Sign Relations, Triadic Relations, Relation Theory • Discussion 5

Re: Cybernetics • Cliff Joslyn
Re: Sign Relations, Triadic Relations, Relation Theory • Discussion (3) (4)

Dear Cliff,

I’m still collecting my wits from the mind-numbing events of the past two weeks so I’ll copy your last remarks here and work through them step by step.

CJ:: I think what you have is sound, and can be described in a number of ways. In years past in seeking ways to both qualify and quantify variety in systems I characterized this distinction as between “dimensional variety” and “cardinal variety”. Thankfully, this seems straightforward from a mathematical perspective, namely in a standard relational system $S = \times_{i=1}^k X_i,$ where the $X_i$ are dimensions (something that can vary), typically cast as sets, so that $\times$ here is Cartesian product.

Relational systems are just the context we need. It is usual to begin at a moderate level of generality by considering a space $X$ of the following form.

$X ~ = ~ \times_{i=1}^k X_i ~ = ~ X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k.$

(I’ll use $X$ instead of $S$ here because I want to save the letter $``S"$ for sign domains when we come to the special case of sign relational systems.)

We can now define a relation $L$ as a subset of a cartesian product.

$L \subseteq X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k.$

There are two common ways of understanding the subset symbol $``\subseteq"$ in this context. Using language from computer science I’ll call them the weak typing and strong typing interpretations.

Under weak typing conventions $L$ is just a set which happens to be a subset of the cartesian product $X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k$ but which could just as easily be cast as a subset of any other qualified superset. The mention of a particular cartesian product is accessory but not necessary to the definition of the relation itself.
Under strong typing conventions the cartesian product $X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k$ in the type-casting $L \subseteq X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k$ is an essential part of the definition of $L.$ Employing a conventional mathematical idiom, a $k$ -adic relation over the nonempty sets $X_1, X_2, \ldots, X_{k-1}, X_k$ is defined as a $(k+1)$ -tuple $(L, X_1, X_2, \ldots, X_{k-1}, X_k)$ where $L$ is a subset of $X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k.$

We have at this point opened two fronts of interest in cybernetics, namely, the generation of variety and the recognition of constraint. There’s more detail on this brand of relation theory in the resource article linked below. I’ll be taking the strong typing approach to relations from this point on, largely because it comports more naturally with category theory and thus enjoys ready applications to systems and their transformations.

But my eye-brain system is going fuzzy on me now, so I’ll break here and continue later …

Resources

cc: Cybernetics • Ontolog • Peirce List (1) (2) (3) • Structural Modeling • Systems Science

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