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:


LOR 1870 Figure 38
(38)

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

This entry was posted in Graph Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , . Bookmark the permalink.

One Response to Peirce’s 1870 “Logic Of Relatives” • Comment 11.8

  1. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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