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.


LOR 1870 Figure 44
(44)

From this we extract the hypergraph picture of relational composition:


LOR 1870 Figure 45
(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.

This entry was posted in Graph Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , . Bookmark the permalink.

One Response to Peirce’s 1870 “Logic Of Relatives” • Comment 11.11

  1. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s