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:
means that
is functional at
means that
is functional at
means that
is pre-functional at
means that
is pre-functional at
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.
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