9 views

Skip to first unread message

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

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

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

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

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

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

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

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

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

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

cc: https://www.academia.edu/community/VvO8Je

cc: https://mathstodon.xyz/@Inquiry/111891382765624469

Reply all

Reply to author

Forward

0 new messages

Search

Clear search

Close search

Google apps

Main menu