Peirce’s 1870 “Logic of Relatives” • Comment 11.9

Peirce’s 1870 “Logic of Relatives”Comment 11.9

Among the variety of regularities affecting dyadic relations we pay special attention to the c-regularity conditions where c is equal to 1.

Let P \subseteq X \times Y be an arbitrary dyadic relation.  The following properties can be defined.

Definitions 1

We previously examined dyadic relations exemplifying each of these regularity conditions.  Then we introduced a few bits of terminology and special-purpose notations for working with tubular relations.

Definitions 2

We arrive by way of this winding stair at the special cases of dyadic relations P \subseteq X \times Y variously described as 1-regular, total and tubular, or total prefunctions on specified domains, X or Y or both, and which are more often celebrated as functions on those domains.

If P is a pre-function P : X \rightharpoonup Y that happens to be total at X, then P is known as a function from X to Y, typically indicated as P : X \to Y.

To say that a relation P \subseteq X \times Y is total and tubular at X is to say that P is 1-regular at X.  Thus, we may formalize the following definitions.

Definitions 3

For example, let X = Y = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \} and let F \subseteq X \times Y be the dyadic relation depicted in the bigraph below.

Dyadic Relation F
\text{Figure 39. Dyadic Relation}~ F

We observe that F is a function at Y and we record this fact in either of the manners F : X \leftarrow Y or F : Y \to X.

Resources

cc: CyberneticsOntolog ForumStructural ModelingSystems Science
cc: FB | Peirce MattersLaws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

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5 Responses to Peirce’s 1870 “Logic of Relatives” • Comment 11.9

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