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DLOG. Note D52
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Enlargement Map of Conjunction (concl.)
Figures 40-a through 40-d present several views of the enlarged proposition EJ.
o---------------------------------------o
| |
| o |
| /%\ |
| /%%%\ |
| /%%%%%\ |
| o%%%%%%%o |
| / \%%%%%/ \ |
| / \%%%/ \ |
| / \%/ \ |
| o o o |
| /%\ / \ /%\ |
| /%%%\ / \ /%%%\ |
| /%%%%%\ / \ /%%%%%\ |
| o%%%%%%%o o%%%%%%%o |
| / \%%%%%/ \ / \%%%%%/ \ |
| / \%%%/ \ / \%%%/ \ |
| / \%/ \ / \%/ \ |
| o o o o o |
| |\ / \ /%\ / \ /| |
| | \ / \ /%%%\ / \ / | |
| | \ / \ /%%%%%\ / \ / | |
| | o o%%%%%%%o o | |
| | |\ / \%%%%%/ \ /| | |
| | | \ / \%%%/ \ / | | |
| | u | \ / \%/ \ / | v | |
| o---+---o o o---+---o |
| | \ / \ / | |
| | \ / \ / | |
| | du \ / \ / dv | |
| o-------o o-------o |
| \ / |
| \ / |
| \ / |
| o |
| |
o---------------------------------------o
Figure 40-a. Enlargement of J (Areal)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| 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 |
| / /`\ @------\-----------@ |%% du %%%| | dv | |
| / /```\ \ | | o%%%%%%%%%o o o |
| / /`````\ \ | | \%%%%%%%%%\ / / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o 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 |
| \ @------\-/---------\---------------@ | du | |%%% dv %%| |
| \ 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%%|
@%%| du | | dv |%%|
|%%o o o o%%|
|%%%\ \ / /%%%|
|%%%%\ o /%%%%|
|%%%%%\ / \ /%%%%%|
|%%%%%%o-----o o-----o%%%%%%|
|%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
o-----------------------------o
Figure 40-b. Enlargement of J (Bundle)
o---------------------------------------------------------------------o
| |
| |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o (du).(dv) o o |
| | | -->-- | | |
| | | \ / | | |
| | dv .(du) | \ / | du .(dv) | |
| | u o---------------->@<----------------o v | |
| | | ^ | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| du . dv |
| | |
| o |
| |
o---------------------------------------------------------------------o
Figure 40-c. Enlargement of J (Compact)
o-----------------------------------------------------------o
| |
| (du).(dv) |
| --->--- |
| \ / |
| \ / |
| \ / |
| u @ v |
| ^^^ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| (du) dv / | \ du (dv) |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| @ | @ |
| u (v) | (u) v |
| | |
| | |
| | |
| | |
| du . dv |
| | |
| | |
| | |
| | |
| | |
| @ |
| |
| (u).(v) |
| |
o-----------------------------------------------------------o
Figure 40-d. Enlargement of J (Digraph)
An intuitive reading of the proposition EJ becomes available at this point,
and may be useful. Recall that propositions in the extended universe EU%
express the "dispositions" of system and the constraints that are placed
on them. In other words, a differential proposition in EU% can be read
as referring to various changes that a system might undergo in and from
its various states. In particular, we can understand EJ as a statement
that tells us what changes need to be made with regard to each state in
the universe of discourse in order to reach the "truth" of J, that is,
the region of the universe where J is true. This interpretation is
visibly clear in the Figures above, and appeals to the imagination
in a satisfying way, but it has the added benefit of giving fresh
meaning to the original name of the shift operator E. Namely, EJ
can be read as a proposition that "enlarges" on the meaning of J,
in the sense of explaining its practical bearings and clarifying
what it means in terms of the available options for differential
action and the consequential effects that result from each choice.
Treated this way, the enlargement EJ has strong ties to the normal use of J,
no matter whether it is understood as a proposition or a function, namely,
to act as a figurative device for indicating the models of J, in effect,
pointing to the interpretive elements in its fiber of truth J^(-1)(1).
It is this kind of "use" that is often compared with the "mention" of
a proposition, and thereby hangs a tale.
Jon Awbrey
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