## Peirce’s 1870 “Logic Of Relatives” • Comment 11.4

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 $v$ as follows:

$v(t) ~:=~ [t] ~=~ \text{the number of the term}~ t.$

To set the dyadic relative term $v$ within a suitable context of interpretation, let us suppose that $v$ corresponds to a relation $V \subseteq \mathbb{R} \times S$ where $\mathbb{R}$ is the set of real numbers and $S$ is a suitable syntactic domain, here described as a set of terms. The dyadic relation $V$ is at first sight a function from $S$ to $\mathbb{R}.$ There is, however, a very great likelihood that we cannot always assign a number to every term in whatever syntactic domain $S$ we happen to choose, so we may eventually be forced to treat the dyadic relation $V$ as a partial function from $S$ to $\mathbb{R}.$ 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:

$q : X \to Y$ means that $q$ is functional at $X.$

$q : X \leftarrow Y$ means that $q$ is functional at $Y.$

$q : X \rightharpoonup Y$ means that $q$ is pre-functional at $X.$

$q : X \leftharpoonup Y$ means that $q$ is pre-functional at $Y.$

Until it becomes necessary to stipulate otherwise, let us assume that $v$ is a function in $\mathbb{R}$ of $S,$ written $v : \mathbb{R} \leftarrow S,$ amounting to the functional alias of the dyadic relation $V \subseteq \mathbb{R} \times S$ and associated with the dyadic relative term $v$ whose relate lies in the set $\mathbb{R}$ of real numbers and whose correlate lies in the set $S$ 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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