Sign Relations
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Jon Awbrey
Feb 7, 2024, 12:36:32 PMFeb 7
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Anthesis
• http://inquiryintoinquiry.com/2024/02/07/sign-relations-anthesis-b/
❝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 —
Semeiotic
• https://oeis.org/wiki/Semeiotic
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Sign Relations
• https://oeis.org/wiki/Sign_relation
Triadic Relations
• https://oeis.org/wiki/Triadic_relation
Relation Theory
• https://oeis.org/wiki/Relation_theory
Regards,
Jon
cc: https://www.academia.edu/community/VqqzoO
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• http://inquiryintoinquiry.com/2024/02/07/sign-relations-anthesis-b/
❝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 —
Semeiotic
• https://oeis.org/wiki/Semeiotic
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Sign Relations
• https://oeis.org/wiki/Sign_relation
Triadic Relations
• https://oeis.org/wiki/Triadic_relation
Relation Theory
• https://oeis.org/wiki/Relation_theory
Regards,
Jon
cc: https://www.academia.edu/community/VqqzoO
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
Jon Awbrey
Feb 8, 2024, 10:36:18 AMFeb 8
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Definition
• https://inquiryintoinquiry.com/2024/02/08/sign-relations-definition-b/
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.
❝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).
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
bandied about in contemporary controversies about “correspondence
theories of truth”.
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 of defining logic,
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.
• https://cspeirce.com/menu/library/bycsp/l75/l75.htm
Resources —
Semeiotic
• https://oeis.org/wiki/Semeiotic
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Sign Relations
• https://oeis.org/wiki/Sign_relation
Triadic Relations
• https://oeis.org/wiki/Triadic_relation
Relation Theory
• https://oeis.org/wiki/Relation_theory
Document History
• https://oeis.org/wiki/Sign_relation#Document_history
Regards,
Jon
cc: https://www.academia.edu/community/lzdB46
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• https://inquiryintoinquiry.com/2024/02/08/sign-relations-definition-b/
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.
❝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).
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
bandied about in contemporary controversies about “correspondence
theories of truth”.
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 of defining logic,
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.
• https://cspeirce.com/menu/library/bycsp/l75/l75.htm
Resources —
Semeiotic
• https://oeis.org/wiki/Semeiotic
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/
Sign Relations
• https://oeis.org/wiki/Sign_relation
Triadic Relations
• https://oeis.org/wiki/Triadic_relation
Relation Theory
• https://oeis.org/wiki/Relation_theory
• https://oeis.org/wiki/Sign_relation#Document_history
Regards,
Jon
cc: https://www.academia.edu/community/lzdB46
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
Jon Awbrey
Feb 9, 2024, 12:16:45 PMFeb 9
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Signs and Inquiry
• https://inquiryintoinquiry.com/2024/02/09/sign-relations-signs-and-inquiry-b/
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” (Dewey, 38).
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 —
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
Peirce, C.S. (1902), “Parts of Carnegie Application” (L 75),
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 —
Inquiry
• https://oeis.org/wiki/Inquiry
Semeiotic
• https://oeis.org/wiki/Semeiotic
Sign Relations
• https://oeis.org/wiki/Sign_relation
cc: https://www.academia.edu/community/Lpen27
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• https://inquiryintoinquiry.com/2024/02/09/sign-relations-signs-and-inquiry-b/
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” (Dewey, 38).
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 —
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
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.
• https://cspeirce.com/menu/library/bycsp/l75/l75.htm
Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk
by Charles S. Peirce, vol. 4, 13–73.
• https://cspeirce.com/menu/library/bycsp/l75/l75.htm
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 —
Inquiry
• https://oeis.org/wiki/Inquiry
Semeiotic
• https://oeis.org/wiki/Semeiotic
Sign Relations
• https://oeis.org/wiki/Sign_relation
cc: https://www.academia.edu/community/Lpen27
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
Jon Awbrey
Feb 10, 2024, 12:12:35 PMFeb 10
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Examples
• https://inquiryintoinquiry.com/2024/02/10/sign-relations-examples-b/
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 cases
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.
Resources —
Sign Relations
• https://oeis.org/wiki/Sign_relation
Examples
• https://oeis.org/wiki/Sign_relation#Examples
cc: https://www.academia.edu/community/LgEZ8b
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• https://inquiryintoinquiry.com/2024/02/10/sign-relations-examples-b/
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 cases
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.
Resources —
Sign Relations
• https://oeis.org/wiki/Sign_relation
Examples
• https://oeis.org/wiki/Sign_relation#Examples
cc: https://www.academia.edu/community/LgEZ8b
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
Jon Awbrey
Feb 11, 2024, 1:30:53 PMFeb 11
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Dyadic Aspects
• https://inquiryintoinquiry.com/2024/02/11/sign-relations-dyadic-aspects-b/
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.
Table 2. Dyadic Aspects of Triadic Relations
• https://inquiryintoinquiry.files.wordpress.com/2020/06/dyadic-projections-of-triadic-relations.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 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.
Naturally, traditions 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 innominate
in current anatomies of sign relations.
Dyadic Aspects
• https://oeis.org/wiki/Sign_relation#Dyadic_Aspects
cc: https://www.academia.edu/community/lPxxxP
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• https://inquiryintoinquiry.com/2024/02/11/sign-relations-dyadic-aspects-b/
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.
Table 2. Dyadic Aspects of Triadic Relations
• https://inquiryintoinquiry.files.wordpress.com/2020/06/dyadic-projections-of-triadic-relations.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 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.
Naturally, traditions 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 innominate
in current anatomies of sign relations.
Dyadic Aspects
• https://oeis.org/wiki/Sign_relation#Dyadic_Aspects
cc: https://www.academia.edu/community/lPxxxP
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
Jon Awbrey
Feb 12, 2024, 11:40:31 AMFeb 12
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Denotation
• https://inquiryintoinquiry.com/2024/02/12/sign-relations-denotation-b/
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, proj₁₂ L, and L₁₂, 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×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.
Denotation
• https://oeis.org/wiki/Sign_relation#Denotation
cc: https://www.academia.edu/community/VXKKxG
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• https://inquiryintoinquiry.com/2024/02/12/sign-relations-denotation-b/
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, proj₁₂ L, and L₁₂, 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×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.
Denotation
• https://oeis.org/wiki/Sign_relation#Denotation
cc: https://www.academia.edu/community/VXKKxG
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
Jon Awbrey
Feb 13, 2024, 9:00:48 AMFeb 13
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Connotation
• https://inquiryintoinquiry.com/2024/02/12/sign-relations-connotation-b/
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
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. 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, proj₂₃ L, and
Tables 4a and 4b show the connotative components of the sign relations
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
Connotation
• https://oeis.org/wiki/Sign_relation#Connotation
cc: https://www.academia.edu/community/LmnnXP
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• https://inquiryintoinquiry.com/2024/02/12/sign-relations-connotation-b/
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
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. 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, proj₂₃ L, and
L₂₃, 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
Connotation
• https://oeis.org/wiki/Sign_relation#Connotation
cc: https://www.academia.edu/community/LmnnXP
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
Jon Awbrey
Feb 14, 2024, 1:30:23 PMFeb 14
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Ennotation
• https://inquiryintoinquiry.com/2024/02/14/sign-relations-ennotation-b/
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, proj₁₃ L, and
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
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
Ennotation
• https://oeis.org/wiki/Sign_relation#Ennotation
cc: https://www.academia.edu/community/lzdd2E
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• https://inquiryintoinquiry.com/2024/02/14/sign-relations-ennotation-b/
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, proj₁₃ L, and
L₁₃, 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
Ennotation
• https://oeis.org/wiki/Sign_relation#Ennotation
cc: https://www.academia.edu/community/lzdd2E
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
Jon Awbrey
Feb 15, 2024, 10:54:20 AMFeb 15
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Semiotic Equivalence Relations 1
• https://inquiryintoinquiry.com/2024/02/15/sign-relations-semiotic-equivalence-relations-1-b/
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
Semiotic Equivalence Relations 1
• https://oeis.org/wiki/Sign_relation#Semiotic_Equivalence_Relations_1
cc: https://www.academia.edu/community/L6PpAM
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• https://inquiryintoinquiry.com/2024/02/15/sign-relations-semiotic-equivalence-relations-1-b/
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
Semiotic Equivalence Relations 1
• https://oeis.org/wiki/Sign_relation#Semiotic_Equivalence_Relations_1
cc: https://www.academia.edu/community/L6PpAM
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
Jon Awbrey
Feb 16, 2024, 1:08:27 PMFeb 16
to Conceptual Graphs, Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Sign Relations • Semiotic Equivalence Relations 2
• https://inquiryintoinquiry.com/2024/02/16/sign-relations-semiotic-equivalence-relations-2-b/
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
In this way it induces the following 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
In this way it induces the following semiotic partition.
• {{“A”, “u”}, {“B”, “i”}}.
Display 9
• https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-9.png
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
Resources —
Sign Relations
• https://oeis.org/wiki/Sign_relation
Semiotic Equivalence Relations 2
• https://oeis.org/wiki/Sign_relation#Semiotic_Equivalence_Relations_2
cc: https://www.academia.edu/community/VvO8Je
cc: https://mathstodon.xyz/@Inquiry/111891382765624469
• https://inquiryintoinquiry.com/2024/02/16/sign-relations-semiotic-equivalence-relations-2-b/
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
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
In this way it induces the following 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
In this way it induces the following semiotic partition.
• {{“A”, “u”}, {“B”, “i”}}.
Display 9
• https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-9.png
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
Resources —
Sign Relations
• https://oeis.org/wiki/Sign_relation
• https://oeis.org/wiki/Sign_relation#Semiotic_Equivalence_Relations_2
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