Triadic Relations • Discussion 2

To everything there is a season,
A time for every purpose under heaven:
A time for building castles in the stratosphere,
A time to mind the anti-gravs that keep us here.

Re: Systems ScienceRob Young
Cf: Conceptual Barriers to Creating Integrative Universities

RY:
The aspiration to a form of knowledge ‘wisdom’ resonated with me, and, not withstanding the ‘university’ context (connotative?) the article was couched in, every time I read the word ‘university’, I mentally substituted it with ‘systems movement’ and the resonance was there.

Dear Rob,

Thanks for your comments and questions.  They took me back to the decade before the turn of the millennium when there was a general trend of thought to embrace chaos and complexity, seeking the order and simplicity on the other side.  (Oliver Wendell Holmes, but it appears in doubt whether Sr. or Jr.)

One thing I’ve learned in the mean time is just how poorly grounded and maintained are many of the abstract concepts and theories we need for grappling with the complexities of communication, computation, experimental information, and scientific inquiry.  So I’ve been doing what I can to reinforce the concrete bases and stabilize the working platforms of what otherwise tend to become empty à priori haunts.

I’ll have to be getting back to that.  For now I’ll just link to a few readings your remarks brought to mind.  The “Conceptual Barriers” paper from 2001 is the journal upgrade of a conference presentation from 1999, “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”.

Your reflex of jumping from universities to systems in general is very much on the mark.  Our work was motivated in large part by the movement toward Learning Organizations, that is, organizations able to apply organizational research to their own organizational development.  To put a fine point on it, all we are saying is, “Shouldn’t a University, as an Organization of Learning, also be a Learning Organization?”

Well, I had a lot more to relate at this point, but our dishwasher just went on the fritz, so I’ll leave it for now with a few links to Susan’s earlier work along these lines and try to get back to it later …

  • Scott, David K., and Awbrey, Susan M. (1993), “Transforming Scholarship”, Change : The Magazine of Higher Learning, 25(4), 38–43. Online (1) (2).
  • Papers by Susan Awbrey and David Scott • University of Massachusetts, Amherst.

Reference

  • Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, pp. 269–284. Abstract. Online.

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Sign Relations • Discussion 1

Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun.

— C.S. Peirce, Collected Papers, CP 2.274

Re: Cybernetic CommunicationsKlaus Krippendorff
Re: Ecology of Systems ThinkingRichard Saunders

I’m working at reviewing and revising some pieces I’ve rewritten two score times over the last … lost count of years … and that bit from Peirce is one of my favorite epigraphs for the work ahead.  But I take it as an allegorical figure whose purpose is to illustrate a certain form of relation, and not to be taken too literally.  So I’m sympathetic to the reactions of several readers who find it clangs a bit if taken at face value.  I think there are clues in the passage, the hypothetical subjunctive construction, the unnatural qualification, “without further condition”, etc., telling us Peirce did not intend it as a truth of botany.  But taken rightly it does point to the shape of a proper definition to come.  So I’ll be getting to that …

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Sign Relations • Anthesis

Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun.

— C.S. Peirce, Collected Papers, CP 2.274

In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or semiosis, Peirce uses the technical term representamen for his concept of a sign, but the shorter word is precise enough, so long as one recognizes its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign.

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Triadic Relations • Discussion 1

Re: CyberneticsLoet Leydesdorff

Loet Leydesdorff mentioned making extensive use of triads in a new paper.

  • Leydesdorff, Loet, and Ivanova, Inga (2020), “The Measurement of ‘Interdisciplinarity’ and ‘Synergy’ in Scientific and Extra-Scientific Collaborations”. Online (1) (2).

Just off-hand this looks like the right ballpark for my long run interests but it will take me a few more posts just dusting off home plate and clearing the base lines.

Here’s a paper Susan Awbrey and I wrote a while back giving some hint of the Big Game in play here, the “scholarship of integration” needed to bring the harvests of information locked away in so many isolated silos to bear on our world of common problems.

  • Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, pp. 269–284. Abstract. Online.

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Triadic Relations • Examples 2

Examples from Semiotics

The study of signs — the full variety of significant forms of expression — in relation to all the affairs signs are significant of, and in relation to all the beings signs are significant to, is known as semiotics or the theory of signs.  As described, semiotics treats of a 3-place relation among signs, their objects, and their interpreters.

The term semiosis refers to any activity or process involving signs.  Studies of semiosis focusing on its abstract form are not concerned with every concrete detail of the entities acting as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among those three roles.  In particular, the formal theory of signs does not consider all the properties of the interpretive agent but only the more striking features of the impressions signs make on a representative interpreter.  From a formal point of view this impact or influence may be treated as just another sign, called the interpretant sign, or the interpretant for short.  A triadic relation of this type, among objects, signs, and interpretants, is called a sign relation.

For example, consider the aspects of sign use involved when two people, say Ann and Bob, use their own proper names, “Ann” and “Bob”, along with the pronouns, “I” and “you”, to refer to themselves and each other.  For brevity, these four signs may be abbreviated to the set \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.  The abstract consideration of how \mathrm{A} and \mathrm{B} use this set of signs leads to the contemplation of a pair of triadic relations, the sign relations L_\mathrm{A} and L_\mathrm{B}, reflecting the differential use of these signs by \mathrm{A} and \mathrm{B}, respectively.

Each of the sign relations L_\mathrm{A} and L_\mathrm{B} consists of eight triples of the form (x, y, z), where the object x belongs to the object domain O = \{ \mathrm{A}, \mathrm{B} \}, the sign y belongs to the sign domain S, the interpretant sign z belongs to the interpretant domain I, and where it happens in this case that S = I = \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.  The union S \cup I is often referred to as the syntactic domain, but in this case S = I = S \cup I.

The set-up so far is summarized as follows:

\begin{array}{ccc}  L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I  \\[8pt]  O & = & \{ \mathrm{A}, \mathrm{B} \}  \\[8pt]  S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}  \\[8pt]  I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}  \end{array}

The relation L_\mathrm{A} is the following set of eight triples in O \times S \times I.

\begin{array}{cccccc}  \{ &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &  \\  &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) &  \}.  \end{array}

The triples in L_\mathrm{A} represent the way interpreter \mathrm{A} uses signs.  For example, the presence of ( \mathrm{B}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ) in L_\mathrm{A} says \mathrm{A} uses {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} to mean the same thing \mathrm{A} uses {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} to mean, namely, \mathrm{B}.

The relation L_\mathrm{B} is the following set of eight triples in O \times S \times I.

\begin{array}{cccccc}  \{ &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &  (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &  \\  &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &  (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) &  \}.  \end{array}

The triples in L_\mathrm{B} represent the way interpreter \mathrm{B} uses signs.  For example, the presence of ( \mathrm{B}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ) in L_\mathrm{B} says \mathrm{B} uses {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} to mean the same thing \mathrm{B} uses {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} to mean, namely, \mathrm{B}.

The triples in the relations L_\mathrm{A} and L_\mathrm{B} are conveniently arranged in the form of relational data tables, as shown below.

LA = Sign Relation of Interpreter A

LB = Sign Relation of Interpreter B

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Triadic Relations • Examples 1

Examples from Mathematics

For the sake of topics to be taken up later, it is useful to examine a pair of triadic relations in tandem.  We will construct two triadic relations, L_0 and L_1, each of which is a subset of the same cartesian product X \times Y \times Z.  The structures of L_0 and L_1 can be described in the following way.

Each space X, Y, Z is isomorphic to the boolean domain \mathbb{B} = \{ 0, 1 \} so L_0 and L_1 are subsets of the cartesian power \mathbb{B} \times \mathbb{B} \times \mathbb{B} or the boolean cube \mathbb{B}^3.

The operation of boolean addition, + : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, is equivalent to addition modulo 2, where 0 acts in the usual manner but 1 + 1 = 0.  In its logical interpretation, the plus sign can be used to indicate either the boolean operation of exclusive disjunction or the boolean relation of logical inequality.

The relation L_0 is defined by the following formula.

L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.

The relation L_0 is the following set of four triples in \mathbb{B}^3.

L_0 ~=~ \{ ~ (0, 0, 0), ~ (0, 1, 1), ~ (1, 0, 1), ~ (1, 1, 0) ~ \}.

The relation L_1 is defined by the following formula.

L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.

The relation L_1 is the following set of four triples in \mathbb{B}^3.

L_1 ~=~ \{ ~ (0, 0, 1), ~ (0, 1, 0), ~ (1, 0, 0), ~ (1, 1, 1) ~ \}.

The triples in the relations L_0 and L_1 are conveniently arranged in the form of relational data tables, as shown below.

Triadic Relation L0 Bit Sum 0

Triadic Relation L1 Bit Sum 1

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Triadic Relations • Preamble

Of triadic Being the multitude of forms is so terrific that I have usually shrunk from the task of enumerating them;  and for the present purpose such an enumeration would be worse than superfluous:  it would be a great inconvenience.

C.S. Peirce, Collected Papers, CP 6.347

A triadic relation (or ternary relation) is a special case of a polyadic or finitary relation, one in which the number of places in the relation is three.  One may also see the adjectives 3‑adic, 3‑ary, 3‑dimensional, or 3‑place being used to describe these relations.

Mathematics is positively rife with examples of triadic relations and the field of semiotics is rich in its harvest of sign relations, which are special cases of triadic relations.  In either subject, as Peirce observes, the multitude of forms is truly terrific, so it’s best to begin with concrete examples just complex enough to illustrate the distinctive features of each type.  The discussion to follow takes up a pair of simple but instructive examples from each of the realms of mathematics and semiotics.

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Relation Theory • Discussion 1

Re: CyberneticsArthur Phillips

Responding to what I’ll abductively interpret as a plea for relevance from the cybernetic galley, let me give a quick review of where we are in this many-oared expedition.

Our reading of Ashby (see Survey of Cybernetics) veered off at a point (Selection 13) where we needed to look more closely at the structures of triadic relations and the ways in which pragmatic, semiotic, and systems thinking all have triadic relations at their core.  As often happens, one side-trip leads to another, but I think our excursions through sign relations, triadic relations, and relations in general will prove useful in the long run as we get back to the question of how signs bear information of use to intelligent systems with a capacity for methodical scientific inquiry.

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Survey of Relation Theory • 4

In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications.  This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Elements

Relational Concepts

Relation Construction Relation Composition Relation Reduction
Relative Term Sign Relation Triadic Relation
Logic of Relatives Hypostatic Abstraction Continuous Predicate

Illustrations

Blog Series

Peirce’s 1870 “Logic of Relatives”

Peirce’s 1880 “Algebra of Logic” Chapter 3

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Peirce’s Categories • 20

Re: Peirce’s Categories • 15

Understanding another person’s thought can be difficult.  Understanding the way another understands a third person’s thought, all the more so, even if that third person is not so formidable a thinker as Charles Sanders Peirce.  Measures of misunderstanding may be moderated if all thoughts and thinkers are guided by common objectives but the proof of the pudding is in the partaking, as they say.  So let’s step carefully and focus on the task of determining whether category theories, old and new, make good tools for understanding sign relations.

The interaction recorded in my last post continued as follows:

RM:
I do not see how we can talk here about an operative relationship that would be a triad relationship.  It is not anything other than the composition of two morphisms and I do not ask for more.  3, 2, and 1 are the “place names” and “involves” are arrow names that I usually call alpha and beta.  Now if you think about the determinations in the sign, I have always assumed after much study of the 76 definitions, this idea that the composition of applications captures the presence in the mind of each of the elements of the sign, in such a way that they are themselves ipso facto connected by a triadic relationship.  There is a relationship of tricoexistence that is established as in this case evoked by Peirce:  “It predicates the genuinely Triadic relationship of tricoexistence, P ~\mathrm{and}~ Q ~\mathrm{and}~ R ~\mathrm{coexist}” (CP 2.318, unfortunately there is a hole in my PDF of CP right after and I [left] my paper edition at the library of my university, inaccessible at the moment).

We have a mutual incomprehension?

JA:
I don’t often join the debates over sign classification so frequently animating the animadversions of the Peirce List.  As more the observer than the participant I see the same pattern over and over, with occasional hints but never any hue of resolution fast enough to last and satisfy every dyehard.

Situations of that sort are no novelty in philosophy, or politics, or even math and science on occasion.  And when they occur it is usually because the “place to stand” from which the subject appears in its proper light has yet to be reached by every viewer.

So I’ll back up a little and say how I see things from where I am.

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