Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.8
http://inquiryintoinquiry.com/2014/05/06/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-8/
Posted : May 6, 2014 at 12:34 pm
Author : Jon Awbrey
Peircers,
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 ( http://intersci.ss.uci.edu/wiki/index.php/Relation_theory ).
For example, L has the property of being c-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)-regular at j, (> c)-regular
at j, and so on. For ease of reference, a few of these definitions are recorded below.
• L is c-regular at j ⇔ |L_{x @ j}| = c for all x in X_j.
• L is (< c)-regular at j ⇔ |L_{x @ j}| < c for all x in X_j.
• L is (> c)-regular at j ⇔ |L_{x @ j}| > c for all x in X_j.
• L is (≤ c)-regular at j ⇔ |L_{x @ j}| ≤ c for all x in X_j.
• L is (≥ c)-regular at j ⇔ |L_{x @ j}| ≥ c for all x in X_j.
Clearly, if any relation is (≤ c)-regular on one of its domains X_j and also (≥ c)-regular on the
same domain, then it must be (= c)-regular on that domain, in short, c-regular at j.
For example, let G = {r, s, t} and H = {1, …, 9} and consider the dyadic relation F ⊆ G × H that is
bigraphed below:
Figure 38. Dyadic Relation Example, Regular on Both Domains
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-38.jpg
We observe that F is 3-regular at G and 1-regular at H.
Regards,
Jon
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