## Forgetfulness Of Purpose • 7

I invited readers to consider Ashby’s example of a regulation game as a triadic relation $G_1 \subseteq D \times R \times O$ whose triples $(d, r, o)$ are given by either one of the following tables.

$\begin{array}{cc|ccc} \multicolumn{5}{c}{\text{Table 11/3/1}} \\[4pt] & & & R & \\ & & \alpha & \beta & \gamma \\ \hline & 1 & b & a & c \\ D & 2 & a & c & b \\ & 3 & c & b & a \end{array}$

$\begin{matrix} D & R & O \\ \hline 1 & \alpha & b \\ 1 & \beta & a \\ 1 & \gamma & c \\ 2 & \alpha & a \\ 2 & \beta & c \\ 2 & \gamma & b \\ 3 & \alpha & c \\ 3 & \beta & b \\ 3 & \gamma & a \end{matrix}$

Sungchul Ji asked a rather good question about the degree of “communication” or “transfer of information” between D and R within $G_1$ and its bearing on the irreducibility of $G_1$ as a triadic relation.  Given the light that concrete examples can throw on abstract questions, I thought it worth the effort to work through a detailed answer and I began as follows:

Ashby cast this example at a high level of abstraction in order to isolate the “bare bones” structure of the situation but we know the intended interpretation makes D the source of disturbances arising from the environment and R the regulator who has to counteract the disturbances in order to maintain the goal.

With regard to the question of communication or transfer of information between D and R, it is clear that R has to make different choices depending on what D does, so there is a degree of information transmission between D and R to that extent.  In effect, R is trying to cancel the variety that D creates.

Careful discussions of irreducibility will require us to work from explicit definitions and here it turns out there are several different notions of irreducibility that are often confounded in common use.

Looking to the case at hand, $G_1 \subseteq D \times R \times O$ is a triadic relation consisting of $9$ triples in the larger set $D \times R \times O$ that consists of $3 \times 3 \times 3 = 27$ triples.

We could reasonably use the ratio $3/9 = 1/3$ as one measure of constraint, determination, information, selection, or “law” involved in carving $G_1$ from the uncarved block $D \times R \times O.$

If we prefer an additive measure, we could extract the exponents from the ratio $3^1/3^3$ and use their difference as a measure of the information it takes to select $G_1$ from $D \times R \times O.$  Peirce pulled this very trick in some manuscripts I saw one time, getting a simple version of our modern day logarithmic measure of information.

To be continued …

### Reference

• Ashby, W.R. (1956), An Introduction to Cybernetics, Chapman and Hall, London, UK.  Republished by Methuen and Company, London, UK, 1964.  Online.
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### 3 Responses to Forgetfulness Of Purpose • 7

1. Jon Awbrey says:

On second or third thought I think Sung is correct in saying that there is no correlation between D and R if we are looking at just those two factors.  What I think I was thinking is that there is a correlation if we condition everything on getting a particular outcome.

2. Jon Awbrey says:

Sung,

Thanks for those thoughts, which will take me a while to work through.  I’m sure I must be wrong about something somewhere but I won’t be able to tell what configuration or sample space is relevant to the problem at hand until I get a good description of what exactly that problem is.

I think it’s still a good question about the degree of “communication” or “transfer of information” between D and R within G1 and its bearing on the irreducibility of G1 as a triadic relation, so I’ll put a marker here and return to the question if and when I get a better handle on it.

At any rate, we can still investigate the irreducibility of G1 independently of the information question, so I will turn to that.

Regards,

Jon

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