## Peirce’s 1870 “Logic Of Relatives” • Comment 11.5

Everyone knows that the right sort of diagram can be a great aid in rendering complex matters comprehensible, so let’s extract what we need from the Relation Theory article to illuminate Peirce’s 1870 “Logic of Relatives” and use it to fashion what icons we can within the current frame of discussion.

For the immediate present, we may start with dyadic relations and describe the most frequently encountered species of relations and functions in terms of their local and numerical incidence properties.

Let $P \subseteq X \times Y$ be an arbitrary dyadic relation. The following properties of $P$ can then be defined: $\begin{array}{lll} P ~\text{is total at}~ X & \iff & P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ X. \\[6pt] P ~\text{is total at}~ Y & \iff & P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ Y. \\[6pt] P ~\text{is tubular at}~ X & \iff & P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ X. \\[6pt] P ~\text{is tubular at}~ Y & \iff & P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y. \end{array}$

If $P \subseteq X \times Y$ is tubular at $X,$ then $P$ is known as a partial function or a pre-function from $X$ to $Y,$ frequently signalized by renaming $P$ with an alternate lower case name, say ${}^{\backprime\backprime} p {}^{\prime\prime},$ and writing $p : X \rightharpoonup Y.$

Just by way of formalizing the definition: $\begin{array}{lll} P ~\text{is a pre-function}~ P : X \rightharpoonup Y & \iff & P ~\text{is tubular at}~ X. \\[6pt] P ~\text{is a pre-function}~ P : X \leftharpoonup Y & \iff & P ~\text{is tubular at}~ Y. \end{array}$

To illustrate these properties, let us fashion a generic enough example of a dyadic relation, $E \subseteq X \times Y,$ where $X = Y = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \},$ and where the bigraph picture of $E$ looks like this:

If we scan along the $X$ dimension from $0$ to $9$ we see that the incidence degrees of the $X$ nodes with the $Y$ domain are $0, 1, 2, 3, 1, 1, 1, 2, 0, 0$ in that order.

If we scan along the $Y$ dimension from $0$ to $9$ we see that the incidence degrees of the $Y$ nodes with the $X$ domain are $0, 0, 3, 2, 1, 1, 2, 1, 1, 0$ in that order.

Thus, $E$ is not total at either $X$ or $Y$ since there are nodes in both $X$ and $Y$ having incidence degrees less than $1.$

Also, $E$ is not tubular at either $X$ or $Y$ since there are nodes in both $X$ and $Y$ having incidence degrees greater than $1.$

Clearly then the relation $E$ cannot qualify as a pre-function, much less as a function on either of its relational domains.

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