[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 33

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I will devote some time to drawing out the relationships
that exist among the different pictures of relations and
relative terms that were shown above, or as redrawn here:

o-------------------------------------------------o
|                                                 |
|                                                 |
|        'l'__$       $'s'__%   %w                |
|             o       o     o   o                 |
|              \     /       \ /                  |
|               \   /         o                   |
|                \ /          %                   |
|                 o                               |
|                 $                               |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 1.  Lover of a Servant of a Woman

o-------------------------------------------------o
|                                                 |
|                                                 |
|        `g`__$__%    $'l'__*   *w   %h           |
|             o  o    o     o   o    o            |
|              \  \  /       \ /    /             |
|               \  \/         o    /              |
|                \ /\         *   /               |
|                 o  ------o------                |
|                 $        %                      |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 2.  Giver of a Horse to a Lover of a Woman

Table 3.  Relational Composition
o---------o---------o---------o---------o
|         #   !1!   |   !1!   |   !1!   |
o=========o=========o=========o=========o
|    L    #    X    |    Y    |         |
o---------o---------o---------o---------o
|    S    #         |    Y    |    Z    |
o---------o---------o---------o---------o
|  L o S  #    X    |         |    Z    |
o---------o---------o---------o---------o

o-------------------------------------------------o
|                                                 |
|                L     L o S     S                |
|                @       @       @                |
|               / \     / \     / \               |
|              o   o   o   o   o   o              |
|              X   Y   X   Z   Y   Z              |
|              o   o   o   o   o   o              |
|               \   \ /     \ /   /               |
|                \   /       \   /                |
|                 \ / \__ __/ \ /                 |
|                  @     @     @                  |
|                 !1!   !1!   !1!                 |
|                                                 |
o-------------------------------------------------o
Figure 4.  Relational Composition

Figures 1 and 2 exhibit examples of relative multiplication
in one of Peirce's styles of syntax, to which I subtended
lines of identity to mark the anaphora of the correlates.
These pictures are adapted to showing the anatomy of the
relative terms, while the forms of analysis illustrated
in Table 3 and Figure 4 are designed to highlight the
structures of the objective relations themselves.

There are many ways that Peirce might have gotten from his 1870 Notation
for the Logic of Relatives to his more evolved systems of Logical Graphs.
For my part, I find it interesting to speculate on how the metamorphosis
might have been accomplished by way of transformations that act on these
nascent forms of syntax and that take place not too far from the pale of
its means, that is, as nearly as possible according to the rules and the
permissions of the initial system itself.

In Existential Graphs, a relation is represented by a node
whose degree is the adicity of that relation, and which is
adjacent via lines of identity to the nodes that represent
its correlative relations, including as a special case any
of its terminal individual arguments.

In the 1870 Logic of Relatives, implicit lines of identity are invoked by
the subjacent numbers and marks of reference only when a correlate of some
relation is the relate of some relation.  Thus, the principal relate, which
is not a correlate of any explicit relation, is not singled out in this way.

Remarkably enough, the comma modifier itself provides us with a mechanism
to abstract the logic of relations from the logic of relatives, and thus
to forge a possible link between the syntax of relative terms and the
more graphical depiction of the objective relations themselves.

Figure 5 demonstrates this possibility, posing a transitional case between
the style of syntax in Figure 1 and the picture of composition in Figure 4.

o-----------------------------------------------------------o
|                                                           |
|                           L o S                           |
|                 ____________ at ____________                 |
|                /                         \                |
|               /      L             S      \               |
|              /       @             @       \              |
|             /       / \           / \       \             |
|            /       /   \         /   \       \            |
|           o       o     o       o     o       o           |
|           X       X     Y       Y     Z       Z           |
|       1,__#       #'l'__$       $'s'__%       %1          |
|           o       o     o       o     o       o           |
|            \     /       \     /       \     /            |
|             \   /         \   /         \   /             |
|              \ /           \ /           \ /              |
|               @             @             @               |
|              !1!           !1!           !1!              |
|                                                           |
o-----------------------------------------------------------o
Figure 5.  Anything that is a Lover of a Servant of Anything

In this composite sketch, the diagonal extension of the universe 1
is invoked up front to anchor an explicit line of identity for the
leading relate of the composition, while the terminal argument "w"
has been generalized to the whole universe 1, in effect, executing
an act of abstraction.  This type of universal bracketing isolates
the composing of the relations L and S to form the composite L o S.
The three relational domains X, Y, Z may be distinguished from one
another, or else rolled up into a single universe of discourse, as
one prefers.

Jon Awbrey

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