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 of P can then be defined:

\begin{array}{lll}  P ~\text{is total at}~ X & \iff & P ~\text{is}~ (\ge 1)\text{-regular at}~ X.  \\[6pt]  P ~\text{is total at}~ Y & \iff & P ~\text{is}~ (\ge 1)\text{-regular at}~ Y.  \\[6pt]  P ~\text{is tubular at}~ X & \iff & P ~\text{is}~ (\le 1)\text{-regular at}~ X.  \\[6pt]  P ~\text{is tubular at}~ Y & \iff & P ~\text{is}~ (\le 1)\text{-regular at}~ Y.  \end{array}

We previously examined dyadic relations that separately exemplified each of these regularity conditions. And we introduced a few bits of terminology and special-purpose notations for working with tubular relations:

\begin{array}{lll}  P ~\text{is a pre-function}~ P : X \rightharpoonup Y & \iff & P ~\text{is tubular at}~ X.  \\[6pt]  P ~\text{is a pre-function}~ P : X \leftharpoonup Y & \iff & P ~\text{is tubular at}~ Y.  \end{array}

We arrive by way of this winding stair at the special stamps of dyadic relations P \subseteq X \times Y that are variously described as 1-regular, total and tubular, or total prefunctions on specified domains, either X or Y or both, and that 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:

\begin{array}{lll}  P ~\text{is a function}~ P : X \to Y & \iff & P ~\text{is}~ 1\text{-regular at}~ X.  \\[6pt]  P ~\text{is a function}~ P : X \leftarrow Y & \iff & P ~\text{is}~ 1\text{-regular at}~ Y.  \end{array}

For example, let X = Y = \{ 0, \ldots, 9 \} and let F \subseteq X \times Y be the dyadic relation depicted in the bigraph below:


LOR 1870 Figure 39
(39)

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.

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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 11.9

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

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