## Peirce’s 1870 “Logic Of Relatives” • Comment 10.8

There’s a critical transition point in sight of Peirce’s 1870 Logic of Relatives and it’s a point that turns on the teridentity relation.

In taking up the next example of relational composition, let’s substitute the relation $\mathit{t} = \text{taker of}\, \underline{~~~~}$ for Peirce’s relation $\mathit{o} = \text{owner of}\, \underline{~~~~},$ simply for the sake of avoiding conflicts in the symbols we use.  In this way, Figure 17 is transformed into Figure 22.

 (22)

The hypergraph picture of the abstract composition is given in Figure 23.

 (23)

If we analyze this in accord with the spreadsheet model of relational composition, the core of it is a particular way of composing a triadic “giving” relation $G \subseteq X \times Y \times Z$ with a dyadic “taking” relation $T \subseteq Y \times Z$ in such a way as to determine a certain dyadic relation $(G \circ T) \subseteq X \times Z.$  Table 24 schematizes the associated constraints on tuples.

$\text{Table 24.} ~~ \text{Another Brand of Composition}$
$\mathit{1}$ $\mathit{1}$ $\mathit{1}$
$G$ $X$ $Y$ $Z$
$T$   $Y$ $Z$
$G \circ T$ $X$   $Z$

So we see that the notorious teridentity relation, which I have left equivocally denoted by the same symbol as the identity relation $\mathit{1},$ is already implicit in Peirce’s discussion at this point.