Triadic Relations

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Jon Awbrey

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Nov 7, 2021, 9:45:12 AMNov 7
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 AMNov 8
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 AMNov 9
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
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