We have reached a suitable place to pause in our reading of Peirce’s text — actually, it’s more like a place to run as fast as we can along a parallel track — where I can pay off a few of the expository IOUs I’ve been using to pave the way to this point.
The more pressing debts that come to mind are concerned with the matter of Peirce’s “number of” function that maps a term into a number and with my justification for calling a certain style of illustration the hypergraph picture of relational composition. As it happens, there is a thematic relation between these topics, and so I can make my way forward by addressing them together.
At this point we have two good pictures of how to compute the relational compositions of arbitrary dyadic relations, namely, the bigraph representation and the matrix representation, each of which has its differential advantages in different types of situations.
But we do not have a comparable picture of how to compute the richer variety of relational compositions that involve triadic or any higher adicity relations. As a matter of fact, we run into a non-trivial classification problem simply to enumerate the different types of compositions that arise in these cases.
Therefore, let us inaugurate a systematic study of relational composition, general enough to articulate the “generative potency” of Peirce’s 1870 Logic of Relatives.