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Jun 29, 2022, 9:40:29 AMJun 29

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Anthesis

http://inquiryintoinquiry.com/2022/06/29/sign-relations-anthesis-2/

<QUOTE CSP:>

Thus, if a sunflower, in turning towards the sun, becomes by that

very act fully capable, without further condition, of reproducing

a sunflower which turns in precisely corresponding ways toward the

sun, and of doing so with the same reproductive power, the sunflower

would become a Representamen of the sun.

— C.S. Peirce, Collected Papers, CP 2.274

</QUOTE>

All,

In his picturesque illustration of a sign relation, along with his tracing

of a corresponding sign process, or “semiosis”, Peirce uses the technical term

“representamen” for his concept of a sign, but the shorter word is precise enough,

so long as one recognizes its meaning in a particular theory of signs is given by

a specific definition of what it means to be a sign.

Resources —

• Semeiotic ( https://oeis.org/wiki/Semeiotic )

• Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )

• Sign Relations ( https://oeis.org/wiki/Sign_relation )

• Triadic Relations ( https://oeis.org/wiki/Triadic_relation )

• Relation Theory ( https://oeis.org/wiki/Relation_theory )

cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum

cc: FB | Semeiotics • Structural Modeling • Systems Science

http://inquiryintoinquiry.com/2022/06/29/sign-relations-anthesis-2/

<QUOTE CSP:>

Thus, if a sunflower, in turning towards the sun, becomes by that

very act fully capable, without further condition, of reproducing

a sunflower which turns in precisely corresponding ways toward the

sun, and of doing so with the same reproductive power, the sunflower

would become a Representamen of the sun.

— C.S. Peirce, Collected Papers, CP 2.274

</QUOTE>

All,

In his picturesque illustration of a sign relation, along with his tracing

of a corresponding sign process, or “semiosis”, Peirce uses the technical term

“representamen” for his concept of a sign, but the shorter word is precise enough,

so long as one recognizes its meaning in a particular theory of signs is given by

a specific definition of what it means to be a sign.

Resources —

• Semeiotic ( https://oeis.org/wiki/Semeiotic )

• Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )

• Sign Relations ( https://oeis.org/wiki/Sign_relation )

• Triadic Relations ( https://oeis.org/wiki/Triadic_relation )

• Relation Theory ( https://oeis.org/wiki/Relation_theory )

cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum

cc: FB | Semeiotics • Structural Modeling • Systems Science

Jun 30, 2022, 11:25:39 AMJun 30

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Definition

http://inquiryintoinquiry.com/2022/06/30/sign-relations-definition-2/

All,

One of Peirce's clearest and most complete definitions of a sign

is one he gives in the context of providing a definition for logic,

and so it is informative to view it in that setting.

<QUOTE CSP:>

Logic will here be defined as formal semiotic. A definition of a sign will be

given which no more refers to human thought than does the definition of a line

as the place which a particle occupies, part by part, during a lapse of time.

Namely, a sign is something, A, which brings something, B, its interpretant sign

determined or created by it, into the same sort of correspondence with something,

C, its object, as that in which itself stands to C. It is from this definition,

together with a definition of “formal”, that I deduce mathematically the principles

of logic.

I also make a historical review of all the definitions and conceptions of logic,

and show, not merely that my definition is no novelty, but that my non-psychological

conception of logic has virtually been quite generally held, though not generally

recognized. (C.S. Peirce, NEM 4, 20–21).

</QUOTE>

In the general discussion of diverse theories of signs, the question frequently

arises whether signhood is an absolute, essential, indelible, or ontological

property of a thing, or whether it is a relational, interpretive, and mutable

role a thing can be said to have only within a particular context of relationships.

Peirce's definition of a sign defines it in relation to its object and

its interpretant sign, and thus defines signhood in relative terms, by

means of a predicate with three places. In this definition, signhood

is a role in a triadic relation, a role a thing bears or plays in a given

context of relationships — it is not an absolute, non-relative property of

a thing-in-itself, a status it maintains independently of all relationships

to other things.

Some of the terms Peirce uses in his definition of a sign

may need to be elaborated for the contemporary reader.

• Correspondence. From the way Peirce uses this term throughout his work it is

clear he means what he elsewhere calls a “triple correspondence”, in short, just

another way of referring to the whole triadic sign relation itself. In particular,

his use of this term should not be taken to imply a dyadic correspondence, as in

the varieties of “mirror image” correspondence between realities and representations

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, 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.

Online ( https://arisbe.sitehost.iu.edu/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

See OEIS Wiki • Sign Relation • Document History

https://oeis.org/wiki/Sign_relation#Document_history

Regards,

Jon

http://inquiryintoinquiry.com/2022/06/30/sign-relations-definition-2/

All,

One of Peirce's clearest and most complete definitions of a sign

is one he gives in the context of providing a definition for logic,

and so it is informative to view it in that setting.

<QUOTE CSP:>

Logic will here be defined as formal semiotic. A definition of a sign will be

given which no more refers to human thought than does the definition of a line

as the place which a particle occupies, part by part, during a lapse of time.

Namely, a sign is something, A, which brings something, B, its interpretant sign

determined or created by it, into the same sort of correspondence with something,

C, its object, as that in which itself stands to C. It is from this definition,

together with a definition of “formal”, that I deduce mathematically the principles

of logic.

I also make a historical review of all the definitions and conceptions of logic,

and show, not merely that my definition is no novelty, but that my non-psychological

conception of logic has virtually been quite generally held, though not generally

recognized. (C.S. Peirce, NEM 4, 20–21).

</QUOTE>

In the general discussion of diverse theories of signs, the question frequently

arises whether signhood is an absolute, essential, indelible, or ontological

property of a thing, or whether it is a relational, interpretive, and mutable

role a thing can be said to have only within a particular context of relationships.

Peirce's definition of a sign defines it in relation to its object and

its interpretant sign, and thus defines signhood in relative terms, by

means of a predicate with three places. In this definition, signhood

is a role in a triadic relation, a role a thing bears or plays in a given

context of relationships — it is not an absolute, non-relative property of

a thing-in-itself, a status it maintains independently of all relationships

to other things.

Some of the terms Peirce uses in his definition of a sign

may need to be elaborated for the contemporary reader.

• Correspondence. From the way Peirce uses this term throughout his work it is

clear he means what he elsewhere calls a “triple correspondence”, in short, just

another way of referring to the whole triadic sign relation itself. In particular,

his use of this term should not be taken to imply a dyadic correspondence, as in

the varieties of “mirror image” correspondence between realities and representations

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, 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.

Online ( https://arisbe.sitehost.iu.edu/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 )

See OEIS Wiki • Sign Relation • Document History

https://oeis.org/wiki/Sign_relation#Document_history

Regards,

Jon

Jul 1, 2022, 10:30:17 AMJul 1

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Signs and Inquiry

http://inquiryintoinquiry.com/2022/06/30/sign-relations-signs-and-inquiry-2/

All,

There is a close relationship between the pragmatic theory of signs and

the pragmatic theory of inquiry. In fact, the correspondence between the

two studies exhibits so many congruences and parallels it is often best to

treat them as integral parts of one and the same subject. In a very real

sense, inquiry is the process by which sign relations come to be established

and continue to evolve. In other words, inquiry, “thinking” in its best sense,

“is a term denoting the various ways in which things acquire significance”

(John Dewey).

Tracing the passage of inquiry through the medium of signs calls for an active,

intricate form of cooperation between our converging modes of investigation.

Its proper character is best understood by realizing the theory of inquiry is

adapted to study the developmental aspects of sign relations, whose evolution

the theory of signs is specialized to treat from comparative and structural

points of view.

References

• Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75),

in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by

Charles S. Peirce, vol. 4, 13–73.

( https://arisbe.sitehost.iu.edu/menu/library/bycsp/L75/l75.htm )

• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”,

Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.

( https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html )

Journal ( https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 )

[doc] ( https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry )

[pdf] ( https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry )

Regards,

Jon

http://inquiryintoinquiry.com/2022/06/30/sign-relations-signs-and-inquiry-2/

All,

There is a close relationship between the pragmatic theory of signs and

the pragmatic theory of inquiry. In fact, the correspondence between the

two studies exhibits so many congruences and parallels it is often best to

treat them as integral parts of one and the same subject. In a very real

sense, inquiry is the process by which sign relations come to be established

and continue to evolve. In other words, inquiry, “thinking” in its best sense,

“is a term denoting the various ways in which things acquire significance”

(John Dewey).

Tracing the passage of inquiry through the medium of signs calls for an active,

intricate form of cooperation between our converging modes of investigation.

Its proper character is best understood by realizing the theory of inquiry is

adapted to study the developmental aspects of sign relations, whose evolution

the theory of signs is specialized to treat from comparative and structural

points of view.

References

• Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75),

in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by

Charles S. Peirce, vol. 4, 13–73.

• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”,

Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.

( https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html )

Journal ( https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 )

[doc] ( https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry )

[pdf] ( https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry )

Regards,

Jon

Jul 2, 2022, 1:20:20 PMJul 2

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Examples

http://inquiryintoinquiry.com/2022/07/02/sign-relations-examples-2/

All,

Soon after I made my third foray into grad school, this time in

Systems Engineering, I was trying to explain sign relations to my

advisor and he — being the very model of a modern systems engineer —

asked me to give a concrete example of a sign relation, as simple

as possible without being trivial. After much cudgeling of the

grey matter I came up with a pair of examples which had the added

benefit of bearing instructive relationships to each other. Despite

their simplicity, the examples to follow have subtleties of their own

and their careful treatment serves to illustrate important issues in

the general theory of signs.

Imagine a discussion between two people, Ann and Bob, and attend only

to the aspects of their interpretive practice involving the use of the

following nouns and pronouns.

• “Ann”, “Bob”, “I”, “you”.

• The “object domain” of their discussion

is the set of two people {Ann, Bob}.

• The “sign domain” of their discussion

is the set of four signs {“Ann”, “Bob”, “I”, “you”}.

Ann and Bob are not only the passive objects of linguistic references but also

the active interpreters of the language they use. The “system of interpretation”

associated with each language user can be represented in the form of an individual

three-place relation known as the “sign relation” of that interpreter.

In terms of its set-theoretic extension, a sign relation L is a subset

of a cartesian product O × S × I. The three sets O, S, I are known as

the “object domain”, the “sign domain”, and the “interpretant domain”,

respectively, of the sign relation L ⊆ O × S × I.

Broadly speaking, the three domains of a sign relation may be any sets

at all but the types of sign relations contemplated in formal settings

are usually constrained to having I ⊆ S. In those situations it becomes

convenient to lump signs and interpretants together in a single class

called the “sign system” or the “syntactic domain”. In the forthcoming

examples S and I are identical as sets, so the same elements manifest

themselves in two different roles of the sign relations in question.

When it becomes necessary to refer to the whole set of objects and signs

in the union of the domains O, S, I for a given sign relation L, we will

call this set the “World of L” and write W = W_L = O ∪ S ∪ I.

To facilitate an interest in the formal structures of sign relations and to

keep notations as simple as possible as the examples become more complicated,

it serves to introduce the following general notations.

• O = Object Domain

• S = Sign Domain

• I = Interpretant Domain

Introducing a few abbreviations for use in this Example,

we have the following data.

• O = {Ann, Bob} = {A, B}

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

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

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

Tables 1a and 1b show the sign relations associated with the

interpreters A and B, respectively. In this arrangement the

rows of each Table list the ordered triples of the form (o, s, i)

belonging to the corresponding sign relations, L_A, L_B ⊆ O × S × I.

Figure. Sign Relation Tables L_A and L_B

https://inquiryintoinquiry.files.wordpress.com/2020/05/sign-relation-twin-tables-la-lb.png

The Tables codify a rudimentary level of interpretive practice for the

agents A and B and provide a basis for formalizing the initial semantics

appropriate to their common syntactic domain. Each row of a Table lists

an object and two co-referent signs, together forming an ordered triple

(o, s, i) called an “elementary sign relation”, that is, one element of

the relation's set-theoretic extension.

Already in this elementary context, there are several meanings which might

attach to the project of a formal semiotics, or a formal theory of meaning

for signs. In the process of discussing the alternatives, it is useful to

introduce a few terms occasionally used in the philosophy of language to

point out the needed distinctions. That is the task we’ll turn to next.

Regards,

Jon

http://inquiryintoinquiry.com/2022/07/02/sign-relations-examples-2/

All,

Soon after I made my third foray into grad school, this time in

Systems Engineering, I was trying to explain sign relations to my

advisor and he — being the very model of a modern systems engineer —

asked me to give a concrete example of a sign relation, as simple

as possible without being trivial. After much cudgeling of the

grey matter I came up with a pair of examples which had the added

benefit of bearing instructive relationships to each other. Despite

their simplicity, the examples to follow have subtleties of their own

and their careful treatment serves to illustrate important issues in

the general theory of signs.

Imagine a discussion between two people, Ann and Bob, and attend only

to the aspects of their interpretive practice involving the use of the

following nouns and pronouns.

• “Ann”, “Bob”, “I”, “you”.

• The “object domain” of their discussion

is the set of two people {Ann, Bob}.

• The “sign domain” of their discussion

is the set of four signs {“Ann”, “Bob”, “I”, “you”}.

Ann and Bob are not only the passive objects of linguistic references but also

the active interpreters of the language they use. The “system of interpretation”

associated with each language user can be represented in the form of an individual

three-place relation known as the “sign relation” of that interpreter.

In terms of its set-theoretic extension, a sign relation L is a subset

of a cartesian product O × S × I. The three sets O, S, I are known as

the “object domain”, the “sign domain”, and the “interpretant domain”,

respectively, of the sign relation L ⊆ O × S × I.

Broadly speaking, the three domains of a sign relation may be any sets

at all but the types of sign relations contemplated in formal settings

are usually constrained to having I ⊆ S. In those situations it becomes

convenient to lump signs and interpretants together in a single class

called the “sign system” or the “syntactic domain”. In the forthcoming

examples S and I are identical as sets, so the same elements manifest

themselves in two different roles of the sign relations in question.

When it becomes necessary to refer to the whole set of objects and signs

in the union of the domains O, S, I for a given sign relation L, we will

call this set the “World of L” and write W = W_L = O ∪ S ∪ I.

To facilitate an interest in the formal structures of sign relations and to

keep notations as simple as possible as the examples become more complicated,

it serves to introduce the following general notations.

• O = Object Domain

• S = Sign Domain

• I = Interpretant Domain

Introducing a few abbreviations for use in this Example,

we have the following data.

• O = {Ann, Bob} = {A, B}

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

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

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

Tables 1a and 1b show the sign relations associated with the

interpreters A and B, respectively. In this arrangement the

rows of each Table list the ordered triples of the form (o, s, i)

belonging to the corresponding sign relations, L_A, L_B ⊆ O × S × I.

Figure. Sign Relation Tables L_A and L_B

https://inquiryintoinquiry.files.wordpress.com/2020/05/sign-relation-twin-tables-la-lb.png

The Tables codify a rudimentary level of interpretive practice for the

agents A and B and provide a basis for formalizing the initial semantics

appropriate to their common syntactic domain. Each row of a Table lists

an object and two co-referent signs, together forming an ordered triple

(o, s, i) called an “elementary sign relation”, that is, one element of

the relation's set-theoretic extension.

Already in this elementary context, there are several meanings which might

attach to the project of a formal semiotics, or a formal theory of meaning

for signs. In the process of discussing the alternatives, it is useful to

introduce a few terms occasionally used in the philosophy of language to

point out the needed distinctions. That is the task we’ll turn to next.

Regards,

Jon

Jul 3, 2022, 11:15:13 AMJul 3

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Dyadic Aspects

http://inquiryintoinquiry.com/2022/07/03/sign-relations-dyadic-aspects-2/

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. Of course, traditions may vary as to the

precise formation and usage of such concepts and terms. Other aspects of meaning

have not received their fair share of attention, and thus remain anonymous on the

contemporary scene of sign studies.

Regards,

Jon

http://inquiryintoinquiry.com/2022/07/03/sign-relations-dyadic-aspects-2/

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. Of course, traditions may vary as to the

precise formation and usage of such concepts and terms. Other aspects of meaning

have not received their fair share of attention, and thus remain anonymous on the

contemporary scene of sign studies.

Regards,

Jon

Jul 3, 2022, 3:28:22 PMJul 3

to cyb...@googlegroups.com

Hey Jon. . .for a given arbitrary triadic relation L \subseteq O \times

S \times I (let's say that O, S, and I are all finite, non-empty sets),

I'm interested to understand what additional axioms you're saying are

necessary and sufficient to make L a sign relation. I checked

https://inquiryintoinquiry.com/2022/06/30/sign-relations-definition-2,

but it wasn't obvious, or at least, not formalized. Thanks.

--

O------------------------------------->

| Cliff Joslyn, Cybernetician at Large

V cajo...@gmail.com

S \times I (let's say that O, S, and I are all finite, non-empty sets),

I'm interested to understand what additional axioms you're saying are

necessary and sufficient to make L a sign relation. I checked

https://inquiryintoinquiry.com/2022/06/30/sign-relations-definition-2,

but it wasn't obvious, or at least, not formalized. Thanks.

O------------------------------------->

| Cliff Joslyn, Cybernetician at Large

V cajo...@gmail.com

Jul 5, 2022, 11:00:46 AMJul 5

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Denotation

http://inquiryintoinquiry.com/2022/07/05/sign-relations-denotation-2/

All,

One aspect of a sign's complete meaning concerns the reference a sign has to

its objects, which objects are collectively known as the “denotation” of the

sign. In the pragmatic theory of sign relations, denotative references fall

within the projection of the sign relation on the plane spanned by its object

domain and its sign domain.

The dyadic relation making up the “denotative”, “referent”, or “semantic”

aspect of a sign relation L is notated as Den(L). Information about the

denotative aspect of meaning is obtained from L by taking its projection

on the object-sign plane. We may visualize this as the “shadow” L casts

on the 2-dimensional space whose axes are the object domain O and the

sign domain S. The denotative component of a sign relation L, variously

written in any of forms, proj_{OS} L, L_OS, proj_{12} L, and L_12, is

defined as follows.

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

Tables 3a and 3b show the denotative components of the sign relations

associated with the interpreters A and B, respectively. The rows of

each Table list the ordered pairs (o, s) in the corresponding projections,

Den(L_A), Den(L_B) ⊆ O \times S.

Tables 3a and 3b. Denotative Components Den(L_A) and Den(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/sign-relation-twin-tables-den-la-den-lb.png

Looking to the denotative aspects of L_A and L_B, various rows of the Tables

specify, for example, that A uses “i” to denote A and “u” to denote B, while

B uses “i” to denote B and “u” to denote A.

Regards,

Jon

http://inquiryintoinquiry.com/2022/07/05/sign-relations-denotation-2/

All,

One aspect of a sign's complete meaning concerns the reference a sign has to

its objects, which objects are collectively known as the “denotation” of the

sign. In the pragmatic theory of sign relations, denotative references fall

within the projection of the sign relation on the plane spanned by its object

domain and its sign domain.

The dyadic relation making up the “denotative”, “referent”, or “semantic”

aspect of a sign relation L is notated as Den(L). Information about the

denotative aspect of meaning is obtained from L by taking its projection

on the object-sign plane. We may visualize this as the “shadow” L casts

on the 2-dimensional space whose axes are the object domain O and the

sign domain S. The denotative component of a sign relation L, variously

written in any of forms, proj_{OS} L, L_OS, proj_{12} L, and L_12, is

defined as follows.

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

Tables 3a and 3b show the denotative components of the sign relations

associated with the interpreters A and B, respectively. The rows of

each Table list the ordered pairs (o, s) in the corresponding projections,

Den(L_A), Den(L_B) ⊆ O \times S.

Tables 3a and 3b. Denotative Components Den(L_A) and Den(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/sign-relation-twin-tables-den-la-den-lb.png

Looking to the denotative aspects of L_A and L_B, various rows of the Tables

specify, for example, that A uses “i” to denote A and “u” to denote B, while

B uses “i” to denote B and “u” to denote A.

Regards,

Jon

Jul 6, 2022, 10:54:08 AMJul 6

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Connotation

https://inquiryintoinquiry.com/2022/07/06/sign-relations-connotation-2/

All,

Another aspect of a sign's complete meaning concerns the reference a sign

has to its interpretants, which interpretants are collectively known as

the “connotation” of the sign. In the pragmatic theory of sign relations,

connotative references fall within the projection of the sign relation on

the plane spanned by its sign domain and its interpretant domain.

In the full theory of sign relations the connotative aspect of meaning

includes the links a sign has to affects, concepts, ideas, impressions,

intentions, and the whole realm of an interpretive agent's mental states

and allied activities, broadly encompassing intellectual associations,

emotional impressions, motivational impulses, and real conduct. Taken

at the full, in the natural setting of semiotic phenomena, this complex

system of references is unlikely ever to find itself mapped in much detail,

much less completely formalized, but the tangible warp of its accumulated

mass is commonly alluded to as the connotative import of language.

Formally speaking, however, the connotative aspect of meaning presents

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, proj_{23} L,

and L_23, is defined as follows.

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

Tables 4a and 4b show the connotative components of the sign relations

Con(L_A), Con(L_B) ⊆ S × I.

Tables 4a and 4b. Connotative Components Con(L_A) and Con(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/sign-relation-twin-tables-con-la-con-lb.png

Regards,

Jon

https://inquiryintoinquiry.com/2022/07/06/sign-relations-connotation-2/

All,

Another aspect of a sign's complete meaning concerns the reference a sign

has to its interpretants, which interpretants are collectively known as

the “connotation” of the sign. In the pragmatic theory of sign relations,

connotative references fall within the projection of the sign relation on

the plane spanned by its sign domain and its interpretant domain.

In the full theory of sign relations the connotative aspect of meaning

includes the links a sign has to affects, concepts, ideas, impressions,

intentions, and the whole realm of an interpretive agent's mental states

and allied activities, broadly encompassing intellectual associations,

emotional impressions, motivational impulses, and real conduct. Taken

at the full, in the natural setting of semiotic phenomena, this complex

system of references is unlikely ever to find itself mapped in much detail,

much less completely formalized, but the tangible warp of its accumulated

mass is commonly alluded to as the connotative import of language.

Formally speaking, however, the connotative aspect of meaning presents

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, proj_{23} L,

and L_23, is defined as follows.

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

Tables 4a and 4b show the connotative components of the sign relations

associated with the interpreters A and B, respectively. The rows of

each Table list the ordered pairs (s, i) in the corresponding projections,
Con(L_A), Con(L_B) ⊆ S × I.

Tables 4a and 4b. Connotative Components Con(L_A) and Con(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/sign-relation-twin-tables-con-la-con-lb.png

Regards,

Jon

Jul 7, 2022, 9:30:20 AMJul 7

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Ennotation

http://inquiryintoinquiry.com/2022/07/06/sign-relations-ennotation-2/

All,

A third aspect of a sign’s complete meaning concerns the relation

between its objects and its interpretants, which has no standard

name in semiotics. It would be called an “induced relation” in

graph theory or the result of “relational composition” in relation

theory. If an interpretant is recognized as a sign in its own right

then its independent reference to an object can be taken as belonging

to another moment of denotation, but this neglects the mediational

character of the whole transaction in which this occurs. Denotation

and connotation have to do with dyadic relations in which the sign

plays an active role but here we are dealing with a dyadic relation

between objects and interpretants mediated by the sign from an

off-stage position, as it were.

As a relation between objects and interpretants mediated by a sign,

this third aspect of meaning may be referred to as the “ennotation”

of a sign and the dyadic relation making up the ennotative aspect

of a sign relation L may be notated as Enn(L). Information about

the ennotative aspect of meaning is obtained from L by taking its

projection on the object-interpretant plane. We may visualize this

component of a sign relation L, variously written in any of the forms,

proj_{OI} L, L_OI, proj_{13} L, and L_13, is defined as follows.

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

As it happens, the sign relations L_A and L_B are fully symmetric

with respect to exchanging signs and interpretants, so all the data

of proj_{OS} L_A is echoed unchanged in proj_{OI} L_A and all the data

of proj_{OS} L_B is echoed unchanged in proj_{OI} L_B.

Tables 5a and 5b show the ennotative components of the sign relations

Enn(L_A), Enn(L_B) ⊆ O × I.

Tables 5a and 5b. Ennotative Components Enn(L_A) and Enn(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/sign-relation-twin-tables-enn-la-enn-lb.png

Regards,

Jon

http://inquiryintoinquiry.com/2022/07/06/sign-relations-ennotation-2/

All,

A third aspect of a sign’s complete meaning concerns the relation

between its objects and its interpretants, which has no standard

name in semiotics. It would be called an “induced relation” in

graph theory or the result of “relational composition” in relation

theory. If an interpretant is recognized as a sign in its own right

then its independent reference to an object can be taken as belonging

to another moment of denotation, but this neglects the mediational

character of the whole transaction in which this occurs. Denotation

and connotation have to do with dyadic relations in which the sign

plays an active role but here we are dealing with a dyadic relation

between objects and interpretants mediated by the sign from an

off-stage position, as it were.

As a relation between objects and interpretants mediated by a sign,

this third aspect of meaning may be referred to as the “ennotation”

of a sign and the dyadic relation making up the ennotative aspect

of a sign relation L may be notated as Enn(L). Information about

the ennotative aspect of meaning is obtained from L by taking its

projection on the object-interpretant plane. We may visualize this

as the “shadow” L casts on the 2-dimensional space whose axes are

the object domain O and the interpretant domain I. The ennotative
component of a sign relation L, variously written in any of the forms,

proj_{OI} L, L_OI, proj_{13} L, and L_13, is defined as follows.

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

As it happens, the sign relations L_A and L_B are fully symmetric

with respect to exchanging signs and interpretants, so all the data

of proj_{OS} L_A is echoed unchanged in proj_{OI} L_A and all the data

of proj_{OS} L_B is echoed unchanged in proj_{OI} L_B.

Tables 5a and 5b show the ennotative components of the sign relations

associated with the interpreters A and B, respectively. The rows of

each Table list the ordered pairs (o, i) in the corresponding projections,
Enn(L_A), Enn(L_B) ⊆ O × I.

Tables 5a and 5b. Ennotative Components Enn(L_A) and Enn(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/sign-relation-twin-tables-enn-la-enn-lb.png

Regards,

Jon

Jul 8, 2022, 8:45:43 AMJul 8

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Semiotic Equivalence Relations 1

http://inquiryintoinquiry.com/2022/07/07/sign-relations-semiotic-equivalence-relations-1-2/

All,

A “semiotic equivalence relation” (SER) is a special type of equivalence

relation arising in the analysis of sign relations. Generally speaking,

any equivalence relation induces a partition of the underlying set of

elements, known as the “domain” or “space” of the relation, into a family

of equivalence classes. In the case of a SER the equivalence classes are

called “semiotic equivalence classes” (SECs) and the partition is called

a “semiotic partition” (SEP).

The sign relations L_A and L_B have many interesting properties over and above

those possessed by sign relations in general. Some of those properties have to

do with the relation between signs and their interpretant signs, as reflected

in the projections of L_A and L_B on the SI-plane, notated as proj_{SI} L_A

and proj_{SI} L_B, respectively. The dyadic relations on S × I induced by

those projections are also referred to as the “connotative components” of the

corresponding sign relations, notated as Con(L_A) and Con(L_B), respectively.

Tables 6a and 6b show the corresponding connotative components.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/connotative-components-con-la-con-lb.png

A nice property of the sign relations L_A and L_B is that their connotative

components Con(L_A) and Con(L_B) form a pair of equivalence relations on

their common syntactic domain S = I. This type of equivalence relation

is called a “semiotic equivalence relation” (SER) because it equates

signs having the same meaning to some interpreter.

Each of the semiotic equivalence relations, Con(L_A), Con(L_B) ⊆ S×I ≅ S×S

partitions the collection of signs into semiotic equivalence classes. This

constitutes a strong form of representation in that the structure of the

interpreters’ common object domain {A, B} is reflected or reconstructed,

part for part, in the structure of each one’s semiotic partition of the

syntactic domain {“A”, “B”, “i”, “u”}.

It’s important to observe the semiotic partitions for interpreters A and B are

not identical, indeed, they are orthogonal to each other. Thus we may regard

the “form” of the partitions as corresponding to an objective structure or

invariant reality, but not the literal sets of signs themselves, independent

of the individual interpreter’s point of view.

Information about the contrasting patterns of semiotic equivalence corresponding to

the interpreters A and B is summarized in Tables 7a and 7b. The form of the Tables

serves to explain what is meant by saying the SEPs for A and B are “orthogonal” to

each other.

Tables 7a and 7b. Semiotic Partitions for Interpreters A and B

https://inquiryintoinquiry.files.wordpress.com/2020/06/semiotic-partitions-for-interpreters-a-b.png

Regards,

Jon

http://inquiryintoinquiry.com/2022/07/07/sign-relations-semiotic-equivalence-relations-1-2/

All,

A “semiotic equivalence relation” (SER) is a special type of equivalence

relation arising in the analysis of sign relations. Generally speaking,

any equivalence relation induces a partition of the underlying set of

elements, known as the “domain” or “space” of the relation, into a family

of equivalence classes. In the case of a SER the equivalence classes are

called “semiotic equivalence classes” (SECs) and the partition is called

a “semiotic partition” (SEP).

The sign relations L_A and L_B have many interesting properties over and above

those possessed by sign relations in general. Some of those properties have to

do with the relation between signs and their interpretant signs, as reflected

in the projections of L_A and L_B on the SI-plane, notated as proj_{SI} L_A

and proj_{SI} L_B, respectively. The dyadic relations on S × I induced by

those projections are also referred to as the “connotative components” of the

corresponding sign relations, notated as Con(L_A) and Con(L_B), respectively.

Tables 6a and 6b show the corresponding connotative components.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/connotative-components-con-la-con-lb.png

A nice property of the sign relations L_A and L_B is that their connotative

components Con(L_A) and Con(L_B) form a pair of equivalence relations on

their common syntactic domain S = I. This type of equivalence relation

is called a “semiotic equivalence relation” (SER) because it equates

signs having the same meaning to some interpreter.

Each of the semiotic equivalence relations, Con(L_A), Con(L_B) ⊆ S×I ≅ S×S

partitions the collection of signs into semiotic equivalence classes. This

constitutes a strong form of representation in that the structure of the

interpreters’ common object domain {A, B} is reflected or reconstructed,

part for part, in the structure of each one’s semiotic partition of the

syntactic domain {“A”, “B”, “i”, “u”}.

It’s important to observe the semiotic partitions for interpreters A and B are

not identical, indeed, they are orthogonal to each other. Thus we may regard

the “form” of the partitions as corresponding to an objective structure or

invariant reality, but not the literal sets of signs themselves, independent

of the individual interpreter’s point of view.

Information about the contrasting patterns of semiotic equivalence corresponding to

the interpreters A and B is summarized in Tables 7a and 7b. The form of the Tables

serves to explain what is meant by saying the SEPs for A and B are “orthogonal” to

each other.

Tables 7a and 7b. Semiotic Partitions for Interpreters A and B

https://inquiryintoinquiry.files.wordpress.com/2020/06/semiotic-partitions-for-interpreters-a-b.png

Regards,

Jon

Jul 9, 2022, 9:54:37 AMJul 9

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Semiotic Equivalence Relations 2

https://inquiryintoinquiry.com/2022/07/08/sign-relations-semiotic-equivalence-relations-2-2/

All,

A few items of notation are useful in discussing equivalence relations

in general and semiotic equivalence relations in particular.

In general, if E is an equivalence relation on a set X then every

element x of X belongs to a unique equivalence class under E called

“the equivalence class of x under E”. Convention provides the “square

bracket notation” for denoting such equivalence classes, in either the

form [x]_E or the simpler form [x] when the subscript E is understood.

A statement that the elements x and y are equivalent under E is called

an “equation” or an “equivalence” and may be expressed in any of the

following ways.

• (x, y) ∈ E

• x ∈ [y]_E

• y ∈ [x]_E

• [x]_E = [y]_E

• x =_E y

Display 1

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

Thus we have the following definitions.

• [x]_E = {y ∈ X : (x, y) ∈ E}

• x =_E y ⇔ (x, y) ∈ E

Display 2

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

In the application to sign relations it is useful to extend the square

bracket notation in the following ways. If L is a sign relation whose

connotative component L_SI is an equivalence relation on S = I, let [s]_L

be the equivalence class of s under L_SI. In short, [s]_L = [s]_{L_{SI}}.

A statement that the signs x and y belong to the same equivalence class

under a semiotic equivalence relation L_SI is called a “semiotic equation”

(SEQ) and may be written in either of the following forms.

• [x]_L = [y]_L

• x =_L y

Display 3

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

In many situations there is one further adaptation of the square bracket

notation for semiotic equivalence classes that can be useful. Namely, when

there is known to exist a particular triple (o, s, i) in a sign relation L,

it is permissible to let [o]_L be defined as [s]_L. This modifications is

designed to make the notation for semiotic equivalence classes harmonize as

well as possible with the frequent use of similar devices for the denotations

of signs and expressions.

Applying the array of equivalence notations to the sign relations for A and B

will serve to illustrate their use and utility.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/connotative-components-con-la-con-lb.png

The semiotic equivalence relation for interpreter A

yields the following semiotic equations.

Display 4

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

or

Display 5

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

Thus it induces the semiotic partition:

• {{“A”, “i”}, {“B”, “u”}}.

Display 6

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

The semiotic equivalence relation for interpreter B

yields the following semiotic equations.

Display 7

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

or

Display 8

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

Thus it induces the semiotic partition:

• {{“A”, “u”}, {“B”, “i”}}.

Display 9

https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-9.png

https://inquiryintoinquiry.com/2022/07/08/sign-relations-semiotic-equivalence-relations-2-2/

All,

A few items of notation are useful in discussing equivalence relations

in general and semiotic equivalence relations in particular.

In general, if E is an equivalence relation on a set X then every

element x of X belongs to a unique equivalence class under E called

“the equivalence class of x under E”. Convention provides the “square

bracket notation” for denoting such equivalence classes, in either the

form [x]_E or the simpler form [x] when the subscript E is understood.

A statement that the elements x and y are equivalent under E is called

an “equation” or an “equivalence” and may be expressed in any of the

following ways.

• (x, y) ∈ E

• x ∈ [y]_E

• y ∈ [x]_E

• [x]_E = [y]_E

• x =_E y

Display 1

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

Thus we have the following definitions.

• [x]_E = {y ∈ X : (x, y) ∈ E}

• x =_E y ⇔ (x, y) ∈ E

Display 2

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

In the application to sign relations it is useful to extend the square

bracket notation in the following ways. If L is a sign relation whose

connotative component L_SI is an equivalence relation on S = I, let [s]_L

be the equivalence class of s under L_SI. In short, [s]_L = [s]_{L_{SI}}.

A statement that the signs x and y belong to the same equivalence class

under a semiotic equivalence relation L_SI is called a “semiotic equation”

(SEQ) and may be written in either of the following forms.

• [x]_L = [y]_L

• x =_L y

Display 3

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

In many situations there is one further adaptation of the square bracket

notation for semiotic equivalence classes that can be useful. Namely, when

there is known to exist a particular triple (o, s, i) in a sign relation L,

it is permissible to let [o]_L be defined as [s]_L. This modifications is

designed to make the notation for semiotic equivalence classes harmonize as

well as possible with the frequent use of similar devices for the denotations

of signs and expressions.

Applying the array of equivalence notations to the sign relations for A and B

will serve to illustrate their use and utility.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)

https://inquiryintoinquiry.files.wordpress.com/2020/06/connotative-components-con-la-con-lb.png

yields the following semiotic equations.

Display 4

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

or

Display 5

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

Thus it induces the semiotic partition:

• {{“A”, “i”}, {“B”, “u”}}.

Display 6

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

The semiotic equivalence relation for interpreter B

yields the following semiotic equations.

Display 7

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

or

Display 8

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

Thus it induces the semiotic partition:

• {{“A”, “u”}, {“B”, “i”}}.

Display 9

https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-9.png

Jul 13, 2022, 12:40:15 PMJul 13

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Discussion 11

https://inquiryintoinquiry.com/2022/07/13/sign-relations-discussion-11/

Re: Cybernetics

https://groups.google.com/g/cybcom/c/TpRK4fxguD0

::: Cliff Joslyn

https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/8mh1CC18EQAJ

Re: Sign Relations • Definition

https://inquiryintoinquiry.com/2022/06/30/sign-relations-definition-2/

<QUOTE CJ:>

For a given arbitrary triadic relation L ⊆ O × S × I (let’s say that

Dear Cliff,

Peirce claims a definition of “logic” as “formal semiotic” and goes on

to define a “sign” in terms of its relation to its “interpretant sign”

and its “object”.

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

<QUOTE CSP:>

Logic will here be defined as formal semiotic. A definition of a sign will be

given which no more refers to human thought than does the definition of a line

as the place which a particle occupies, part by part, during a lapse of time.

Namely, a sign is something, A, which brings something, B, its interpretant sign

determined or created by it, into the same sort of correspondence with something,

C, its object, as that in which itself stands to C. It is from this definition,

together with a definition of “formal”, that I deduce mathematically the principles

of logic. I also make a historical review of all the definitions and conceptions

of logic, and show, not merely that my definition is no novelty, but that my

non-psychological conception of logic has virtually been quite generally held,

though not generally recognized. (C.S. Peirce, NEM 4, 20–21).

</QUOTE>

Let me cut to the chase and say what I see in that passage. Peirce draws

our attention to a category of mathematical structures of use in understanding

various domains of complex phenomena by capturing aspects of objective structure

immanent in those domains.

The domains of complex phenomena of interest to “logic” in its broadest sense

encompass all that appears on the “discourse” side of any universe of discourse

we happen to discuss. That's a big enough sky for anyone to live under, but for the

moment I am focusing on the ways we transform signs in activities like communication,

computation, inquiry, learning, proof, and reasoning in general. I'm especially focused

on the ways we do now and may yet use computation to advance the other pursuits on that list.

To be continued ...

Sorry, Cliff, it took me a week to write that set-up, most of which

I spent deleting previous drafts ... I did start out with a pretty

direct reply to your question but I kept being forced to back-track

into deeper backgrounders to explain why I thought it was an answer.

So I'll continue when I rest up ...

Regards,

Jon

Reference

* Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75),

* C.S. Peirce • On the Definition of Logic

( https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/ )

* C.S. Peirce • Logic as Semiotic

( https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/ )

* C.S. Peirce • Objective Logic

( https://inquiryintoinquiry.com/2012/03/09/c-s-peirce-objective-logic/ )

https://inquiryintoinquiry.com/2022/07/13/sign-relations-discussion-11/

Re: Cybernetics

https://groups.google.com/g/cybcom/c/TpRK4fxguD0

::: Cliff Joslyn

https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/8mh1CC18EQAJ

Re: Sign Relations • Definition

https://inquiryintoinquiry.com/2022/06/30/sign-relations-definition-2/

<QUOTE CJ:>

For a given arbitrary triadic relation L ⊆ O × S × I (let’s say that

O, S, and I are all finite, non-empty sets), I’m interested to understand

what additional axioms you’re saying are necessary and sufficient to make

L a sign relation. I checked Sign Relations • Definition, but it wasn’t
what additional axioms you’re saying are necessary and sufficient to make

obvious, or at least, not formalized.

</QUOTE>\
Dear Cliff,

Peirce claims a definition of “logic” as “formal semiotic” and goes on

to define a “sign” in terms of its relation to its “interpretant sign”

and its “object”.

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

<QUOTE CSP:>

Logic will here be defined as formal semiotic. A definition of a sign will be

given which no more refers to human thought than does the definition of a line

as the place which a particle occupies, part by part, during a lapse of time.

Namely, a sign is something, A, which brings something, B, its interpretant sign

determined or created by it, into the same sort of correspondence with something,

C, its object, as that in which itself stands to C. It is from this definition,

together with a definition of “formal”, that I deduce mathematically the principles

of logic. I also make a historical review of all the definitions and conceptions

of logic, and show, not merely that my definition is no novelty, but that my

non-psychological conception of logic has virtually been quite generally held,

though not generally recognized. (C.S. Peirce, NEM 4, 20–21).

</QUOTE>

our attention to a category of mathematical structures of use in understanding

various domains of complex phenomena by capturing aspects of objective structure

immanent in those domains.

The domains of complex phenomena of interest to “logic” in its broadest sense

encompass all that appears on the “discourse” side of any universe of discourse

we happen to discuss. That's a big enough sky for anyone to live under, but for the

moment I am focusing on the ways we transform signs in activities like communication,

computation, inquiry, learning, proof, and reasoning in general. I'm especially focused

on the ways we do now and may yet use computation to advance the other pursuits on that list.

To be continued ...

Sorry, Cliff, it took me a week to write that set-up, most of which

I spent deleting previous drafts ... I did start out with a pretty

direct reply to your question but I kept being forced to back-track

into deeper backgrounders to explain why I thought it was an answer.

So I'll continue when I rest up ...

Regards,

Jon

Reference

* Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75),

in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by

Charles S. Peirce, vol. 4, 13–73.

Sources
Charles S. Peirce, vol. 4, 13–73.

* C.S. Peirce • On the Definition of Logic

( https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/ )

* C.S. Peirce • Logic as Semiotic

( https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/ )

* C.S. Peirce • Objective Logic

( https://inquiryintoinquiry.com/2012/03/09/c-s-peirce-objective-logic/ )

Jul 16, 2022, 1:36:11 PMJul 16

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Discussion 12

http://inquiryintoinquiry.com/2022/07/16/sign-relations-discussion-12/

Re: Cybernetics

https://groups.google.com/g/cybcom/c/TpRK4fxguD0

::: Cliff Joslyn

https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/8mh1CC18EQAJ

Dear Cliff,

From a purely speculative point of view, any triadic relation L ⊆ X×X×X

on any set X might be capable of capturing aspects of objective structure

immanent in the conduct of logical reasoning. At least I can think of no

reason to exclude those possibilities à priori.

When we turn to the task of developing computational adjuncts to inquiry

there is still no harm in keeping arbitrary triadic relations in mind, as

entire hosts of them will turn up on the “universe” side of many universes

of discourse we happen to encounter, if nowhere else.

Peirce's use of the word “definition” understandably leads us to anticipate a

strictly apodictic development, say, along the lines of abstract group theory

or axiomatic geometry. In that light I often look to group theory for hints

on how to go about tackling a category of triadic relations such as we find

in semiotics. The comparison makes for a very rough guide but the contrasts

are also instructive.

More than that, the history of group theory, springing as it did as yet unnamed

from the ground of pressing mathematical problems, from Newton's use of symmetric

functions and Galois' application of permutation groups to the theory of equations

among other sources, tells us what state of development we might reasonably expect

from the current still early days of semiotics.

To be continued …

Regards,

Jon

http://inquiryintoinquiry.com/2022/07/16/sign-relations-discussion-12/

Re: Cybernetics

https://groups.google.com/g/cybcom/c/TpRK4fxguD0

::: Cliff Joslyn

https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/8mh1CC18EQAJ

From a purely speculative point of view, any triadic relation L ⊆ X×X×X

on any set X might be capable of capturing aspects of objective structure

immanent in the conduct of logical reasoning. At least I can think of no

reason to exclude those possibilities à priori.

When we turn to the task of developing computational adjuncts to inquiry

there is still no harm in keeping arbitrary triadic relations in mind, as

entire hosts of them will turn up on the “universe” side of many universes

of discourse we happen to encounter, if nowhere else.

Peirce's use of the word “definition” understandably leads us to anticipate a

strictly apodictic development, say, along the lines of abstract group theory

or axiomatic geometry. In that light I often look to group theory for hints

on how to go about tackling a category of triadic relations such as we find

in semiotics. The comparison makes for a very rough guide but the contrasts

are also instructive.

More than that, the history of group theory, springing as it did as yet unnamed

from the ground of pressing mathematical problems, from Newton's use of symmetric

functions and Galois' application of permutation groups to the theory of equations

among other sources, tells us what state of development we might reasonably expect

from the current still early days of semiotics.

To be continued …

Regards,

Jon

Jul 16, 2022, 6:15:31 PMJul 16

to cyb...@googlegroups.com

Jon: Thanks for the response. I have a response to your other response

as well.

First off, the perspective that I'm taking is mathematical. Not

exclusively so, as the interpretation and use of mathematical models is

essential for their understanding, but at least at the moment, there is

a formal question which I'm focusing on, not a philosophical or

methodological question.

Second, I understand that you and Peirce are saying that a sign relation

*is* a ternary relation L \subseteq O \times S \times I. In other words,

if L is a sign relation, then L is ternary.

My formal question is: given that all sign relations L are ternary, is

it also the case that any ternary relation can be taken as a sign

relation? And if not, then what are the mathematical features which

distinguish sign ternary relations from non-sign ternary relations?

as well.

First off, the perspective that I'm taking is mathematical. Not

exclusively so, as the interpretation and use of mathematical models is

essential for their understanding, but at least at the moment, there is

a formal question which I'm focusing on, not a philosophical or

methodological question.

Second, I understand that you and Peirce are saying that a sign relation

*is* a ternary relation L \subseteq O \times S \times I. In other words,

if L is a sign relation, then L is ternary.

My formal question is: given that all sign relations L are ternary, is

it also the case that any ternary relation can be taken as a sign

relation? And if not, then what are the mathematical features which

distinguish sign ternary relations from non-sign ternary relations?

Jul 16, 2022, 6:15:35 PMJul 16

to cyb...@googlegroups.com

Jon: Thanks for the response. I have a response to your other response

as well.

as well.

On 7/16/2022 10:36 AM, Jon Awbrey wrote:

> From a purely speculative point of view, any triadic relation L ⊆ X×X×X

> on any set X might be capable of capturing aspects of objective structure

> immanent in the conduct of logical reasoning. At least I can think of no

> reason to exclude those possibilities à priori.

Please note that previously you'd said L \subseteq O \times S \times I,
> From a purely speculative point of view, any triadic relation L ⊆ X×X×X

> on any set X might be capable of capturing aspects of objective structure

> immanent in the conduct of logical reasoning. At least I can think of no

> reason to exclude those possibilities à priori.

so a relation on three distinct sets (presumably disjoint, although you

didn't say that). A *self*-relation L \subseteq X^3 is *quite* a

different thing.

So here you seem to answering my other response to you by saying that,

yes, not only are all sign relations ternary, but any ternary relation

can be a sign relation. Except about this difference between a ternary

relation and a ternary self-relation.

> Peirce's use of the word “definition” understandably leads us to

> anticipate a

> strictly apodictic development, say, along the lines of abstract group

> theory

> or axiomatic geometry. In that light I often look to group theory for

> hints

> on how to go about tackling a category of triadic relations such as we

> find

> in semiotics. The comparison makes for a very rough guide but the

> contrasts

> are also instructive.

> More than that, the history of group theory, springing as it did as

> yet unnamed

> from the ground of pressing mathematical problems, from Newton's use

> of symmetric

> functions and Galois' application of permutation groups to the theory

> of equations

> among other sources, tells us what state of development we might

> reasonably expect

> from the current still early days of semiotics.

Interesting. I'm working now with both binary and ternary relations
> yet unnamed

> from the ground of pressing mathematical problems, from Newton's use

> of symmetric

> functions and Galois' application of permutation groups to the theory

> of equations

> among other sources, tells us what state of development we might

> reasonably expect

> from the current still early days of semiotics.

trying to bridge together Galois correspondences on set systems,

hypergraph incidence matrices, concept lattices, and Dowker complexes,

while maybe also engaging with the singular values of Boolean matrices.

Most all of that is for binary relations, but expanding to ternary

relations is also of great interest, both in their own right, and also

in terms of the mappings between their {3 \choose 2} = 6 binary

projections, thinking of L as a ternary incidence tensor. It would be

really interesting to think about how to interpret these complexities

semiotically as well, if you had more insights into the classes of

ternary relations, if any, which may NOT be sign relations.

>

> To be continued …

>

> Regards,

>

> Jon

>

> To be continued …

>

> Regards,

>

> Jon

>

Jul 17, 2022, 8:24:17 AMJul 17

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Discussion 13

https://inquiryintoinquiry.com/2022/07/17/sign-relations-discussion-13/

(1) https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/8mh1CC18EQAJ

(2) https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/a1IhmXFEAQAJ

(3) https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/WXz1R3JEAQAJ

Dear Cliff,

Backing up a little —

Whether a thing qualifies as a sign is not an ontological question,

a matter of what it is in itself, but a pragmatic question, a matter

of what role it plays in a particular application.

By extension, whether a triadic relation qualifies as a sign relation

is not just a question of its abstract structure but a question of its

potential applications, of its fitness for a particular purpose, namely,

whether we can imagine it capturing aspects of objective structure immanent

“what we can imagine finding a use for”, we probably can't, or shouldn't try,

to reduce pragmatic definitions to ontological definitions. That's why I feel

bound to leave the boundaries a bit fuzzy.

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

It's not a bad idea to cast an oversized net at the outset, and the à priori

method can take us a way with that, but developing semiotics beyond its first

principles and early stages will depend on gathering more significant examples

of sign relations and sign transformations approaching the level we actually

employ in the practice of communication, computation, inquiry, learning, proof,

and reasoning in general. I think that's probably the best way to see the real

sense and utility of Peirce's double definition of logic and signs.

Regards,

Jon

https://inquiryintoinquiry.com/2022/07/17/sign-relations-discussion-13/

(1) https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/8mh1CC18EQAJ

(2) https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/a1IhmXFEAQAJ

(3) https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/WXz1R3JEAQAJ

Dear Cliff,

Backing up a little —

Whether a thing qualifies as a sign is not an ontological question,

a matter of what it is in itself, but a pragmatic question, a matter

of what role it plays in a particular application.

By extension, whether a triadic relation qualifies as a sign relation

is not just a question of its abstract structure but a question of its

potential applications, of its fitness for a particular purpose, namely,

whether we can imagine it capturing aspects of objective structure immanent

in the conduct of logical reasoning.

Because it's difficult, and not even desirable, to place prior limits on
“what we can imagine finding a use for”, we probably can't, or shouldn't try,

to reduce pragmatic definitions to ontological definitions. That's why I feel

bound to leave the boundaries a bit fuzzy.

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

It's not a bad idea to cast an oversized net at the outset, and the à priori

method can take us a way with that, but developing semiotics beyond its first

principles and early stages will depend on gathering more significant examples

of sign relations and sign transformations approaching the level we actually

employ in the practice of communication, computation, inquiry, learning, proof,

and reasoning in general. I think that's probably the best way to see the real

sense and utility of Peirce's double definition of logic and signs.

Regards,

Jon

Jul 17, 2022, 1:50:09 PMJul 17

to cyb...@googlegroups.com

Jon: Thanks for the continuing discussion. I very much appreciate your

perspective below, and don't necessarily disagree with it.

But that said, I think you're not engaging me the way I'm seeking. Not

that you have to, of course, your choice, but I think it's valuable to

verify that I'm making myself clear.

My perspective is that of a formal modeler. There are many ways of doing

modeling, of course, but I was following your cue that we wanted to

build models of semiotic systems, models of sign relations, which are

formal. Commonly this means mathematical, which is how you initiated

this, but they could also be just formal natural language, effectively

philosophical, aiming towards precision, logical forms of consistency

and coherence, and wherever possible, denotations and explicit external

references. Towards that end, mathematical notation can be an effective

tool, and highly efficient, but isn't strictly necessary.

Anyway, on that basis, you said that a sign relation can be modeled as a

ternary relation which you expressed as L \subseteq O \times S \times I.

This notation has an explicit meaning in mathematics in terms of the

possible relationships between three sets O, S, and I which are

identified as at least somewhat different from each other. What I then

asked you is if this is a sufficient definition, whether ANY possible

ternary relation between ANY three sets could be used to model a sign

relation, or whether there were some ternary relations between some sets

which could NOT be used to model sign relations. So far, you haven't

answered, although you suggested that some mathematical groups over L

could be brought to bear.

Now if you don't know, or don't care, or if Perice doesn't say, or we

can't tell, then all that's fine, of course. I just want to make sure

that you truly understand my question.

Thanks for engaging.

perspective below, and don't necessarily disagree with it.

But that said, I think you're not engaging me the way I'm seeking. Not

that you have to, of course, your choice, but I think it's valuable to

verify that I'm making myself clear.

My perspective is that of a formal modeler. There are many ways of doing

modeling, of course, but I was following your cue that we wanted to

build models of semiotic systems, models of sign relations, which are

formal. Commonly this means mathematical, which is how you initiated

this, but they could also be just formal natural language, effectively

philosophical, aiming towards precision, logical forms of consistency

and coherence, and wherever possible, denotations and explicit external

references. Towards that end, mathematical notation can be an effective

tool, and highly efficient, but isn't strictly necessary.

Anyway, on that basis, you said that a sign relation can be modeled as a

ternary relation which you expressed as L \subseteq O \times S \times I.

This notation has an explicit meaning in mathematics in terms of the

possible relationships between three sets O, S, and I which are

identified as at least somewhat different from each other. What I then

asked you is if this is a sufficient definition, whether ANY possible

ternary relation between ANY three sets could be used to model a sign

relation, or whether there were some ternary relations between some sets

which could NOT be used to model sign relations. So far, you haven't

answered, although you suggested that some mathematical groups over L

could be brought to bear.

Now if you don't know, or don't care, or if Perice doesn't say, or we

can't tell, then all that's fine, of course. I just want to make sure

that you truly understand my question.

Thanks for engaging.

Jul 18, 2022, 11:20:27 AMJul 18

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Sign Relations • Discussion 14

https://inquiryintoinquiry.com/2022/07/17/sign-relations-discussion-14/

https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/iNl_yoqEAQAJ

Dear Cliff,

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

First, to clear up one point of notation, in writing L ⊆ O × S × I,

there is no assumption on my part the relational domains O, S, I are

necessarily disjoint. They may intersect or even be identical, as

O = S = I. Of course we rarely need to contemplate limiting cases of

that type but I find it useful to keep then in our categorical catalogue.

(Other writers will differ on that score.) On the other hand, we very

often consider cases where S = I, as in the following two examples of

sign relations discussed in a previous post of this series.

Sign Relations • Examples

https://inquiryintoinquiry.com/2022/07/02/sign-relations-examples-2/

Tables 1a and 1b. Sign Relation Tables L_A and L_B

https://inquiryintoinquiry.files.wordpress.com/2020/05/sign-relation-twin-tables-la-lb.png

We have the following data.

O = {A, B}

S = {“A”, “B”, “i”, “u”}

I = {“A”, “B”, “i”, “u”}

As I mentioned, those examples were deliberately constructed to be as simple

as possible but they do exemplify many typical features of sign relations in

general. Until the time my advisor asked me for cases of that order I had

always contemplated formal languages with countable numbers of signs and

never really thought about finite sign relations at all.

Regards,

Jon

https://inquiryintoinquiry.com/2022/07/17/sign-relations-discussion-14/

https://groups.google.com/g/cybcom/c/TpRK4fxguD0/m/iNl_yoqEAQAJ

Dear Cliff,

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

First, to clear up one point of notation, in writing L ⊆ O × S × I,

there is no assumption on my part the relational domains O, S, I are

necessarily disjoint. They may intersect or even be identical, as

O = S = I. Of course we rarely need to contemplate limiting cases of

that type but I find it useful to keep then in our categorical catalogue.

(Other writers will differ on that score.) On the other hand, we very

often consider cases where S = I, as in the following two examples of

sign relations discussed in a previous post of this series.

Sign Relations • Examples

https://inquiryintoinquiry.com/2022/07/02/sign-relations-examples-2/

Tables 1a and 1b. Sign Relation Tables L_A and L_B

https://inquiryintoinquiry.files.wordpress.com/2020/05/sign-relation-twin-tables-la-lb.png

We have the following data.

O = {A, B}

S = {“A”, “B”, “i”, “u”}

I = {“A”, “B”, “i”, “u”}

As I mentioned, those examples were deliberately constructed to be as simple

as possible but they do exemplify many typical features of sign relations in

general. Until the time my advisor asked me for cases of that order I had

always contemplated formal languages with countable numbers of signs and

never really thought about finite sign relations at all.

Regards,

Jon

Jul 19, 2022, 11:43:03 PMJul 19

to cyb...@googlegroups.com

Thank you, my understanding is that you hold that any ternary relation

can be taken as a sign relation, that's fine.

can be taken as a sign relation, that's fine.

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