[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 38

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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.

o-------------------------------------------------o
|                                                 |
|                                                 |
|         `g`__$__%    $'t'__*   *%h              |
|              o  o    o     o   oo               |
|               \  \  /       \ //                |
|                \  \/         @/                 |
|                 \ /\____ ____/                  |
|                  @      @                       |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 11.  Giver of a Horse to a Trainer of It

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.

o---------------------------------------------------------------------o
|                                                                     |
|                                G o T                                |
|                 _________________ at _________________                 |
|                /                                   \                |
|               /        G              T             \               |
|              /         @              @              \              |
|             /         /|\            / \              \             |
|            /         / | \          /   \              \            |
|           /         /  |  \        /     \              \           |
|          /         /   |   \      /       \              \          |
|         o         o    o    o    o         o              o         |
|         X         X    Y    Z    Y         Z              Z         |
|      1,_#         #`g`_$____%    $'t'______%              %1        |
|         o         o    o    o    o         o              o         |
|          \       /      \    \  /          |             /          |
|           \     /        \    \/           |            /           |
|            \   /          \   /\           |           /            |
|             \ /            \ /  \__________|__________/             |
|              @              @              @                        |
|             !1!            !1!            !1!                       |
|                                                                     |
o---------------------------------------------------------------------o
Figure 12.  Anything that is a Giver of Anything to a Trainer of It

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 c X x Y x Z
with a 2-adic "training" relation T c Y x Z in such a way
as to determine a certain 2-adic relation (G o T) c X x Z.
Table 13 schematizes the associated constraints on tuples.

Table 13.  Another Brand of Composition
o---------o---------o---------o---------o
|         #   !1!   |   !1!   |   !1!   |
o=========o=========o=========o=========o
|    G    #    X    |    Y    |    Z    |
o---------o---------o---------o---------o
|    T    #         |    Y    |    Z    |
o---------o---------o---------o---------o
|  G o T  #    X    |         |    Z    |
o---------o---------o---------o---------o

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.

Jon Awbrey

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