[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 32

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To say that a relative term "imparts a relation"
is to say that it conveys information about the
space of tuples in a cartesian product, that is,
it determines a particular subset of that space.

When we study the combinations of relative terms, from the most
elementary forms of composition to the most complex patterns of
correlation, we are considering the ways that these constraints,
determinations, and informations, as imparted by relative terms,
can be compounded in the formation of syntax.

Let us go back and look more carefully at just how it happens that
Peirce's jacent terms and subjacent indices manage to impart their
respective measures of information about relations.

I will begin with the two examples illustrated in Figures 1 and 2,
where I have drawn in the corresponding lines of identity between
the subjacent marks of reference #, $, %.

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

One way to approach the problem of "information fusion"
in Peirce's syntax is to soften the distinction between
jacent terms and subjacent signs, and to treat the types
of constraints that they separately signify more on a par
with each other.

To that purpose, I will set forth a way of thinking about
relational composition that emphasizes the set-theoretic
constraints involved in the construction of a composite.

For example, suppose that we are given the relations L c X x Y, M c Y x Z.
Table 3 and Figure 4 present a couple of ways of picturing the constraints
that are involved in constructing the relational composition L o M c X x Z.

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
|    M    #         |    Y    |    Z    |
o---------o---------o---------o---------o
|  L o M  #    X    |         |    Z    |
o---------o---------o---------o---------o

The way to read Table 3 is to imagine that you are
playing a game that involves placing tokens on the
squares of a board that is marked in just this way.
The rules are that you have to place a single token
on each marked square in the middle of the board in
such a way that all of the indicated constraints are
satisfied.  That is to say, you have to place a token
whose denomination is a value in the set X on each of
the squares marked "X", and similarly for the squares
marked "Y" and "Z", meanwhile leaving all of the blank
squares empty.  Furthermore, the tokens placed in each
row and column have to obey the relational constraints
that are indicated at the heads of the corresponding
row and column.  Thus, the two tokens from X have to
denominate the very same value from X, and likewise
for Y and Z, while the pairs of tokens on the rows
marked "L" and "M" are required to denote elements
that are in the relations L and M, respectively.
The upshot is that when just this much is done,
that is, when the L, M, and !1! relations are
satisfied, then the row marked "L o M" will
automatically bear the tokens of a pair of
elements in the composite relation L o M.

o-------------------------------------------------o
|                                                 |
|                L     L o M     M                |
|                @       @       @                |
|               / \     / \     / \               |
|              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

Figure 4 merely shows a different way of viewing the same situation.

Jon Awbrey

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