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

Posted on May 6, 2014 by Jon Awbrey

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.

Definitions

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.

Dyadic Relation F
\text{Figure 38. Dyadic Relation}~ F

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

Resources

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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