessed by a system.
o-----------------------------o o-------------------o
| | | |
| | | o-------o |
| o---------o | | / \ |
| / \ | | o o |
| / o------------------------| | dx | |
| / \ | | o o |
| / \ | | \ / |
| o o | | o-------o |
| | | | | |
| | | | o-------------------o
| | x | |
| | | | o-------------------o
| | | | | |
| o o | | o-------o |
| \ / | | / \ |
| \ / | | o o |
| \ / o------------| | dx | |
| \ / | | o o |
| o---------o | | \ / |
| | | o-------o |
| | | |
o-----------------------------o o-------------------o
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
Figure 18-c shows the same extension in a "compact" style of venn diagram,
where the differential features at each position are represented by arrows
from that position that cross the corresponding feature boundaries.
o-----------------------------------------------------------o
| |
| |
| o-----------------o |
| / o \ |
| / (dx) / \ \ dx |
| / v o--------------------->o |
| / \ / \ |
| / o \ |
| o o |
| | | |
| | | |
| | x | (x) |
| | | |
| | | |
| o o |
| \ / o |
| \ / / \ |
| \ o<---------------------o v |
| \ / dx \ / (dx) |
| \ / o |
| o-----------------o |
| |
| |
o-----------------------------------------------------------o
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
Figure 18-d compresses the picture of the differential extension even further,
yielding a directed graph or "digraph" form of representation. (Notice that
my definition of a digraph allows for loops or "slings" at individual points,
in addition to arcs or "arrows" between the points.)
o-----------------------------------------------------------o
| |
| |
| dx |
| o o------>>-------o o |
| / \ / \ / \ |
| (dx) ^ @ x (x) @ ^ (dx) |
| \ / \ / \ / |
| o o------<<-------o o |
| dx |
| |
| |
o-----------------------------------------------------------o
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
Jon Awbrey
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