Changes

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:

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:

<pre>

<pre>

o-------------------------------------------------o

o-------------------------------------------------o

|                                                |

|                                                |

|                                                |

|                                                |

|        'l'__$       $'s'__%   %w                |

|        'l'__!       !'s'__@   @w                |

|            o      o    o  o                |

|            o      o    o  o                |

|              \    /      \ /                  |

|              \    /      \ /                  |

|              \  /        o                  |

|              \  /        o                  |

|                \ /          %                   |

|                \ /          @                   |

|                o                              |

|                o                              |

|                $                               |

|                !                               |

|                                                |

|                                                |

|                                                |

|                                                |

Figure 1.  Lover of a Servant of a Woman

Figure 1.  Lover of a Servant of a Woman

</pre>

</pre>

<pre>

<pre>

o-------------------------------------------------o

o-------------------------------------------------o

|                                                |

|                                                |

|                                                |

|                                                |

|        `g`__$__%   $'l'__*   *w  %h          |

|        `g`__!__@   !'l'__#   #w  @h          |

|            o  o    o    o  o    o            |

|            o  o    o    o  o    o            |

|              \  \  /      \ /    /            |

|              \  \  /      \ /    /            |

|              \  \/        o    /              |

|              \  \/        o    /              |

|                \ /\        *   /              |

|                \ /\        #   /              |

|                o  ------o------                |

|                o  ------o------                |

|                $       %                     |

|                !       @                     |

|                                                |

|                                                |

|                                                |

|                                                |

Figure 2.  Giver of a Horse to a Lover of a Woman

Figure 2.  Giver of a Horse to a Lover of a Woman

</pre>

</pre>

<pre>

<pre>

Table 3.  Relational Composition

Table 3.  Relational Composition

o---------o---------o---------o---------o

o---------o---------o---------o---------o

</pre>

</pre>

<pre>

<pre>

o-------------------------------------------------o

o-------------------------------------------------o

Figure 4.  Relational Composition

Figure 4.  Relational Composition

</pre>

</pre>

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.

Figures&nbsp;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&nbsp;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.

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.

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.

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.

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

<pre>

<pre>

o-----------------------------------------------------------o

o-----------------------------------------------------------o

|                                                          |

|                                                          |

|                          L o S                          |

|                          L o S                          |

|                ____________@____________                 |

|                ____________O____________                 |

|                /                        \                |

|                /                        \                |

|              /      L            S      \              |

|              /      L            S      \              |

|          o      o    o      o    o      o          |

|          o      o    o      o    o      o          |

|          X      X    Y      Y    Z      Z          |

|          X      X    Y      Y    Z      Z          |

|      1,__#       #'l'__$       $'s'__%       %1          |

|      1,__!       !'l'__@       @'s'__#       #1          |

|          o      o    o      o    o      o          |

|          o      o    o      o    o      o          |

|            \    /      \    /      \    /            |

|            \    /      \    /      \    /            |

|            \  /        \  /        \  /            |

|            \  /        \  /        \  /            |

|              \ /          \ /          \ /              |

|              \ /          \ /          \ /              |

|              @             @             @               |

|              O             O             O               |

|              !1!          !1!          !1!              |

|              !1!          !1!          !1!              |

|                                                          |

|                                                          |

Figure 5.  Anything that is a Lover of a Servant of Anything

Figure 5.  Anything that is a Lover of a Servant of Anything

</pre>

</pre>

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''&nbsp;o&nbsp;''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.

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

===Commentary Note 10.4===

===Commentary Note 10.4===