Re: Relations And Their Divisitudes
Jon Awbrey <[email protected]>
| Newsgroups | gmane.comp.inquiry,gmane.comp.misc.ontology.general |
|---|---|
| Message-ID | <[email protected]> |
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RATD. Note 26
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Compositional Analysis of Relations (concl.)
We can render the picture of our composition example a little
less impressionistic and a little more realistic in the style
of its representation through the introduction of coordinates,
in other words, concrete names for the objects that we relate
through various orders of relations, 2-adic and 3-adic in the
present instance.
Figure 12 shows the Example with a suitable set of coordinates added:
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ 7\/// | \\\/7 /| |
| | \ 6// | \\6 / | |
| | \ //5\ | /5\\ / | |
| | \ /// 4\ | /4 \\\ / | |
| | \/// 3\ | /3 \\\/ | |
| | G /\/ 2\ | /2 \/\ H | |
| | *//\ 1\|/1 /\\* | |
| X *\ Y o Y /* Z |
| 7\ *\\ |7 7| //* /7 |
| 6\ |\\\|6 6|///| /6 |
| 5\| \\.5 5.// |/5 |
| 4. \.4 4./ .4 |
| 3\ .3 3. /3 |
| 2\ |2 2| /2 |
| 1\|1 1|/1 |
| o o |
| |
o-------------------------------------------------o
Figure 12. F as the Intersection of te(G) and te(H)
In the manner of representation that is accorded to us by means
of these coordinates, we have the following data with regard to
F c X x Y x Z, G c X x Y, H c Y x Z.
F = 4:3:4 + 4:4:4 + 4:5:4 c X x Y x Z
G = 4:3 + 4:4 + 4:5 c X x Y
H = 3:4 + 4:4 + 5:4 c Y x Z
Here I have written out the ordered pairs and ordered triples in the syntax
that C.S. Peirce frequently used, where a:b = <a, b> and x:y:z = <x, y, z>.
Let us now verify that all of the proposed definitions, formulas, and other
relationships check out against the concrete data of the composition example.
The ultimate goal is to develop a clearer picture of what is going on in the
formula that expresses the relational composition of 2-adic relations by way
of the projection of the intersection of their respective tacit extensions:
G o H = Proj_XZ (TE_XY.Z (G) |^| TE_YZ.X (H)).
Figure 13 gives the big picture of the composition,
with all of the facets and insets set in one frame:
o-------------------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| / G o H \ |
| / \ |
| X * Z |
| 7\ /^\ /7 |
| 6\ / | \ /6 |
| 5\ / | \ /5 |
| 4. | .4 |
| 3\ | /3 |
| 2\ | /2 |
| 1\|/1 |
| | |
| | |
| | |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o | o |
| |\ /|\ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | G /\/ \ | / \/\ H | |
| | *//\ \|/ /\\* | |
| X *\ Y o Y /* Z |
| 7\ *\\ |7 7| //* /7 |
| 6\ |\\\|6 6|///| /6 |
| 5\| \\.5 5.// |/5 |
| 4. \.4 4./ @4 |
| 3\ .3 3. /3 |
| 2\ |2 2| /2 |
| 1\|1 1|/1 |
| o o |
| |
o-------------------------------------------------o
Figure 13. G o H = Proj_XZ (TE(G) |^| TE(H))
All that remains to do now is to check the following
collection of data and derivations against Figure 13.
F = 4:3:4 + 4:4:4 + 4:5:4
G = 4:3 + 4:4 + 4:5
H = 3:4 + 4:4 + 5:4
G o H = (4:3 + 4:4 + 4:5)(3:4 + 4:4 + 5:4)
= 4:4
TE(G) = TE_XY.Z (G)
= Sum_(z = 1 to 7) (4:3:z + 4:4:z + 4:5:z)
= 4:3:1 + 4:4:1 + 4:5:1 +
4:3:2 + 4:4:2 + 4:5:2 +
4:3:3 + 4:4:3 + 4:5:3 +
4:3:4 + 4:4:4 + 4:5:4 +
4:3:5 + 4:4:5 + 4:5:5 +
4:3:6 + 4:4:6 + 4:5:6 +
4:3:7 + 4:4:7 + 4:5:7
TE(H) = TE_YZ.X (H)
= Sum_(x = 1 to 7) (x:3:4 + x:4:4 + x:5:4)
= 1:3:4 + 1:4:4 + 1:5:4 +
2:3:4 + 2:4:4 + 2:5:4 +
3:3:4 + 3:4:4 + 3:5:4 +
4:3:4 + 4:4:4 + 4:5:4 +
5:3:4 + 5:4:4 + 5:5:4 +
6:3:4 + 6:4:4 + 6:5:4 +
7:3:4 + 7:4:4 + 7:5:4
TE(G) |^| TE(H) = 4:3:4 + 4:4:4 + 4:5:4
G o H = Proj_XZ (TE(G) |^| TE(H))
= Proj_XZ (4:3:4 + 4:4:4 + 4:5:4)
= 4:4
By my lights, anyway, it all checks.
Jon Awbrey
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