ONT Re: Differential Logic B -- Discussion
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG B. Discussion Note 3
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Re: DLOG B17. http://suo.ieee.org/ontology/msg05452.html
HT = Hugh Trenchard
Hi Hugh,
Thanks for waking me from my documental stupors --
I'm beginning to remember why I put off doing this
for 15 years -- that estimate of 10^3 words/picture
really is a gross under-statement.
HT: I'm not clear on at least a couple of points here.
Is the "region bordering the origin cell" that part
of the Venn diagram where all three circles overlap?
If so, is the "point omitted neighbourhood" those
regions of the three circles which do not overlap
with any of their neighbours?
I called that outer region where all the predicates x, y, z
are false the "origin" because its coordinates are <0, 0, 0>.
So the cells adjacent to the origin are the acutely accented
ones in the following venn diagram:
o-----------------------------------------------------------o
| U |
| |
| o-------------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| /```````````````````````\ |
| o`````````````````````````o |
| |``````````` X ```````````| |
| |`````````````````````````| |
| |```````````100```````````| |
| |`````````````````````````| |
| |`````````````````````````| |
| o--o----------o```o----------o--o |
| /````\ \`/ /````\ |
| /``````\ 110 o 101 /``````\ |
| /````````\ / \ /````````\ |
| /``````````\ / \ /``````````\ |
| /````````````\ / 111 \ /````````````\ |
| o``````````````o--o-------o--o``````````````o |
| |`````````````````| |`````````````````| |
| |`````````````````| |`````````````````| |
| |`````````````````| |`````````````````| |
| |``````` Y ```````| |`````` Z ````````| |
| |`````````````````| |`````````````````| |
| o```````010```````o 011 o``````001````````o |
| \`````````````````\ /`````````````````/ |
| \`````````````````\ /`````````````````/ |
| \`````````````````\ /`````````````````/ |
| \`````````````````o`````````````````/ |
| \```````````````/ \```````````````/ |
| o-------------o o-------------o |
| |
| 000 |
| |
o-----------------------------------------------------------o
Figure 11. Venn Diagram for ((x),(y),(z))
HT: You note that the phrase "just one of three is true" describes
the Venn diagram. This is obviously different from the intuitive
sense that "there is a part of x and a part of y and a part of z
that are common, or shared, among x, y, and z". So if the centre,
common region, represents "one of three is true" (if I have that
much right, which I might not), does that mean as between y and z,
"one of two is true", and as between x and z "one of two is true",
and as between x and y "one of two is true"?
No, the center cell is the one where all 3 are true,
that is, its coordinates are <x, y, z> = <1, 1, 1>.
The shaded region that is indicated by ((x),(y),(z))
has the 3 cells whose coordinates are 100, 010, 001.
If I had overlapping gels, these would be the pure
primary colors, thus, the cells where just 1 of 3
properties applies, a different 1 in each case.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o