[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 40

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Figure 8 depicts the last of the three examples involving
the composition of 3-adic relatives with 2-adic relatives:

o-------------------------------------------------o
|                                                 |
|                                                 |
|        'l',__$__%    $'s'__*   *%w              |
|              o  o    o     o   oo               |
|               \  \  /       \ //                |
|                \  \/         @/                 |
|                 \ /\____ ____/                  |
|                  @      @                       |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 8.  Lover that is a Servant of a Woman

The hypergraph picture of the abstract composition is given in Figure 14.

o---------------------------------------------------------------------o
|                                                                     |
|                                L , S                                |
|                __________________ at __________________                |
|               /                                     \               |
|              /       L_,              S              \              |
|             /         @               @               \             |
|            /         /|\             / \               \            |
|           /         / | \           /   \               \           |
|          /         /  |  \         /     \               \          |
|         /         /   |   \       /       \               \         |
|        /         /    |    \     /         \               \        |
|       o         o     o     o   o           o               o       |
|       X         X     X     Y   X           Y               Y       |
|    1,_#         #'l',_$_____%   $'t'________%               %1      |
|       o         o     o     o   o           o               o       |
|        \       /       \     \ /            |              /        |
|         \     /         \     \             |             /         |
|          \   /           \   / \            |            /          |
|           \ /             \ /   \___________|___________/           |
|            @               @                @                       |
|           !1!             !1!              !1!                      |
|                                                                     |
o---------------------------------------------------------------------o
Figure 14.  Anything that's a Lover of Anything and that's a Servant of It

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 c X x Y to the corresponding 3-adic
"loving and being" relation L_, c X x X x Y, which is then composed
in a specific way with a 2-adic "serving" relation S c X x Y, so as
to determine the 2-adic relation L,S c X x Y.  Table 15 schematizes
the associated constraints on tuples.

Table 15.  Conjunction Via Composition
o---------o---------o---------o---------o
|         #   !1!   |   !1!   |   !1!   |
o=========o=========o=========o=========o
|    L,   #    X    |    X    |    Y    |
o---------o---------o---------o---------o
|    S    #         |    X    |    Y    |
o---------o---------o---------o---------o
|  L , S  #    X    |         |    Y    |
o---------o---------o---------o---------o

Jon Awbrey

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