## Peirce’s 1870 “Logic Of Relatives” • Comment 11.11

The preceding exercises were intended to beef-up our “functional literacy” skills to the point where we can read our functional alphabets backwards and forwards and recognize the local functionalities that are immanent in relative terms no matter where they locate themselves within the domains of relations. These skills will serve us in good stead as we work to build a catwalk from Peirce’s platform of 1870 to contemporary scenes on the logic of relatives, and back again.

By way of extending a few very tentative planks, let us experiment with the following definitions:

• A relative term $p"$ and the corresponding relation $P \subseteq X \times Y$ are both called functional on relates if and only if $P$ is a function at $X.$ We write this in symbols as $P : X \to Y.$
• A relative term $p"$ and the corresponding relation $P \subseteq X \times Y$ are both called functional on correlates if and only if $P$ is a function at $Y.$ We write this in symbols as $P : X \gets Y.$

When a relation happens to be a function, it may be excusable to use the same name for it in both applications, writing out explicit type markers like $P : X \times Y,$   $P : X \to Y,$   $P : X \gets Y,$ as the case may be, when and if it serves to clarify matters.

From this current, perhaps transient, perspective, it appears that our next task is to examine how the known properties of relations are modified when an aspect of functionality is spied in the mix. Let us then return to our various ways of looking at relational composition, and see what changes and what stays the same when the relations in question happen to be functions of various kinds at some of their domains. Here is one generic picture of relational composition, cast in a style that hews pretty close to the line of potentials inherent in Peirce’s syntax of this period.

 (44)

From this we extract the hypergraph picture of relational composition:

 (45)

All of the information contained in these Figures can be expressed in the form of a constraint satisfaction table, or spreadsheet picture of relational composition:

$\text{Table 46.} ~~ \text{Relational Composition}~ P \circ Q$
$\mathit{1}$ $\mathit{1}$ $\mathit{1}$
$P$ $X$ $Y$
$Q$   $Y$ $Z$
$P \circ Q$ $X$   $Z$

The following plan of study then presents itself, to see what easy mileage we can get in our exploration of functions by adopting the above templates as the primers of a paradigm.

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