12,136
edits
Changes
Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 17:11, 10 April 2009
, 17:11, 10 April 2009→Commentary Note 10.10
===Commentary Note 10.10===
===Commentary Note 10.10===
Figure 8 depicts the last of the three examples involving the composition of 3-adic relatives with 2-adic relatives:
Figure 8 depicts the last of the three examples involving the composition of 3-adic relatives with 2-adic relatives:
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| |
| |
| 'l',__$__% $'s'__* *%w |
| 'l',__!__@ !'s'__# #@w |
| o o o o oo |
| o o o o oo |
| \ \ / \ // |
| \ \ / \ // |
| \ \/ @/ |
| \ \/ @/ |
| \ /\____ ____/ |
| \ /\____ ____/ |
| @ @ |
| O O |
| |
| |
| |
| |
Figure 8. Lover that is a Servant of a Woman
Figure 8. Lover that is a Servant of a Woman
</pre>
</pre>
|}
The hypergraph picture of the abstract composition is given in Figure 14.
The hypergraph picture of the abstract composition is given in Figure 14.
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
<pre>
o---------------------------------------------------------------------o
o---------------------------------------------------------------------o
| |
| |
| L , S |
| L , S |
| __________________^__________________ |
| __________________O__________________ |
| / \ |
| / \ |
| / L_, S \ |
| / L_, S \ |
| / @ @ \ |
| / O O \ |
| / /|\ / \ \ |
| / /|\ / \ \ |
| / / | \ / \ \ |
| / / | \ / \ \ |
| o o o o o o o |
| o o o o o o o |
| X X X Y X Y Y |
| X X X Y X Y Y |
| 1,_# #'l',_$_____% $'t'________% %1 |
| 1,_! !'l',_@_____# @'t'________$ #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 14. Anything that's a Lover of Anything and that's a Servant of It
Figure 14. Anything that's a Lover of Anything and that's a Servant of It
</pre>
</pre>
|}
This example illustrates the way that Peirce analyzes the logical conjunction, we might even say the "parallel conjunction", of a couple of 2-adic relatives in terms of the comma extension and the same style of composition that we saw in the last example, that is, according to a pattern of anaphora that invokes the teridentity relation.
This example illustrates the way that Peirce analyzes the logical conjunction, we might even say the "parallel conjunction", of a couple of 2-adic relatives in terms of the comma extension and the same style of composition that we saw in the last example, that is, according to a pattern of anaphora that invokes the teridentity relation.
If we lay out this analysis of conjunction on the spreadsheet model of relational composition, the gist of it is the diagonal extension of a 2-adic "loving" relation ''L'' ⊆ ''X'' × ''Y'' to the corresponding 3-adic "loving and being" relation ''L'', ⊆ ''X'' × ''X'' × ''Y'', which is then composed in a specific way with a 2-adic "serving" relation ''S'' ⊆ ''X'' × ''Y'', so as to determine the 2-adic relation ''L'',''S'' ⊆ ''X'' × ''Y''. Table 15 schematizes the associated constraints on tuples.
If we lay out this analysis of conjunction on the spreadsheet model of relational composition, the gist of it is the diagonal extension of a 2-adic "loving" relation ''L'' ⊆ ''X'' × ''Y'' to the corresponding 3-adic "loving and being" relation ''L'', ⊆ ''X'' × ''X'' × ''Y'', which is then composed in a specific way with a 2-adic "serving" relation ''S'' ⊆ ''X'' × ''Y'', so as to determine the 2-adic relation ''L'',''S'' ⊆ ''X'' × ''Y''. Table 15 schematizes the associated constraints on tuples.
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
<pre>
Table 15. Conjunction Via Composition
Table 15. Conjunction Via Composition
o---------o---------o---------o---------o
o---------o---------o---------o---------o
</pre>
</pre>
|}
===Commentary Note 10.11===
===Commentary Note 10.11===