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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 14:12, 10 April 2009
, 14:12, 10 April 2009→Commentary Note 10.7
Here is what I get when I try to analyze Peirce's "giver of a horse to a lover of a woman" example along the same lines as the 2-adic compositions.
Here is what I get when I try to analyze Peirce's "giver of a horse to a lover of a woman" example along the same lines as the 2-adic compositions.
We may begin with the mark-up shown in Figure 6.
We may begin with the mark-up shown in Figure 6.
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| |
| |
| `g`__$__% $'l'__* *w %h |
| `g`__!__@ !'l'__# #w @h |
| o o o o o o |
| o o o o o o |
| \ \ / \ / / |
| \ \ / \ / / |
| \ \/ @ / |
| \ \/ O / |
| \ /\______ ______/ |
| \ /\______ ______/ |
| @ @ |
| O O |
| |
| |
| |
| |
Figure 6. Giver of a Horse to a Lover of a Woman
Figure 6. Giver of a Horse to a Lover of a Woman
</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'' ⊆ ''T'' × ''U'' × ''V'' with a 2-adic "loving" relation ''L'' ⊆ ''U'' × ''W'' so as to obtain a specialized sort of 3-adic relation (''G'' o ''L'') ⊆ ''T'' × ''W'' × ''V''. The applicable constraints on tuples are shown in Table 9.
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 <math>G \subseteq T \times U \times V</math> with a 2-adic ''loving'' relation <math>L \subseteq U \times W</math> so as to obtain a specialized sort of 3-adic relation <math>(G \circ L) \subseteq T \times W \times V.</math> The applicable constraints on tuples are shown in Table 9.
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
<pre>
Table 9. Composite of Triadic and Dyadic Relations
Table 9. Composite of Triadic and Dyadic Relations
o---------o---------o---------o---------o---------o
o---------o---------o---------o---------o---------o
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</pre>
|}
The hypergraph picture of the abstract composition is given in Figure 10.
The hypergraph picture of the abstract composition is given in Figure 10.
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
<pre>
o---------------------------------------------------------------------o
o---------------------------------------------------------------------o
| |
| |
| G o L |
| G o L |
| ___________@___________ |
| ___________O___________ |
| / \ \ |
| / \ \ |
| / G L \ \ |
| / G L \ \ |
| / @ @ \ \ |
| / O O \ \ |
| / /|\ / \ \ \ |
| / /|\ / \ \ \ |
| / / | \ / \ \ \ |
| / / | \ / \ \ \ |
| o o o o o o o o |
| o o o o o o o o |
| T T U V U W W V |
| T T U V U W W V |
| 1,_# #`g`_$____% $'l'______* *1 %1 |
| 1,_! !`g`_@____# @'l'______$ $1 #1 |
| o o o o o o o o |
| o o o o o o o o |
| \ / \ \ / \ / / |
| \ / \ \ / \ / / |
| @ \ \/ @ / |
| O \ \/ O / |
| !1! \ /\ !1! / |
| !1! \ /\ !1! / |
| \ / \_______ _______/ |
| \ / \_______ _______/ |
| @ @ |
| O O |
| !1! !1! |
| !1! !1! |
| |
| |
Figure 10. Anything that is a Giver of Anything to a Lover of Anything
Figure 10. Anything that is a Giver of Anything to a Lover of Anything
</pre>
</pre>
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===Commentary Note 10.8===
===Commentary Note 10.8===