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

### 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 number of such definitions are recorded below. 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 $H = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$ and consider the dyadic relation $F \subseteq G \times H$ bigraphed below.  $\text{Figure 38. Dyadic Relation}~ F$

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

### Resources

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