# Sign Relations

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### Jon Awbrey

Jun 29, 2022, 9:40:29 AMJun 29
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Anthesis
http://inquiryintoinquiry.com/2022/06/29/sign-relations-anthesis-2/

<QUOTE CSP:>
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
</QUOTE>

All,

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.

Resources —

• Semeiotic ( https://oeis.org/wiki/Semeiotic )
• Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )
• Sign Relations ( https://oeis.org/wiki/Sign_relation )
• Relation Theory ( https://oeis.org/wiki/Relation_theory )

cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum
cc: FB | Semeiotics • Structural Modeling • Systems Science

### Jon Awbrey

Jun 30, 2022, 11:25:39 AMJun 30
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Definition
http://inquiryintoinquiry.com/2022/06/30/sign-relations-definition-2/

All,

One of Peirce's clearest and most complete definitions of a sign
is one he gives in the context of providing a definition for logic,
and so it is informative to view it in that setting.

<QUOTE CSP:>
Logic will here be defined as formal semiotic. A definition of a sign will be
given which no more refers to human thought than does the definition of a line
as the place which a particle occupies, part by part, during a lapse of time.

Namely, a sign is something, A, which brings something, B, its interpretant sign
determined or created by it, into the same sort of correspondence with something,
C, its object, as that in which itself stands to C. It is from this definition,
together with a definition of “formal”, that I deduce mathematically the principles
of logic.

I also make a historical review of all the definitions and conceptions of logic,
and show, not merely that my definition is no novelty, but that my non-psychological
conception of logic has virtually been quite generally held, though not generally
recognized. (C.S. Peirce, NEM 4, 20–21).
</QUOTE>

In the general discussion of diverse theories of signs, the question frequently
arises whether signhood is an absolute, essential, indelible, or ontological
property of a thing, or whether it is a relational, interpretive, and mutable
role a thing can be said to have only within a particular context of relationships.

Peirce's definition of a sign defines it in relation to its object and
its interpretant sign, and thus defines signhood in relative terms, by
means of a predicate with three places. In this definition, signhood
is a role in a triadic relation, a role a thing bears or plays in a given
context of relationships — it is not an absolute, non-relative property of
a thing-in-itself, a status it maintains independently of all relationships
to other things.

Some of the terms Peirce uses in his definition of a sign
may need to be elaborated for the contemporary reader.

• Correspondence. From the way Peirce uses this term throughout his work it is
clear he means what he elsewhere calls a “triple correspondence”, in short, just
another way of referring to the whole triadic sign relation itself. In particular,
his use of this term should not be taken to imply a dyadic correspondence, as in
the varieties of “mirror image” correspondence between realities and representations

• Determination. Peirce's concept of determination is broader in several ways
than the sense of the word referring to strictly deterministic causal-temporal
processes. First, and especially in this context, he uses a more general concept
of determination, what is known as formal or informational determination, as we use
in geometry when we say “two points determine a line”, rather than the more special
cases of causal or temporal determinisms. Second, he characteristically allows for
the broader concept of determination in measure, that is, an order of determinism
admitting a full spectrum of more and less determined relationships.

• Non-psychological. Peirce's “non-psychological conception of logic” must be
distinguished from any variety of anti-psychologism. He was quite interested in
matters of psychology and had much of import to say about them. But logic and
psychology operate on different planes of study even when they happen to view
the same data, as logic is a normative science where psychology is a descriptive
science. Thus they have distinct aims, methods, and rationales.

Reference

• Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75),
in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by
Charles S. Peirce, vol. 4, 13–73.

Resources

• Semeiotic ( https://oeis.org/wiki/Semeiotic )
• Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )
• Sign Relations ( https://oeis.org/wiki/Sign_relation )
• Relation Theory ( https://oeis.org/wiki/Relation_theory )

Document History

See OEIS Wiki • Sign Relation • Document History
https://oeis.org/wiki/Sign_relation#Document_history

Regards,

Jon

### Jon Awbrey

Jul 1, 2022, 10:30:17 AMJul 1
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Signs and Inquiry
http://inquiryintoinquiry.com/2022/06/30/sign-relations-signs-and-inquiry-2/

All,

There is a close relationship between the pragmatic theory of signs and
the pragmatic theory of inquiry. In fact, the correspondence between the
two studies exhibits so many congruences and parallels it is often best to
treat them as integral parts of one and the same subject. In a very real
sense, inquiry is the process by which sign relations come to be established
and continue to evolve. In other words, inquiry, “thinking” in its best sense,
“is a term denoting the various ways in which things acquire significance”
(John Dewey).

Tracing the passage of inquiry through the medium of signs calls for an active,
intricate form of cooperation between our converging modes of investigation.
Its proper character is best understood by realizing the theory of inquiry is
adapted to study the developmental aspects of sign relations, whose evolution
the theory of signs is specialized to treat from comparative and structural
points of view.

References

• Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75),
in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by
Charles S. Peirce, vol. 4, 13–73.

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

Regards,

Jon

### Jon Awbrey

Jul 2, 2022, 1:20:20 PMJul 2
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Examples
http://inquiryintoinquiry.com/2022/07/02/sign-relations-examples-2/

All,

Soon after I made my third foray into grad school, this time in
Systems Engineering, I was trying to explain sign relations to my
advisor and he — being the very model of a modern systems engineer —
asked me to give a concrete example of a sign relation, as simple
as possible without being trivial. After much cudgeling of the
grey matter I came up with a pair of examples which had the added
benefit of bearing instructive relationships to each other. Despite
their simplicity, the examples to follow have subtleties of their own
and their careful treatment serves to illustrate important issues in
the general theory of signs.

Imagine a discussion between two people, Ann and Bob, and attend only
to the aspects of their interpretive practice involving the use of the
following nouns and pronouns.

• “Ann”, “Bob”, “I”, “you”.

• The “object domain” of their discussion
is the set of two people {Ann, Bob}.

• The “sign domain” of their discussion
is the set of four signs {“Ann”, “Bob”, “I”, “you”}.

Ann and Bob are not only the passive objects of linguistic references but also
the active interpreters of the language they use. The “system of interpretation”
associated with each language user can be represented in the form of an individual
three-place relation known as the “sign relation” of that interpreter.

In terms of its set-theoretic extension, a sign relation L is a subset
of a cartesian product O × S × I. The three sets O, S, I are known as
the “object domain”, the “sign domain”, and the “interpretant domain”,
respectively, of the sign relation L ⊆ O × S × I.

Broadly speaking, the three domains of a sign relation may be any sets
at all but the types of sign relations contemplated in formal settings
are usually constrained to having I ⊆ S. In those situations it becomes
convenient to lump signs and interpretants together in a single class
called the “sign system” or the “syntactic domain”. In the forthcoming
examples S and I are identical as sets, so the same elements manifest
themselves in two different roles of the sign relations in question.

When it becomes necessary to refer to the whole set of objects and signs
in the union of the domains O, S, I for a given sign relation L, we will
call this set the “World of L” and write W = W_L = O ∪ S ∪ I.

To facilitate an interest in the formal structures of sign relations and to
keep notations as simple as possible as the examples become more complicated,
it serves to introduce the following general notations.

• O = Object Domain
• S = Sign Domain
• I = Interpretant Domain

Introducing a few abbreviations for use in this Example,
we have the following data.

• O = {Ann, Bob} = {A, B}
• S = {“Ann”, “Bob”, “I”, “you”} = {“A”, “B”, “i”, “u”}
• I = {“Ann”, “Bob”, “I”, “you”} = {“A”, “B”, “i”, “u”}

In the present example, S = I = Syntactic Domain.

Tables 1a and 1b show the sign relations associated with the
interpreters A and B, respectively. In this arrangement the
rows of each Table list the ordered triples of the form (o, s, i)
belonging to the corresponding sign relations, L_A, L_B ⊆ O × S × I.

Figure. Sign Relation Tables L_A and L_B
https://inquiryintoinquiry.files.wordpress.com/2020/05/sign-relation-twin-tables-la-lb.png

The Tables codify a rudimentary level of interpretive practice for the
agents A and B and provide a basis for formalizing the initial semantics
appropriate to their common syntactic domain. Each row of a Table lists
an object and two co-referent signs, together forming an ordered triple
(o, s, i) called an “elementary sign relation”, that is, one element of
the relation's set-theoretic extension.

Already in this elementary context, there are several meanings which might
attach to the project of a formal semiotics, or a formal theory of meaning
for signs. In the process of discussing the alternatives, it is useful to
introduce a few terms occasionally used in the philosophy of language to
point out the needed distinctions. That is the task we’ll turn to next.

Regards,

Jon
Sign Relation Twin Tables LA & LB.png

### Jon Awbrey

Jul 3, 2022, 11:15:13 AMJul 3
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Dyadic Aspects

All,

For an arbitrary triadic relation L ⊆ O × S × I, whether it happens
to be a sign relation or not, there are 6 dyadic relations obtained
by projecting L on one of the planes of the OSI-space O × S × I.
The 6 dyadic projections of a triadic relation L are defined and
notated as shown in Table 2.

By way of unpacking the set-theoretic notation, here is what the
first definition says in ordinary language.

The dyadic relation resulting from the projection of L on the OS-plane
O × S is written briefly as L_OS or written more fully as proj_{OS}(L)
and is defined as the set of all ordered pairs (o, s) in the cartesian
product O × S for which there exists an ordered triple (o, s, i) in L
for some element i in the set I.

In the case where L is a sign relation, which it becomes by satisfying one of the
definitions of a sign relation, some of the dyadic aspects of L can be recognized
as formalizing aspects of sign meaning which have received their share of attention
from students of signs over the centuries, and thus they can be associated with
traditional concepts and terminology. Of course, traditions may vary as to the
precise formation and usage of such concepts and terms. Other aspects of meaning
have not received their fair share of attention, and thus remain anonymous on the
contemporary scene of sign studies.

Regards,

Jon

### Cliff Joslyn

Jul 3, 2022, 3:28:22 PMJul 3
Hey Jon. . .for a given arbitrary triadic relation L \subseteq O \times
S \times I (let's say that O, S, and I are all finite, non-empty sets),
I'm interested to understand what additional axioms you're saying are
necessary and sufficient to make L a sign relation. I checked
https://inquiryintoinquiry.com/2022/06/30/sign-relations-definition-2,
but it wasn't obvious, or at least, not formalized. Thanks.
--
O------------------------------------->
| Cliff Joslyn, Cybernetician at Large
V cajo...@gmail.com

### Jon Awbrey

Jul 5, 2022, 11:00:46 AMJul 5
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Denotation
http://inquiryintoinquiry.com/2022/07/05/sign-relations-denotation-2/

All,

One aspect of a sign's complete meaning concerns the reference a sign has to
its objects, which objects are collectively known as the “denotation” of the
sign. In the pragmatic theory of sign relations, denotative references fall
within the projection of the sign relation on the plane spanned by its object
domain and its sign domain.

The dyadic relation making up the “denotative”, “referent”, or “semantic”
aspect of a sign relation L is notated as Den(L). Information about the
denotative aspect of meaning is obtained from L by taking its projection
on the object-sign plane. We may visualize this as the “shadow” L casts
on the 2-dimensional space whose axes are the object domain O and the
sign domain S. The denotative component of a sign relation L, variously
written in any of forms, proj_{OS} L, L_OS, proj_{12} L, and L_12, is
defined as follows.

• Den(L) = proj_{OS} L = {(o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I}.

Tables 3a and 3b show the denotative components of the sign relations
associated with the interpreters A and B, respectively. The rows of
each Table list the ordered pairs (o, s) in the corresponding projections,
Den(L_A), Den(L_B) ⊆ O \times S.

Tables 3a and 3b. Denotative Components Den(L_A) and Den(L_B)
https://inquiryintoinquiry.files.wordpress.com/2020/06/sign-relation-twin-tables-den-la-den-lb.png

Looking to the denotative aspects of L_A and L_B, various rows of the Tables
specify, for example, that A uses “i” to denote A and “u” to denote B, while
B uses “i” to denote B and “u” to denote A.

Regards,

Jon
Sign Relation Twin Tables Den LA & Den LB.png

### Jon Awbrey

Jul 6, 2022, 10:54:08 AMJul 6
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Connotation
https://inquiryintoinquiry.com/2022/07/06/sign-relations-connotation-2/

All,

Another aspect of a sign's complete meaning concerns the reference a sign
has to its interpretants, which interpretants are collectively known as
the “connotation” of the sign. In the pragmatic theory of sign relations,
connotative references fall within the projection of the sign relation on
the plane spanned by its sign domain and its interpretant domain.

In the full theory of sign relations the connotative aspect of meaning
includes the links a sign has to affects, concepts, ideas, impressions,
intentions, and the whole realm of an interpretive agent's mental states
and allied activities, broadly encompassing intellectual associations,
emotional impressions, motivational impulses, and real conduct. Taken
at the full, in the natural setting of semiotic phenomena, this complex
system of references is unlikely ever to find itself mapped in much detail,
much less completely formalized, but the tangible warp of its accumulated
mass is commonly alluded to as the connotative import of language.

Formally speaking, however, the connotative aspect of meaning presents
aspect of a sign relation L is notated as Con(L). Information about the
connotative aspect of meaning is obtained from L by taking its projection
on the sign-interpretant plane. We may visualize this as the “shadow” L
casts on the 2-dimensional space whose axes are the sign domain S and the
interpretant domain I. The connotative component of a sign relation L,
alternatively written in any of forms, proj_{SI} L, L_SI, proj_{23} L,
and L_23, is defined as follows.

• Con(L) = proj}_{SI} L = {(s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O}.

Tables 4a and 4b show the connotative components of the sign relations
associated with the interpreters A and B, respectively. The rows of
each Table list the ordered pairs (s, i) in the corresponding projections,
Con(L_A), Con(L_B) ⊆ S × I.

Tables 4a and 4b. Connotative Components Con(L_A) and Con(L_B)
https://inquiryintoinquiry.files.wordpress.com/2020/06/sign-relation-twin-tables-con-la-con-lb.png

Regards,

Jon
Sign Relation Twin Tables Con LA & Con LB.png

### Jon Awbrey

Jul 7, 2022, 9:30:20 AMJul 7
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Ennotation
http://inquiryintoinquiry.com/2022/07/06/sign-relations-ennotation-2/

All,

A third aspect of a sign’s complete meaning concerns the relation
between its objects and its interpretants, which has no standard
name in semiotics. It would be called an “induced relation” in
graph theory or the result of “relational composition” in relation
theory. If an interpretant is recognized as a sign in its own right
then its independent reference to an object can be taken as belonging
to another moment of denotation, but this neglects the mediational
character of the whole transaction in which this occurs. Denotation
and connotation have to do with dyadic relations in which the sign
plays an active role but here we are dealing with a dyadic relation
between objects and interpretants mediated by the sign from an
off-stage position, as it were.

As a relation between objects and interpretants mediated by a sign,
this third aspect of meaning may be referred to as the “ennotation”
of a sign and the dyadic relation making up the ennotative aspect
of a sign relation L may be notated as Enn(L). Information about
the ennotative aspect of meaning is obtained from L by taking its
projection on the object-interpretant plane. We may visualize this
as the “shadow” L casts on the 2-dimensional space whose axes are
the object domain O and the interpretant domain I. The ennotative
component of a sign relation L, variously written in any of the forms,
proj_{OI} L, L_OI, proj_{13} L, and L_13, is defined as follows.

• Enn(L) = proj_{OI} L = {(o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S}.

As it happens, the sign relations L_A and L_B are fully symmetric
with respect to exchanging signs and interpretants, so all the data
of proj_{OS} L_A is echoed unchanged in proj_{OI} L_A and all the data
of proj_{OS} L_B is echoed unchanged in proj_{OI} L_B.

Tables 5a and 5b show the ennotative components of the sign relations
associated with the interpreters A and B, respectively. The rows of
each Table list the ordered pairs (o, i) in the corresponding projections,
Enn(L_A), Enn(L_B) ⊆ O × I.

Tables 5a and 5b. Ennotative Components Enn(L_A) and Enn(L_B)
https://inquiryintoinquiry.files.wordpress.com/2020/06/sign-relation-twin-tables-enn-la-enn-lb.png

Regards,

Jon
Sign Relation Twin Tables Enn LA & Enn LB.png

### Jon Awbrey

Jul 8, 2022, 8:45:43 AMJul 8
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Semiotic Equivalence Relations 1
http://inquiryintoinquiry.com/2022/07/07/sign-relations-semiotic-equivalence-relations-1-2/

All,

A “semiotic equivalence relation” (SER) is a special type of equivalence
relation arising in the analysis of sign relations. Generally speaking,
any equivalence relation induces a partition of the underlying set of
elements, known as the “domain” or “space” of the relation, into a family
of equivalence classes. In the case of a SER the equivalence classes are
called “semiotic equivalence classes” (SECs) and the partition is called
a “semiotic partition” (SEP).

The sign relations L_A and L_B have many interesting properties over and above
those possessed by sign relations in general. Some of those properties have to
do with the relation between signs and their interpretant signs, as reflected
in the projections of L_A and L_B on the SI-plane, notated as proj_{SI} L_A
and proj_{SI} L_B, respectively. The dyadic relations on S × I induced by
those projections are also referred to as the “connotative components” of the
corresponding sign relations, notated as Con(L_A) and Con(L_B), respectively.
Tables 6a and 6b show the corresponding connotative components.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)
https://inquiryintoinquiry.files.wordpress.com/2020/06/connotative-components-con-la-con-lb.png

A nice property of the sign relations L_A and L_B is that their connotative
components Con(L_A) and Con(L_B) form a pair of equivalence relations on
their common syntactic domain S = I. This type of equivalence relation
is called a “semiotic equivalence relation” (SER) because it equates
signs having the same meaning to some interpreter.

Each of the semiotic equivalence relations, Con(L_A), Con(L_B) ⊆ S×I ≅ S×S
partitions the collection of signs into semiotic equivalence classes. This
constitutes a strong form of representation in that the structure of the
interpreters’ common object domain {A, B} is reflected or reconstructed,
part for part, in the structure of each one’s semiotic partition of the
syntactic domain {“A”, “B”, “i”, “u”}.

It’s important to observe the semiotic partitions for interpreters A and B are
not identical, indeed, they are orthogonal to each other. Thus we may regard
the “form” of the partitions as corresponding to an objective structure or
invariant reality, but not the literal sets of signs themselves, independent
of the individual interpreter’s point of view.

Information about the contrasting patterns of semiotic equivalence corresponding to
the interpreters A and B is summarized in Tables 7a and 7b. The form of the Tables
serves to explain what is meant by saying the SEPs for A and B are “orthogonal” to
each other.

Tables 7a and 7b. Semiotic Partitions for Interpreters A and B
https://inquiryintoinquiry.files.wordpress.com/2020/06/semiotic-partitions-for-interpreters-a-b.png

Regards,

Jon
Connotative Components Con LA & Con LB.png
Semiotic Partitions For Interpreters A & B.png

### Jon Awbrey

Jul 9, 2022, 9:54:37 AMJul 9
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Semiotic Equivalence Relations 2
https://inquiryintoinquiry.com/2022/07/08/sign-relations-semiotic-equivalence-relations-2-2/

All,

A few items of notation are useful in discussing equivalence relations
in general and semiotic equivalence relations in particular.

In general, if E is an equivalence relation on a set X then every
element x of X belongs to a unique equivalence class under E called
“the equivalence class of x under E”. Convention provides the “square
bracket notation” for denoting such equivalence classes, in either the
form [x]_E or the simpler form [x] when the subscript E is understood.
A statement that the elements x and y are equivalent under E is called
an “equation” or an “equivalence” and may be expressed in any of the
following ways.

• (x, y) ∈ E

• x ∈ [y]_E

• y ∈ [x]_E

• [x]_E = [y]_E

• x =_E y

Display 1
https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-1.png

Thus we have the following definitions.

• [x]_E = {y ∈ X : (x, y) ∈ E}

• x =_E y ⇔ (x, y) ∈ E

Display 2
https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-2.png

In the application to sign relations it is useful to extend the square
bracket notation in the following ways. If L is a sign relation whose
connotative component L_SI is an equivalence relation on S = I, let [s]_L
be the equivalence class of s under L_SI. In short, [s]_L = [s]_{L_{SI}}.
A statement that the signs x and y belong to the same equivalence class
under a semiotic equivalence relation L_SI is called a “semiotic equation”
(SEQ) and may be written in either of the following forms.

• [x]_L = [y]_L

• x =_L y

Display 3
https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-3.png

In many situations there is one further adaptation of the square bracket
notation for semiotic equivalence classes that can be useful. Namely, when
there is known to exist a particular triple (o, s, i) in a sign relation L,
it is permissible to let [o]_L be defined as [s]_L. This modifications is
designed to make the notation for semiotic equivalence classes harmonize as
well as possible with the frequent use of similar devices for the denotations
of signs and expressions.

Applying the array of equivalence notations to the sign relations for A and B
will serve to illustrate their use and utility.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)
https://inquiryintoinquiry.files.wordpress.com/2020/06/connotative-components-con-la-con-lb.png

The semiotic equivalence relation for interpreter A
yields the following semiotic equations.

Display 4
https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-4.png

or

Display 5
https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-5.png

Thus it induces the semiotic partition:

• {{“A”, “i”}, {“B”, “u”}}.

Display 6
https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-6.png

The semiotic equivalence relation for interpreter B
yields the following semiotic equations.

Display 7
https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-7.png

or

Display 8
https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-8.png

Thus it induces the semiotic partition:

• {{“A”, “u”}, {“B”, “i”}}.

Display 9
https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-9.png
SER 2 Display 1.png
SER 2 Display 9.png
Semiotic Partitions For Interpreters A & B.png
SER 2 Display 2.png
SER 2 Display 3.png
Connotative Components Con LA & Con LB.png
SER 2 Display 4.png
SER 2 Display 5.png
SER 2 Display 6.png
SER 2 Display 7.png
SER 2 Display 8.png

### Jon Awbrey

Jul 13, 2022, 12:40:15 PMJul 13
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 11
https://inquiryintoinquiry.com/2022/07/13/sign-relations-discussion-11/

Re: Cybernetics
::: Cliff Joslyn
Re: Sign Relations • Definition
https://inquiryintoinquiry.com/2022/06/30/sign-relations-definition-2/

<QUOTE CJ:>
For a given arbitrary triadic relation L ⊆ O × S × I (let’s say that
O, S, and I are all finite, non-empty sets), I’m interested to understand
what additional axioms you’re saying are necessary and sufficient to make
L a sign relation. I checked Sign Relations • Definition, but it wasn’t
obvious, or at least, not formalized.
</QUOTE>\

Dear Cliff,

Peirce claims a definition of “logic” as “formal semiotic” and goes on
to define a “sign” in terms of its relation to its “interpretant sign”
and its “object”.

For ease of reference, here's the cited paragraph again.

<QUOTE CSP:>
Logic will here be defined as formal semiotic. A definition of a sign will be
given which no more refers to human thought than does the definition of a line
as the place which a particle occupies, part by part, during a lapse of time.
Namely, a sign is something, A, which brings something, B, its interpretant sign
determined or created by it, into the same sort of correspondence with something,
C, its object, as that in which itself stands to C. It is from this definition,
together with a definition of “formal”, that I deduce mathematically the principles
of logic. I also make a historical review of all the definitions and conceptions
of logic, and show, not merely that my definition is no novelty, but that my
non-psychological conception of logic has virtually been quite generally held,
though not generally recognized. (C.S. Peirce, NEM 4, 20–21).
</QUOTE>

Let me cut to the chase and say what I see in that passage. Peirce draws
our attention to a category of mathematical structures of use in understanding
various domains of complex phenomena by capturing aspects of objective structure
immanent in those domains.

The domains of complex phenomena of interest to “logic” in its broadest sense
encompass all that appears on the “discourse” side of any universe of discourse
we happen to discuss. That's a big enough sky for anyone to live under, but for the
moment I am focusing on the ways we transform signs in activities like communication,
computation, inquiry, learning, proof, and reasoning in general. I'm especially focused
on the ways we do now and may yet use computation to advance the other pursuits on that list.

To be continued ...

Sorry, Cliff, it took me a week to write that set-up, most of which
I spent deleting previous drafts ... I did start out with a pretty
direct reply to your question but I kept being forced to back-track
into deeper backgrounders to explain why I thought it was an answer.
So I'll continue when I rest up ...

Regards,

Jon

Reference

* Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75),
in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by
Charles S. Peirce, vol. 4, 13–73.
Sources

* C.S. Peirce • On the Definition of Logic
( https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/ )

* C.S. Peirce • Logic as Semiotic
( https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/ )

* C.S. Peirce • Objective Logic
( https://inquiryintoinquiry.com/2012/03/09/c-s-peirce-objective-logic/ )

### Jon Awbrey

Jul 16, 2022, 1:36:11 PMJul 16
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 12
http://inquiryintoinquiry.com/2022/07/16/sign-relations-discussion-12/
Dear Cliff,

From a purely speculative point of view, any triadic relation L ⊆ X×X×X
on any set X might be capable of capturing aspects of objective structure
immanent in the conduct of logical reasoning. At least I can think of no
reason to exclude those possibilities à priori.

When we turn to the task of developing computational adjuncts to inquiry
there is still no harm in keeping arbitrary triadic relations in mind, as
entire hosts of them will turn up on the “universe” side of many universes
of discourse we happen to encounter, if nowhere else.

Peirce's use of the word “definition” understandably leads us to anticipate a
strictly apodictic development, say, along the lines of abstract group theory
or axiomatic geometry. In that light I often look to group theory for hints
on how to go about tackling a category of triadic relations such as we find
in semiotics. The comparison makes for a very rough guide but the contrasts
are also instructive.

More than that, the history of group theory, springing as it did as yet unnamed
from the ground of pressing mathematical problems, from Newton's use of symmetric
functions and Galois' application of permutation groups to the theory of equations
among other sources, tells us what state of development we might reasonably expect
from the current still early days of semiotics.

To be continued …

Regards,

Jon

### Cliff Joslyn

Jul 16, 2022, 6:15:31 PMJul 16
Jon: Thanks for the response. I have a response to your other response
as well.

First off, the perspective that I'm taking is mathematical. Not
exclusively so, as the interpretation and use of mathematical models is
essential for their understanding, but at least at the moment, there is
a formal question which I'm focusing on, not a philosophical or
methodological question.

Second, I understand that you and Peirce are saying that a sign relation
*is* a ternary relation L \subseteq O \times S \times I. In other words,
if L is a sign relation, then L is ternary.

My formal question is: given that all sign relations L are ternary, is
it also the case that any ternary relation can be taken as a sign
relation? And if not, then what are the mathematical features which
distinguish sign ternary relations from non-sign ternary relations?

### Cliff Joslyn

Jul 16, 2022, 6:15:35 PMJul 16
Jon: Thanks for the response. I have a response to your other response
as well.

On 7/16/2022 10:36 AM, Jon Awbrey wrote:
> From a purely speculative point of view, any triadic relation L ⊆ X×X×X
> on any set X might be capable of capturing aspects of objective structure
> immanent in the conduct of logical reasoning.  At least I can think of no
> reason to exclude those possibilities à priori.

Please note that previously you'd said L \subseteq O \times S \times I,
so a relation on three distinct sets (presumably disjoint, although you
didn't say that). A *self*-relation L \subseteq X^3 is *quite* a
different thing.

So here you seem to answering my other response to you by saying that,
yes, not only are all sign relations ternary, but any ternary relation
relation and a ternary self-relation.

> Peirce's use of the word “definition” understandably leads us to
> anticipate a
> strictly apodictic development, say, along the lines of abstract group
> theory
> or axiomatic geometry.  In that light I often look to group theory for
> hints
> on how to go about tackling a category of triadic relations such as we
> find
> in semiotics.  The comparison makes for a very rough guide but the
> contrasts
> are also instructive.
Very good, any sense of what groups might be involved, specifically?
> More than that, the history of group theory, springing as it did as
> yet unnamed
> from the ground of pressing mathematical problems, from Newton's use
> of symmetric
> functions and Galois' application of permutation groups to the theory
> of equations
> among other sources, tells us what state of development we might
> reasonably expect
> from the current still early days of semiotics.
Interesting. I'm working now with both binary and ternary relations
trying to bridge together Galois correspondences on set systems,
hypergraph incidence matrices, concept lattices, and Dowker complexes,
while maybe also engaging with the singular values of Boolean matrices.
Most all of that is for binary relations, but expanding to ternary
relations is also of great interest, both in their own right, and also
in terms of the mappings between their {3 \choose 2} = 6 binary
projections, thinking of L as a ternary incidence tensor. It would be
really interesting to think about how to interpret these complexities
semiotically as well, if you had more insights into the classes of
ternary relations, if any, which may NOT be sign relations.
>
> To be continued …
>
> Regards,
>
> Jon
>

### Jon Awbrey

Jul 17, 2022, 8:24:17 AMJul 17
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 13
https://inquiryintoinquiry.com/2022/07/17/sign-relations-discussion-13/

Re: Cybernetics
::: Cliff Joslyn

Dear Cliff,

Backing up a little —

Whether a thing qualifies as a sign is not an ontological question,
a matter of what it is in itself, but a pragmatic question, a matter
of what role it plays in a particular application.

By extension, whether a triadic relation qualifies as a sign relation
is not just a question of its abstract structure but a question of its
potential applications, of its fitness for a particular purpose, namely,
whether we can imagine it capturing aspects of objective structure immanent
in the conduct of logical reasoning.

Because it's difficult, and not even desirable, to place prior limits on
“what we can imagine finding a use for”, we probably can't, or shouldn't try,
to reduce pragmatic definitions to ontological definitions. That's why I feel
bound to leave the boundaries a bit fuzzy.

Just to sum up what I've been struggling to say here —

It's not a bad idea to cast an oversized net at the outset, and the à priori
method can take us a way with that, but developing semiotics beyond its first
principles and early stages will depend on gathering more significant examples
of sign relations and sign transformations approaching the level we actually
employ in the practice of communication, computation, inquiry, learning, proof,
and reasoning in general. I think that's probably the best way to see the real
sense and utility of Peirce's double definition of logic and signs.

Regards,

Jon

### Cliff Joslyn

Jul 17, 2022, 1:50:09 PMJul 17
Jon: Thanks for the continuing discussion. I very much appreciate your
perspective below, and don't necessarily disagree with it.

But that said, I think you're not engaging me the way I'm seeking. Not
that you have to, of course, your choice, but I think it's valuable to
verify that I'm making myself clear.

My perspective is that of a formal modeler. There are many ways of doing
modeling, of course, but I was following your cue that we wanted to
build models of semiotic systems, models of sign relations, which are
formal. Commonly this means mathematical, which is how you initiated
this, but they could also be just formal natural language, effectively
philosophical, aiming towards precision, logical forms of consistency
and coherence, and wherever possible, denotations and explicit external
references. Towards that end, mathematical notation can be an effective
tool, and highly efficient, but isn't strictly necessary.

Anyway, on that basis, you said that a sign relation can be modeled as a
ternary relation which you expressed as L \subseteq O \times S \times I.
This notation has an explicit meaning in mathematics in terms of the
possible relationships between three sets O, S, and I which are
identified as at least somewhat different from each other. What I then
asked you is if this is a sufficient definition, whether ANY possible
ternary relation between ANY three sets could be used to model a sign
relation, or whether there were some ternary relations between some sets
which could NOT be used to model sign relations. So far, you haven't
answered, although you suggested that some mathematical groups over L
could be brought to bear.

Now if you don't know, or don't care, or if Perice doesn't say, or we
can't tell, then all that's fine, of course. I just want to make sure
that you truly understand my question.

Thanks for engaging.

### Jon Awbrey

Jul 18, 2022, 11:20:27 AMJul 18
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 14
https://inquiryintoinquiry.com/2022/07/17/sign-relations-discussion-14/

Re: Cybernetics
:: Cliff Joslyn

Dear Cliff,

Let me see if I can illustrate the problem of definition with a few examples.

First, to clear up one point of notation, in writing L ⊆ O × S × I,
there is no assumption on my part the relational domains O, S, I are
necessarily disjoint. They may intersect or even be identical, as
O = S = I. Of course we rarely need to contemplate limiting cases of
that type but I find it useful to keep then in our categorical catalogue.
(Other writers will differ on that score.) On the other hand, we very
often consider cases where S = I, as in the following two examples of
sign relations discussed in a previous post of this series.

Sign Relations • Examples
https://inquiryintoinquiry.com/2022/07/02/sign-relations-examples-2/

Tables 1a and 1b. Sign Relation Tables L_A and L_B
https://inquiryintoinquiry.files.wordpress.com/2020/05/sign-relation-twin-tables-la-lb.png

We have the following data.

O = {A, B}
S = {“A”, “B”, “i”, “u”}
I = {“A”, “B”, “i”, “u”}

As I mentioned, those examples were deliberately constructed to be as simple
as possible but they do exemplify many typical features of sign relations in
always contemplated formal languages with countable numbers of signs and
never really thought about finite sign relations at all.

Regards,

Jon
Sign Relation Twin Tables LA & LB.png