## Charles Sanders Peirce, George Spencer Brown, and Me • 11

There’s a new Laws of Form group in town.  James Bowery et al. have just revived the earlier group on a new platform and everything looks pretty handy so far.  There’s an honest-to-goodness 60s vibe about it for me since it’s well-known to those in the know how Spencer Brown’s work builds on Peirce’s, not just because his calculus of indications resurrects aspects of Peirce’s alpha level logical graphs but because the broader scope of his interests touched on inductive reasoning and the whole welter of knotty tangles in the pragmatics of communication, computation, concept formation which Arbib, Ashby, Bateson, Korzybski, R.D. Laing, McCulloch, Peirce, Polanyi, Watzlawick, and others probed in the matrix of quasi-paradoxes and games people play with symbols.

All of which inspires me to revise and extend the series of posts I shared with the old group a few years back, fixing in passing the large number of now broken links.

## Sign Relations, Triadic Relations, Relation Theory • Discussion 5

Re: CyberneticsCliff 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: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science

## Icon Index Symbol • 20

### Questions Concerning Certain Faculties Claimed For Signs

JA:
Icon, Index, Symbol and all other classifications are ideal types abstracted from concrete signs and there are no pure types in actual existence.  However, it is a consequence of triadic relation irreducibility that symbols are in a genuine sense the generic type while icons and indices are specializations or so-called “degenerate” cases.
DL:
I think the first sentence answers the question brilliantly.  However, I disagree with your assertion about the “degenerate cases”.  It is my understanding that iconicity is the aspect of a sign that represents its Firstness, which is incapable of degeneracy.  This also leads me to the notion that fully degenerate Thirdness, as applied to a Symbol, is not an Icon.  I would be very interested to read your thoughts on this.

The way I see Categories applying to Peirce’s logic and semiotics may be gleaned from the following Survey page.

The series beginning with the following post might be a good place to start.

cc: Peirce List (1) (2) (3) (4) (5) (6) (7)

## Icon Index Symbol • 19

### Questions Concerning Certain Faculties Claimed For Signs

SS:
As far as visual semiotics is concerned, it is helpful to think of a “terrain” or map of a semantic mode territory in which of icon-index-symbol form the points of the triangular map, or gamut.
Visual entities (and we can extend this beyond that sense modality but that is what I do most of my work in) can be plotted on this gamut.  For instance, an extremely legible typographic word would be in the upper right corner, while an illegible scribble would appear down near the bottom.  But expressive calligraphy, or perhaps a graffiti tag, might well occupy somewhere in between.  The late works of Paul Klee would mostly be placed in the center.  A passport photograph the upper left corner, etc.

cc: Peirce List (1) (2) (3) (4) (5) (6) (7)

## Icon Index Symbol • 18

### Questions Concerning Certain Faculties Claimed For Signs

Another one of those recurring questions just came up in a Facebook group devoted to Semiotics and I thought it would be useful to try my hand at a fresh attempt to answer it — or at least promote further discussion.

• MA:  Can index become symbol?  Why or why not?

Icon, Index, Symbol and all other classifications are ideal types abstracted from concrete signs and there are no pure types in actual existence.  However, it is a consequence of triadic relation irreducibility that symbols are in a genuine sense the generic type while icons and indices are specializations or so-called degenerate cases.

cc: Peirce List (1) (2) (3) (4) (5) (6) (7)

## Riffs and Rotes • Happy New Year 2021

Apart from their abstract beauty, Riffs and Rotes are structures I discovered while playing around with Gödel numberings of graphs and digraphs.  To my way of thinking they bear a deep connection to the mathematical infrastructure of logic.  Here are the Riff and Rote for 2021.

$\text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}.$

$\text{Then} ~ 2021 = 43 \cdot 47 = p_{14} \cdot p_{15} = p_{2 \cdot 7} \cdot p_{3 \cdot 5} = p_{p_1 \cdot p_4} \cdot p_{p_2 \cdot p_3}.$

## Sign Relations, Triadic Relations, Relation Theory • Discussion 4

Dear Cliff,

Many thanks for your thoughtful reply.  I copied a transcript to my blog to take up first thing next year.  Here’s hoping we all have a better one!

Regards,

Jon

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.  Here $k$ is the dimensional variety (number of dimensions, $k$-adicity), while $n_i = |X_i|$ is the cardinal variety (cardinality of dimension $i,$ $n_i$-tomicity ($n_i$-tonicity, actually?)).  One might think of the two most classic examples:
• Multiadic diatom/nic:  Maximal (finite) dimensionality, minimal non-trivial cardinality:  The bit string $(b_1, b_2, \ldots, b_k)$ where there are $k$ Boolean dimensions $X_i = \{ 0, 1 \}.$  One can imagine $k \to \infty,$ an infinite bit string, even moreso.
• Diadic infini-omic:  Minimal non-trivial dimensionality, maximal cardinality:  The Cartesian plane $\mathbb{R}^2,$ where there are $2$ real dimensions.
There’s another quantity you didn’t mention, which is the overall “variety” or size of the system, so $\prod_{i=1}^k n_i,$ which is itself a well-formed expression (only) if there are a finite number of finite dimensions.

### Resources

cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science

## Sign Relations, Triadic Relations, Relation Theory • Discussion 3

HR:
As Peircean semiotics is a three-valued logic, I think it bears relevance for the discussion about multiple-valued logic.

Dear Helmut,

The distinction between “k-adic” (involving a span of k dimensions) and “k-tomic” (involving a range of k values) is one of the earliest questions I remember discussing on the Peirce List and the panoply of other lists we ranged across in those heady surfer days.  It is critical not to confuse the two aspects of multiplicity.  In some cases it is possible to observe what mathematicians call a projective relationship between the two aspects but that does not make them identical.

I’m adding a lightly edited excerpt from one of those earlier discussions as I think it introduces the issues about as well as I could manage today.

Here is an old note I’ve been looking for since we started on this bit about isms, as I feel I managed to express in it my point of view that the key to integrating variant perspectives is to treat their contrasting values as axes or dimensions rather than so many points on a line to be selected among, each in exclusion of all the others.  To express it briefly, it is the difference between k-tomic decisions among terminal values and k-adic dimensions of extended variation.

Jon Awbrey:
I think we need to distinguish “dichotomous thinking” from “dyadic thinking”.  One has to do with the number of values, {0, 1}, {F, T}, {evil, good}, and so on, one imposes on the cosmos, the other with the number of dimensions a person puts on the face of the deep, that is to say, the number of independent axes in the frame of reference one projects on the scene or otherwise puts up to put the cosmos on.
Tom Gollier:
Your transmission kind of faded out after the “number of values”, but do you mean a difference between, say, two values of truth and falsity on the one hand, and all things being divided into subjects and predicates, functions and arguments, and such as that on the other?  If so, I’d like to second the notion, as not only are the two values much less odious, if no less rigorous, in their applications, but they’re often maligned as naive or simplistic by arguments which actually should be applied to the idea, naive and simplistic in the extreme, that there are only two kinds of things.
Jon Awbrey:
There may be a connection — I will have to think about it — but trichotomic, dichotomic, monocotyledonic, whatever, refer to the number of values, 3, 2, 1, whatever, in the range of a function.  In contrast, triadic, dyadic, monadic, as a series, refer to the number of independent dimensions involved in a relation, which could be represented as the axes of a coordinate frame or the columns of a data table.  As the appearance of the word “independent” should clue you in, this will be one of those parti-colored woods in which the interpretive paths of mathematicians and normal folks are likely to diverge.
A particular type of misunderstanding may arise when people imbued in the different ways of thinking try to communicate with each other.  The following figure illustrates the situation for the case where k = 2.
This shows how the “number of values” thinker projects the indications of the “number of axes” thinker onto the linear spectrum of admitted directions, oppositions, or values, tending to reduce the mutually complementing dimensions to a tug-of-war of strife-torn exclusions and polarizations.
Even when the tomic thinker tries to achieve a balance, a form of equilibrium, or a compromising harmony, the distortion due to this style of projection will always render the resulting system untenable.
Probably my bias is evident.
But I think it is safe to say, for whatever else it might be good, tomic thinking is of limited use in trying to understand Peirce’s thought.
Just to mention one of the settings where this theme has arisen in my studies recently, you may enjoy the exercise of reading, in the light of this projective template, Susan Haack’s Evidence and Inquiry, where she strives to achieve a balance or a compromise between foundationalism and coherentism, that is, more or less, objectivism and relativism, and with some attempt to incorporate the insights of Peirce’s POV.  But a tomic thinker, per se, will not be able to comprehend what the heck Peirce was talking about.

### Resources

cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science

## Survey of Inquiry Driven Systems • 3

This is a Survey of blog and wiki posts on Inquiry Driven Systems, material I plan to refine toward a more compact and systematic treatment of the subject.

An inquiry driven system is a system having among its state variables some representing its state of information with respect to various topics of interest, for example, its own state and the states of any potential object systems.  Thus it has a component of state tracing a trajectory though an information state space.

## Inquiry Driven Systems • Comment 6

Dear Peter,

It’s funny you should mention Tennyson’s poem in the context of an author’s view of publication as I once laid out a detailed interpretation of the poem as a metaphor on the poet’s quest to communicate.  I know I wrote a shorter, sweeter essay on that somewhere I can’t find right now but here’s one of my more turgid dilatations where I used the poem as an “epitext” — a connected series of epigraphs — for a discussion of what I called Ostensibly Recursive Texts (ORTs).

Tennyson’s poem The Lady of Shalott is akin to an ORT, but a bit more remote, since the name styled as “The Lady of Shalott”, that the author invokes over the course of the text, is not at first sight the title of a poem, but a title its character adopts and afterwards adapts as the name of a boat.  It is only on a deeper reading that this text can be related to or transformed into a proper ORT.  Operating on a general principle of interpretation, the reader is entitled to suspect the author is trying to say something about himself, his life, and his work, and that he is likely to be exploiting for this purpose the figure of his ostensible character and the vehicle of his manifest text.  If this is an aspect of the author’s intention, whether conscious or unconscious, then the reader has a right to expect several forms of analogy are key to understanding the full intention of the text.

cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science