Triadic Relations

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Jon Awbrey

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Nov 7, 2021, 9:45:13 AM11/7/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Triadic Relations • 1
https://inquiryintoinquiry.com/2021/11/07/triadic-relations-1/

| 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
( https://inquiryintoinquiry.com/2012/06/14/c-s-peirce-of-triadic-being/ )

All,

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.

Resources
=========

• Relation Theory ( https://oeis.org/wiki/Relation_theory )
• Triadic Relations ( https://oeis.org/wiki/Triadic_relation )
• Sign Relations ( https://oeis.org/wiki/Sign_relation )
• Survey of Relation Theory
( https://inquiryintoinquiry.com/2020/05/15/survey-of-relation-theory-4/ )

Regards,

Jon

Jon Awbrey

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Nov 8, 2021, 9:06:33 AM11/8/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Triadic Relations • 2
https://inquiryintoinquiry.com/2021/11/08/triadic-relations-2/

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.

Figure 0. Triadic Relation L₀
https://inquiryintoinquiry.files.wordpress.com/2020/05/triadic-relation-l0-bit-sum-0.png

Figure 1. Triadic Relation L₁
https://inquiryintoinquiry.files.wordpress.com/2020/05/triadic-relation-l1-bit-sum-1.png

References
==========

Boolean Domain
https://oeis.org/wiki/Boolean_domain

Exclusive Disjunction
https://oeis.org/wiki/Exclusive_disjunction

Regards,

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

Jon Awbrey

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Nov 9, 2021, 7:24:12 AM11/9/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Triadic Relations • 3
https://inquiryintoinquiry.com/2021/11/08/triadic-relations-3/

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 impactor
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.

Table A. L_A = Sign Relation of Interpreter A
https://inquiryintoinquiry.files.wordpress.com/2020/05/sign-relation-la-interpreter-a.png

Table B. L_B = Sign Relation of Interpreter B
https://inquiryintoinquiry.files.wordpress.com/2020/05/sign-relation-lb-interpreter-b.png

Resources
=========

Survey of Relation Theory
https://inquiryintoinquiry.com/2021/11/08/survey-of-relation-theory-5/

Survey of Semiotics, Semiosis, Sign Relations
https://inquiryintoinquiry.com/2019/10/29/survey-of-semiotics-semiosis-sign-relations-1/

Regards,

Jon
Sign Relation LA Interpreter A.png
Sign Relation LB Interpreter B.png

Jon Awbrey

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Dec 6, 2021, 1:00:19 PM12/6/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG, Conceptual Graphs
Cf: Triadic Relations • Discussion 3
https://inquiryintoinquiry.com/2021/12/06/triadic-relations-discussion-3/

Re: Conceptual Graphs
https://lists.cs.uni-kassel.de/hyperkitty/list/c...@lists.iccs-conference.org/
::: Edwina Taborsky
https://lists.cs.uni-kassel.de/hyperkitty/list/c...@lists.iccs-conference.org/message/AUJBY3M7UV5KU4SKZPTONQSBNRUERFXT/

<QUOTE ET:>
A few comments on your outline of the Sign.
I think one has to be careful not to set up
a Saussurian linguistic dyad. …
</QUOTE>

Dear Edwina,

I copied your comments to a draft page and will take them up
in the fullness of time, but a few remarks by way of general
orientation to relations, triadic relations, sign relations,
and sign transformations, partly prompted by the earlier
discussion of complex systems, may be useful at this point.

One does not come to terms with systems of any complexity — adaptive,
anticipatory, intelligent systems, and those with a capacity to support
scientific inquiry, whether as autonomous agents or assistive utilities —
without the use of mathematical models to negotiate the gap between our
naturally evolved linguistic capacities and the just barely scrutable
realities manifesting in phenomena.

Peirce's quest to understand how science works takes its first big steps with his
lectures on the Logic of Science at Harvard and the Lowell Institute (1865–1866),
where he traces the bearings of deduction, induction, and hypothesis on the conduct
of scientific inquiry. There Peirce makes a good beginning by taking up Boole's
functional recasting of logic, a major advance over traditional logic rooted in the
paradigms of historical grammars. But developing a minimal adequate mathematical
basis for the logic of science will take drilling down to a deeper core.

The mathematics we need to build models of inquiry as a sign-relational
process appears for the first time in history with Peirce's early work,
especially his 1870 Logic of Relatives. It has its sources in the
mathematical realism of Leibniz and De Morgan, the functional logic of
Boole, and the algebraic research of Peirce's own father, Benjamin Peirce
( https://mathshistory.st-andrews.ac.uk/Biographies/Peirce_Benjamin/ ),
whose “Linear Associative Algebra” Charles edited for publication in the
American Journal of Mathematics (1881).

My own contributions to this pursuit I've collected over the years
under the heading of Inquiry Driven Systems, portions of which I've
shared here and there across the Web for lo! this whole millennium
in progress. A few resources along those lines are listed below.

References
==========

* 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, 269–284.
https://journals.sagepub.com/doi/abs/10.1177/1350508401082013
https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities

* Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action • The Risk
of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html
https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052
doc: https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry
pdf: https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry

* Awbrey, S.M., and Awbrey, J.L. (1991), “An Architecture for Inquiry • Building
Computer Platforms for Discovery”, Proceedings of the Eighth International Conference
on Technology and Education, Toronto, Canada, 874–875.
https://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery

* Awbrey, J.L., and Awbrey, S.M. (1990), “Exploring Research Data Interactively.
Theme One : A Program of Inquiry”, Proceedings of the Sixth Annual Conference
on Applications of Artificial Intelligence and CD-ROM in Education and Training,
Society for Applied Learning Technology, Washington, DC, 9–15.
https://academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry

Resources
=========

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
https://inquiryintoinquiry.com/2020/12/16/survey-of-abduction-deduction-induction-analogy-inquiry-2/

Survey of Cybernetics
https://inquiryintoinquiry.com/2020/03/18/survey-of-cybernetics-1/

Survey of Inquiry Driven Systems
https://inquiryintoinquiry.com/2020/12/27/survey-of-inquiry-driven-systems-3/

Survey of Pragmatic Semiotic Information
https://inquiryintoinquiry.com/2020/11/01/survey-of-pragmatic-semiotic-information-5/

Survey of Semiotics, Semiosis, Sign Relations
https://inquiryintoinquiry.com/2021/12/02/survey-of-semiotics-semiosis-sign-relations-2/

Survey of Theme One Program
https://inquiryintoinquiry.com/2020/08/28/survey-of-theme-one-program-3/

Regards,

Jon

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