The task before us is to clarify the relationships among relative terms, relations, and the special cases of relations that are given by equivalence relations, functions, and so on.

The first obstacle to get past is the order convention that Peirce’s orientation to relative terms causes him to use for functions. To focus on a concrete example of immediate use in this discussion, let’s take the “number of” function that Peirce denotes by means of square brackets and re-formulate it as a dyadic relative term as follows:

To set the dyadic relative term within a suitable context of interpretation, let us suppose that corresponds to a relation where is the set of real numbers and is a suitable syntactic domain, here described as a set of terms. The dyadic relation is at first sight a function from to There is, however, a very great likelihood that we cannot always assign a number to every term in whatever syntactic domain we happen to choose, so we may eventually be forced to treat the dyadic relation as a partial function from to All things considered, then, let me try out the following impedimentaria of strategies and compromises.

First, I adapt the functional arrow notation so that it allows us to detach the functional orientation from the order in which the names of domains are written on the page. Second, I change the notation for partial functions, or pre-functions, to one that is less likely to be confounded. This gives the scheme:

Until it becomes necessary to stipulate otherwise, let us assume that is a function in of written amounting to the functional alias of the dyadic relation and associated with the dyadic relative term whose relate lies in the set of real numbers and whose correlate lies in the set of syntactic terms.

Note.  See the article “Relation Theory” for the definitions of functions and pre-functions used in the above discussion.