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 given by equivalence relations, functions, and so on.

The first obstacle to get past is the order convention 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 Peirce denotes by means of square brackets and re-formulate it as a dyadic relative term $v$ in the following way.

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

To set the dyadic relative term $v$ within a suitable context of interpretation, let’s suppose $v$ 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 $\mathbb{R}.$ It is, however, not always possible to assign a number to every term in whatever syntactic domain $S$ we happen to pick, 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’s try the following budget of strategies and compromises.

First, let’s adapt the arrow notation for functions in such a way as to allow detaching the functional orientation from the order in which the names of domains are written on the page. Second, let’s change the notation for partial functions, or pre-functions, to mark more clearly their distinction from functions. This produces the following scheme.

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

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

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

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

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

Note. Please refer to the article on Relation Theory for the definitions of functions and pre‑functions used in the above discussion.

Resources

Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

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