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Revision as of 14:55, 10 April 2009
, 14:55, 10 April 2009→Commentary Note 10.8
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===Commentary Note 10.8===
===Commentary Note 10.8===
−In taking up the next example of relational composition, let's exchange the relation 't' = "trainer of ---" for Peirce's relation 'o' = "owner of ---", simply for the sake of avoiding conflicts in the symbols that we use. In this way, Figure 7 is transformed into Figure 11.
+In taking up the next example of relational composition, let's exchange the relation <math>\mathit{t} = \text{trainer of}\, \underline{~~~~}</math> for Peirce's relation <math>\mathit{o} = \text{owner of}\, \underline{~~~~},</math> simply for the sake of avoiding conflicts in the symbols that we use. In this way, Figure 7 is transformed into Figure 11.
+{| align="center" cellspacing="6" width="90%"
+| align="center" |
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| |
| |
−| `g`__$__% $'t'__* *%h |
+| `g`__!__@ !'t'__# #@h |
| o o o o oo |
| o o o o oo |
| \ \ / \ // |
| \ \ / \ // |
−| \ \/ @/ |
+| \ \/ O/ |
| \ /\____ ____/ |
| \ /\____ ____/ |
−| @ @ |
+| O O |
| |
| |
| |
| |
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Figure 11. Giver of a Horse to a Trainer of It
Figure 11. Giver of a Horse to a Trainer of It
</pre>
</pre>
+|}
Now here's an interesting point, in fact, a critical transition point, that we see resting in potential but a stone's throw removed from the chronism, the secular neigborhood, the temporal vicinity of Peirce's 1870 LOR, and it's a vertex that turns on the teridentity relation.
Now here's an interesting point, in fact, a critical transition point, that we see resting in potential but a stone's throw removed from the chronism, the secular neigborhood, the temporal vicinity of Peirce's 1870 LOR, and it's a vertex that turns on the teridentity relation.
−The hypergraph picture of the abstract composition is given in Figure 12.
+The hypergraph picture of the abstract composition is given in Figure 12.
+{| align="center" cellspacing="6" width="90%"
+| align="center" |
<pre>
<pre>
o---------------------------------------------------------------------o
o---------------------------------------------------------------------o
| |
| |
| G o T |
| G o T |
−| _________________@_________________ |
+| _________________O_________________ |
| / \ |
| / \ |
| / G T \ |
| / G T \ |
−| / @ @ \ |
+| / O O \ |
| / /|\ / \ \ |
| / /|\ / \ \ |
| / / | \ / \ \ |
| / / | \ / \ \ |
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| o o o o o o o |
| o o o o o o o |
| X X Y Z Y Z Z |
| X X Y Z Y Z Z |
−| 1,_# #`g`_$____% $'t'______% %1 |
+| 1,_! !`g`_@____# @'t'______$ #1 |
| o o o o o o o |
| o o o o o o o |
| \ / \ \ / | / |
| \ / \ \ / | / |
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| \ / \ /\ | / |
| \ / \ /\ | / |
| \ / \ / \__________|__________/ |
| \ / \ / \__________|__________/ |
−| @ @ @ |
+| O O O |
| !1! !1! !1! |
| !1! !1! !1! |
| |
| |
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Figure 12. Anything that is a Giver of Anything to a Trainer of It
Figure 12. Anything that is a Giver of Anything to a Trainer of It
</pre>
</pre>
+|}
If we analyze this in accord with the "spreadsheet" model of relational composition, the core of it is a particular way of composing a 3-adic "giving" relation ''G'' ⊆ ''X'' × ''Y'' × ''Z'' with a 2-adic "training" relation ''T'' ⊆ ''Y'' × ''Z'' in such a way as to determine a certain 2-adic relation (''G'' o ''T'') ⊆ ''X'' × ''Z''. Table 13 schematizes the associated constraints on tuples.
If we analyze this in accord with the "spreadsheet" model of relational composition, the core of it is a particular way of composing a 3-adic "giving" relation ''G'' ⊆ ''X'' × ''Y'' × ''Z'' with a 2-adic "training" relation ''T'' ⊆ ''Y'' × ''Z'' in such a way as to determine a certain 2-adic relation (''G'' o ''T'') ⊆ ''X'' × ''Z''. Table 13 schematizes the associated constraints on tuples.
+{| align="center" cellspacing="6" width="90%"
+| align="center" |
<pre>
<pre>
Table 13. Another Brand of Composition
Table 13. Another Brand of Composition
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o---------o---------o---------o---------o
o---------o---------o---------o---------o
</pre>
</pre>
+|}
−So we see that the notorious teridentity relation, which I have left equivocally denoted by the same symbol as the identity relation !1!, is already implicit in Peirce's discussion at this point.
+So we see that the notorious teridentity relation, which I have left equivocally denoted by the same symbol as the identity relation <math>\mathit{1},\!</math> is already implicit in Peirce's discussion at this point.
===Commentary Note 10.9===
===Commentary Note 10.9===
