Sign Relations

[Jon Awbrey]

Greetings All,
I've been upgrading the formats and graphics of the more
often resorted to resources I've put on the web over the
years and I'll be hammering away at that into the future.

The various MediaWiki sites where I put copies of this work
are apparently going through some kind of transition as far
their math formatting goes, all of them in different stages
of change. As things stand, it looks like the Wikiversity
formatting is slightly improved over the others … for now.

Regards,
JonL

Sign Relations • Anthesis
• https://inquiryintoinquiry.com/2025/12/12/sign-relations-anthesis-c/

❝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.

Resources —

Sign Relation • OEIS • MyWikiBiz • Wikiversity
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/LGxrpW
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements

[Jon Awbrey]

Sign Relations • Definition
• https://inquiryintoinquiry.com/2025/12/14/sign-relations-definition-c/

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.

❝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, New Elements of Mathematics, vol. 4, 20–21

In the general discussion of diverse theories of signs, the
question 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
may be said to have only within a particular context of
relationships.

Peirce's definition of a sign defines it in relation to its
objects and its interpretant signs, and thus defines signhood
in relative terms, by means of a predicate with three places.

In that definition, signhood is a role in a triadic relation,
a role a thing bears or plays in a determinate context of
relationships — it is not an absolute or non‑relative property
of a thing‑in‑itself, one it possesses 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 the term throughout his work, it is
clear he means what he elsewhere calls a “triple correspondence”,
and thus it is just another way of referring to the whole triadic
sign relation itself. In particular, his use of the term should
not be taken to imply a dyadic correspondence, like the kinds of
“mirror image” correspondence between realities and representations
bandied about in contemporary controversies about “correspondence
theories of truth”.

Determination —

Peirce's concept of determination is broader in several directions
than the sense of the word referring to strictly deterministic
causal‑temporal processes.

First, and especially in this context, he is invoking a more general
concept of determination, what is called a formal or informational
determination, as in saying “two points determine a line”, rather
than the more special cases of causal and temporal determinisms.

Second, he characteristically allows for what is called 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 have
occasion to view the same data, as logic is a normative
science where psychology is a descriptive science, and
so they have very different aims, methods, and rationales.

Reference —

Peirce, C.S. (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.
• Online ( https://cspeirce.com/menu/library/bycsp/l75/l75.htm )

Resources —

Sign Relation • OEIS • MyWikiBiz • Wikiversity
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

Regards,

Jon

cc: https://www.academia.edu/community/Vj32R6
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2025/12/15/sign-relations-definition/

[Jon Awbrey]

Sign Relations • Signs and Inquiry
• https://inquiryintoinquiry.com/2025/12/16/sign-relations-signs-and-inquiry-c/

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”
(Dewey, 38).

Tracing the passage of inquiry through the medium of signs calls for
an active, intricate form of cooperation between the 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, a subject the theory of signs is specialized to treat
from comparative and structural points of view.

References —

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.
Reprinted (1991), Prometheus Books, Buffalo, NY.
• https://www.gutenberg.org/files/37423/37423-h/37423-h.htm

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
• https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry
• https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry

Resources —

Sign Relation • OEIS • MyWikiBiz • Wikiversity
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

Regards,

Jon

cc: https://www.academia.edu/community/lQk7Z2
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2025/12/16/sign-relations-signs-and-inquiry/

[Jon Awbrey]

Sign Relations • Examples
• https://inquiryintoinquiry.com/2025/12/18/sign-relations-examples-c/

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 i
ndividual 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 cases
it becomes convenient to lump signs and interpretants together
in a single class called a “sign system” or “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

Display 1 • Domains of a Triadic Sign Relation
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-display-1.png

Introducing a few abbreviations for use in the 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”}

Display 2 • Domains and Elements of Two Sign Relation Examples
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-display-2.png

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.

Sign Relation Tables L(A) and L(B)
• https://inquiryintoinquiry.com/wp-content/uploads/2025/11/sign-relation-twin-tables-la-lb-2.0.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”, in other words, 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.

Resources —

Sign Relation • https://en.wikiversity.org/wiki/Sign_relation

[Jon Awbrey]

Sign Relations • Dyadic Aspects
• https://inquiryintoinquiry.com/2025/12/25/sign-relations-dyadic-aspects-c/

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

Table 2. Dyadic Aspects of Triadic Relations
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/dyadic-projections-of-triadic-relations-osi-2.m.png

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 L₁₂ or written more fully as
proj₁₂(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 vary with respect 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
innominate in current anatomies of sign relations.

Resource —

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

Regards,

Jon

cc: https://www.academia.edu/community/VX9MkR
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2025/12/25/sign-relations-dyadic-aspects/

[Jon Awbrey]

Sign Relations • Denotation
• https://inquiryintoinquiry.com/2025/12/27/sign-relations-denotation-c/

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. The result may be
visualized 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 as proj_{OS} L,
L_OS, proj₁₂ L, or L₁₂, is defined as follows.

• Den(L) = proj_{OS} L = {(o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I}.
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-display-3.png

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×S.

• Tables 3a and 3b. Denotative Components Den(L_A) and Den(L_B)
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-twin-tables-den-la-den-lb-2.0.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.

Resource —

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

Regards,

Jon

cc: https://www.academia.edu/community/V0rb3O
cc: https://stream.syscoi.com/2025/12/27/sign-relations-denotation/

[Jon Awbrey]

Sign Relations • Connotation
• https://inquiryintoinquiry.com/2025/12/28/sign-relations-connotation-c/

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 no additional difficulty. The dyadic relation making up
the connotative 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
and visualized 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, variously written
as proj_{SI} L, L_SI, proj₂₃ L, or L₂₃, is defined as follows.

• Con(L) = proj_{SI} L = {(s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O}.
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-display-4.png

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.com/wp-content/uploads/2025/12/sign-relation-twin-tables-con-la-con-lb-2.0.png

Resource —

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

Regards,

Jon

cc: https://www.academia.edu/community/VqeB0k
cc: https://stream.syscoi.com/2025/12/28/sign-relations-connotation/

[Jon Awbrey]

Sign Relations • Ennotation
• https://inquiryintoinquiry.com/2025/12/29/sign-relations-ennotation-c/

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 and visualized 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
as proj_{OI} L, L_OI, proj₁₃ L, or L₁₃, is defined as follows.

• Enn(L) = proj_{OI} L = {(o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S}.
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-display-5.png

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.com/wp-content/uploads/2025/12/sign-relation-twin-tables-enn-la-enn-lb-2.0.png

Resource —

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

Regards,

Jon

cc: https://www.academia.edu/community/V0rbOx
cc: https://stream.syscoi.com/2025/12/29/sign-relations-ennotation/

[Jon Awbrey]

Sign Relations • Semiotic Equivalence Relations 1
• https://inquiryintoinquiry.com/2025/12/30/sign-relations-semiotic-equivalence-relations-1-c/

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.com/wp-content/uploads/2025/12/connotative-components-con-la-con-lb-3.0.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.com/wp-content/uploads/2025/12/semiotic-partitions-for-interpreters-a-b-2.0.png

Resource —

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

Regards,

Jon

cc: https://www.academia.edu/community/Lm48yP
cc: https://stream.syscoi.com/2025/12/30/sign-relations-semiotic-equivalence-relations-1/

[Jon Awbrey]

Sign Relations • Semiotic Equivalence Relations 2
• https://inquiryintoinquiry.com/2025/12/31/sign-relations-semiotic-equivalence-relations-2-c/

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.com/wp-content/uploads/2025/12/sign-relation-ser-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.com/wp-content/uploads/2025/12/sign-relation-ser-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.com/wp-content/uploads/2025/12/sign-relation-ser-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.com/wp-content/uploads/2025/12/connotative-components-con-la-con-lb-3.0.png

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

• [“A”]_A = [“i”]_A

• [“B”]_A = [“u”]_A

Display 4
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-4.png

or

• “A” =_A “i”

• “B” =_A “u”

Display 5
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-5.png

In this way the SER for A induces the following semiotic partition.

• {{“A”, “i”}, {“B”, “u”}}.

Display 6
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-6.png

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

• [“A”]_B = [“u”]_B

• [“B”]_B = [“i”]_B

Display 7
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-7.png

or

• “A” =_B “u”

• “B” =_B “i”

Display 8
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-8.png

In this way the SER for B induces the following semiotic partition.

• {{“A”, “u”}, {“B”, “i”}}.

Display 9
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-9.png

Taken all together we have the following picture.

Tables 7a and 7b. Semiotic Partitions for Interpreters A and B
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/semiotic-partitions-for-interpreters-a-b-2.0.png

Resource —

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

Regards,

Jon

cc: https://www.academia.edu/community/VBAXbj
cc: https://stream.syscoi.com/2026/01/01/sign-relations-semiotic-equivalence-relations-2/