Peirce’s 1870 “Logic of Relatives” • Comment 11.7

Peirce’s 1870 “Logic of Relatives”Comment 11.7

We come now to the special cases of dyadic relations known as functions.  It will serve a dual purpose in the present exposition to take the class of functions as a source of object examples for clarifying the more abstruse concepts of Relation Theory.

To begin, let us recall the definition of a local flag L_{a @ j} of a k-adic relation L.

Display 1

For a dyadic relation L \subseteq X \times Y the notation for local flags can be simplified in two ways.  First, the local flags L_{u @ 1} and L_{v @ 2} are often more conveniently notated as L_{u @ X} and L_{v @ Y}, respectively.  Second, the notation may be streamlined even further by making the following definitions.

Display 2

In light of these conventions, the local flags of a dyadic relation L \subseteq X \times Y may be comprehended under the following descriptions.

Display 3

The following definitions are also useful.

Display 4

A sufficient illustration is supplied by the earlier example E.

Dyadic Relation E
\text{Figure 35. Dyadic Relation}~ E

Figure 36 shows the local flag E_{3 @ X} of E.

Local Flag E_{3 @ X}
\text{Figure 36. Local Flag}~ E_{3 @ X}

Figure 37 shows the local flag E_{2 @ Y} of E.

Local Flag E_{2 @ Y}
\text{Figure 37. Local Flag}~ E_{2 @ Y}

Resources

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

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

6 Responses to Peirce’s 1870 “Logic of Relatives” • Comment 11.7

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