Peirce’s 1870 “Logic Of Relatives” • Comment 10.3

We’ve been using several different styles of picture to illustrate relative terms and the relations they denote. Let us now examine the relationships that exist among the variety of visual schemes. Two examples of relative multiplication that we considered before are diagrammed again in Figures 11 and 12.

 (11) (12)

Figures 11 and 12 employ one style of syntax Peirce used for relative multiplication, to which I added lines of identity to connect the corresponding marks of reference. These forms are adapted to showing the anatomy of relative terms themselves, while the forms of analysis in Table 13 and Figure 14 are designed to highlight the structures of the objective relations they denote.

$\text{Table 13.} ~~ \text{Relational Composition}$
$\mathit{1}$ $\mathit{1}$ $\mathit{1}$
$L$ $X$ $Y$
$S$   $Y$ $Z$
$L \circ S$ $X$   $Z$

 (14)

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. It is 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 15 demonstrates this possibility, posing a transitional case between the style of syntax in Figure 11 and the picture of composition in Figure 14.

 (15)

In this composite sketch the diagonal extension $\mathit{1}$ of the universe $\mathbf{1}$ is invoked up front to anchor an explicit line of identity for the leading relate of the composition, while the terminal argument $\mathrm{w}$ has been generalized to the whole universe $\mathbf{1}.$ Doing this amounts to an act of abstraction from the particular application to $\mathrm{w}.$ This form of universal bracketing isolates the serial composition of the relations $L$ and $S$ to form the composite $L \circ S.$

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