Peirce’s 1870 “Logic Of Relatives” • Comment 10.2

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, are compounded in the formation of syntax.

Let us go back and look more carefully at just how it happens that Peirce’s adjacent terms and subjacent indices manage to impart their respective measures of information about relations.  Consider the examples shown in Figures 7 and 8, where connecting lines of identity have been drawn between the corresponding occurrences of the subjacent marks of reference:  $\dagger\, \ddagger\, \parallel\, \S\, \P.$

 (7) (8)

One way to approach the problem of “information fusion” in Peirce’s syntax is to soften the distinction between adjacent terms and subjacent signs and treat the types of constraints they separately signify more on a par with each other.  To that purpose, let us consider a way of thinking about relational composition that emphasizes the set-theoretic constraints involved in the construction of a composite relation.

For example, given the relations $L \subseteq X \times Y$ and $M \subseteq Y \times Z,$ Table 9 and Figure 10 present two ways of picturing the constraints that are involved in constructing the relational composition $L \circ M \subseteq X \times Z.$

$\text{Table 9.} ~~ \text{Relational Composition}$
$\mathit{1}$ $\mathit{1}$ $\mathit{1}$
$L$ $X$ $Y$
$M$   $Y$ $Z$
$L \circ M$ $X$   $Z$

The way to read Table 9 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, 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 denote 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 $\mathit{1}$ relations are satisfied, then the row marked $L \circ M$ will automatically bear the tokens of a pair of elements in the composite relation $L \circ M.$

Figure 10 shows a different way of viewing the same situation.

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