Triadic Relations

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Jon Awbrey

May 20, 2020, 11:05:53 AM5/20/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: 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 truly terrific, so it's best to begin with concrete examples of their vast array. The discussion to follow takes
up a pair of simple, just barely non-trivial, but instructive examples from each of the realms of mathematics and semiotics.


• Logic Syllabus ( )
• Sign Relations ( )
• Triadic Relations ( )
• Relation Theory ( )



Jon Awbrey

May 22, 2020, 9:45:12 AM5/22/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Triadic Relations • 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, L₀ and L₁, each of which is a subset of the same cartesian
product X × Y × Z. The structures of L₀ and L₁ can be described in
the following way.

Each space X, Y, Z is isomorphic to the boolean domain B = {0, 1}
so L₀ and L₁ are subsets of the cartesian power B × B × B or the
boolean cube B³.

The operation of boolean addition, + : B × B → 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 L₀ is defined by the following formula.

L₀ = { (x, y, z) ∈ B³ : x + y + z = 0 }.

The relation L₀ is the following set of four triples in B³.

L₀ = { (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) }.

The relation L₁ is defined by the following formula.

L₁ = { (x, y, z) ∈ B³ : x + y + z = 1 }.

The relation L₁ is the following set of four triples in B³.

L₁ = { (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) }.

The triples in the relations L₀ and L₁ are conveniently arranged
in the form of relational data tables, as shown below.

Triadic Relation L₀

Triadic Relation L₁
Triadic Relation L0 Bit Sum 0.png
Triadic Relation L1 Bit Sum 1.png

Jon Awbrey

May 23, 2020, 4:06:24 PM5/23/20
to Cybernetic Communications, Loet Leydesdorff, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Loet, All ...

Many thanks for the link to your paper! 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 chalking in some 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.



On 5/23/2020 9:18 AM, Loet Leydesdorff wrote:
> Dear Jon,
> I extensively use triads in a new paper:
> Leydesdorff, Loet and Ivanova, Inga, The Measurement of 'Interdisciplinarity' and 'Synergy' in Scientific and
> Extra-Scientific Collaborations (May 17, 2020). Available at SSRN: or
> <>
> You may find it interesting.
> Best,
> Loet
> --------------------------------------------------------------------------------
> Loet Leydesdorff
> Professor emeritus, University of Amsterdam
> Amsterdam School of Communication Research (ASCoR)
> <>;
> Associate Faculty, SPRU, <>University of Sussex;
> Guest Professor Zhejiang Univ. <>, Hangzhou; Visiting Professor, ISTIC,
> <>Beijing;
> Visiting Fellow, Birkbeck <>, University of London;

Jon Awbrey

May 25, 2020, 9:30:07 AM5/25/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Triadic Relations • 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 {“A”, “B”, “i”, “u”}. The abstract consideration of how A and B use this set of
signs leads to the contemplation of a pair of triadic relations, the sign relations L_A and L_B, reflecting the
differential use of these signs by A and B, respectively.

Each of the sign relations L_A and L_B consists of eight triples of the form (x, y, z), where the “object” x belongs to
the “object domain” O = {A, 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 = {“A”, “B”, “i”, “u”}. The union S ∪ I is often
referred to as the “syntactic domain”, but in this case S = I = S ∪ I.

The set-up so far is summarized as follows:

• L_A, L_B ⊆ O × S × I

• O = {A, B}

• S = {“A”, “B”, “i”, “u”}

• I = {“A”, “B”, “i”, “u”}

The relation L_A is the following set of eight triples in O × S × I.

{ (A, “A”, “A”), (A, “A”, “i”), (A, “i”, “A”), (A, “i”, “i”),
(B, “B”, “B”), (B, “B”, “u”), (B, “u”, “B”), (B, “u”, “u”) }

The triples in L_A represent the way interpreter A uses signs. For example, the presence of (B, “u”, “B”) in L_A says A
uses “B” to mean the same thing A uses “u” to mean, namely, B.

The relation L_B is the following set of eight triples in O × S × I.

{ (A, “A”, “A”), (A, “A”, “u”), (A, “u”, “A”), (A, “u”, “u”),
(B, “B”, “B”), (B, “B”, “i”), (B, “i”, “B”), (B, “i”, “i”) }

The triples in L_B represent the way interpreter B uses signs. For example, the presence of (B, “i”, “B”) in L_B says B
uses “B” to mean the same thing B uses “i” to mean, namely, B.

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

L_A = Sign Relation of Interpreter A

L_B = Sign Relation of Interpreter B


• Semiotics ( )
Sign Relation LA Interpreter A.png
Sign Relation LB Interpreter B.png

Jon Awbrey

May 30, 2020, 9:54:57 AM5/30/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: 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 Science ( )
::: Rob Young ( )
Cf: Conceptual Barriers to Creating Integrative Universities


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 were saying was, “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
( ) .


• 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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