## Peirce’s 1870 “Logic Of Relatives” • Comment 11.8

Let’s take a closer look at the numerical incidence properties of relations, concentrating on the assorted regularity conditions defined in the article on Relation Theory.

For example, $L$ has the property of being $c\text{-regular at}~ j$ if and only if the cardinality of the local flag $L_{x @ j}$ is equal to $c$ for all $x$ in $X_j,$ coded in symbols, if and only if $|L_{x @ j}| = c$ for all $x$ in $X_j.$

In like fashion, one may define the numerical incidence properties $(< c)\text{-regular at}~ j,$ $(> c)\text{-regular at}~ j,$ and so on. For ease of reference, a few of these definitions are recorded below.

$\begin{array}{lll} L ~\text{is}~ c\text{-regular at}~ j & \iff & |L_{x @ j}| = c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (< c)\text{-regular at}~ j & \iff & |L_{x @ j}| < c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (> c)\text{-regular at}~ j & \iff & |L_{x @ j}| > c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (\le c)\text{-regular at}~ j & \iff & |L_{x @ j}| \le c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (\ge c)\text{-regular at}~ j & \iff & |L_{x @ j}| \ge c ~\text{for all}~ x \in X_j. \end{array}$

Clearly, if any relation is $(\le c)\text{-regular}$ on one of its domains $X_j$ and also $(\ge c)\text{-regular}$ on the same domain, then it must be $(= c)\text{-regular}$ on that domain, in short, $c\text{-regular}$ at $j.$

For example, let $G = \{ r, s, t \}$ and $H = \{ 1, \ldots, 9 \}$ and consider the dyadic relation $F \subseteq G \times H$ that is bigraphed below:

 (38)

We observe that $F$ is 3-regular at $G$ and 1-regular at $H.$