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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 14:55, 10 April 2009
, 14:55, 10 April 2009→Commentary Note 10.8
===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`__$__% $'t'__* *%h |
| `g`__!__@ !'t'__# #@h |
| o o o o oo |
| o o o o oo |
| \ \ / \ // |
| \ \ / \ // |
| \ \/ @/ |
| \ \/ O/ |
| \ /\____ ____/ |
| \ /\____ ____/ |
| @ @ |
| O O |
| |
| |
| |
| |
Figure 11. Giver of a Horse to a Trainer of It
Figure 11. Giver of a Horse to a Trainer of It
</pre>
</pre>
|}
Now here's an interesting point, in fact, a critical transition point, that we see resting in potential but a stone's throw removed from the chronism, the secular neigborhood, the temporal vicinity of Peirce's 1870 LOR, and it's a vertex that turns on the teridentity relation.
Now here's an interesting point, in fact, a critical transition point, that we see resting in potential but a stone's throw removed from the chronism, the secular neigborhood, the temporal vicinity of Peirce's 1870 LOR, and it's a vertex that turns on the teridentity relation.
The hypergraph picture of the abstract composition is given in Figure 12.
The hypergraph picture of the abstract composition is given in Figure 12.
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
<pre>
o---------------------------------------------------------------------o
o---------------------------------------------------------------------o
| |
| |
| G o T |
| G o T |
| _________________@_________________ |
| _________________O_________________ |
| / \ |
| / \ |
| / G T \ |
| / G T \ |
| / @ @ \ |
| / O O \ |
| / /|\ / \ \ |
| / /|\ / \ \ |
| / / | \ / \ \ |
| / / | \ / \ \ |
| o o o o o o o |
| o o o o o o o |
| X X Y Z Y Z Z |
| X X Y Z Y Z Z |
| 1,_# #`g`_$____% $'t'______% %1 |
| 1,_! !`g`_@____# @'t'______$ #1 |
| o o o o o o o |
| o o o o o o o |
| \ / \ \ / | / |
| \ / \ \ / | / |
| \ / \ /\ | / |
| \ / \ /\ | / |
| \ / \ / \__________|__________/ |
| \ / \ / \__________|__________/ |
| @ @ @ |
| O O O |
| !1! !1! !1! |
| !1! !1! !1! |
| |
| |
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
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===