Sign Relations

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Jon Awbrey

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May 28, 2020, 7:00:14 AM5/28/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Anthesis
At: http://inquiryintoinquiry.com/2020/05/28/sign-relations-%e2%80%a2-anthesis/

<QUOTE>

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>

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

inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache

Jon Awbrey

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May 28, 2020, 4:10:26 PM5/28/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Klaus, All ...

I'm working at reviewing and revising some pieces I've rewritten
two score times over the last ... lost count of years ... and that
bit from Peirce is one of my favorite epigraphs for the work ahead.
But I take it as an allegorical figure whose purpose is to illustrate
a certain form of relation, and not to be taken too literally. So I'd
agree with your reaction that taking it literally clangs a bit. I think
there are clues in the passage, the hypothetical subjunctive construction,
the unnatural qualification, “without further condition”, etc., telling us
Peirce did not intend it a truth of botany. But taken rightly it does point
to the shape of a proper definition to come. So I'll be getting to that ...

Regards,

Jon

On 5/28/2020 12:30 PM, Krippendorff, Klaus wrote:
> This interesting quote reveals three things to me:
>
> 1. Peirce observed And described what cyberneticians call an adaptive system. The sunflower trying to maximize
exposure to the moving sun by turning.
>
> 2. This behavior may well have an evolutionary advantage in the sense that other flowers that cannot take advantage
of this ability might be less capable of reproduction. While all flowers end up bearing seeds, not all flowers that have
survived today have that capability.
>
> 3. Projecting the triadic sign conception on this process is ludicrous. It demonstrates Peirce’s single-minded
projection of a simplistic conception on everything in the world.
>
> To me signs are the result of interpretations of perceptions. Even if the connection between what one sees and what
it means Is explainable as a circular causality, it always Is conceptualized as such. It seems to me that Peirce
confuses his conception with what IS the case. His triadic explanations do no not cover the dynamics of the sunflower’s
behavior. It favors static descriptions which cybernetics is fundamentally oppose to, moreover including the
cybernetician as enactor of his or her conceptual system.
>
> Klaus
>
> Sent from my iPhone
>
>> On May 28, 2020, at 7:00 AM, Jon Awbrey <jaw...@att.net> wrote:
>>
>> Cf: Sign Relations • Anthesis

Jon Awbrey

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Jun 1, 2020, 11:40:15 AM6/1/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Definition
At: http://inquiryintoinquiry.com/2020/06/01/sign-relations-%e2%80%a2-definition/

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>

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 as 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/ )

• Logic of Relatives ( https://oeis.org/wiki/Logic_of_relatives )

Jon Awbrey

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Jun 10, 2020, 1:20:16 PM6/10/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 2
At: http://inquiryintoinquiry.com/2020/06/10/sign-relations-%e2%80%a2-discussion-2/

Re: Cybernetics ( https://groups.google.com/d/topic/cybcom/Xb7_CYkMLwA/overview )
::: Bernard C.E. Scott ( https://groups.google.com/d/msg/cybcom/Xb7_CYkMLwA/3aTHnBIzAwAJ )
Re: Sign Relations • Definition
At: https://inquiryintoinquiry.com/2020/06/01/sign-relations-%e2%80%a2-definition/

Regarding Peirce's definition of a sign given in the previous post, Bernard Scott writes:

<QUOTE>

It is very helpful [to] distinguish Peirce's formal semiotic (his logic)
from psychological, and by extension, ‘biosemiotic’ understandings of ‘sign’.

</QUOTE>

Dear Bernard,

You raise a very important point. It is critical to distinguish
the abstract theory from its concrete applications. The power of
a great theory lies in the diversity of applications it bears.
But that very power comes with a warning, as the diversity it
generates can be the source of dispute and dissension among
its appliers and interpreters.

We all know the parable of the seven sightless sages and
the polymorphous pachyderm they ponder, so I don't need to
spend a lot of words on the moral of that story here. But
it may be useful to say more about the major misunderstandings
occasioned by, the schisms, sects, and splinter groups spawned
by Peirce's extremely general and powerful theory of triadic
sign relations. I'll attend to that when I next get a chance.

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

Jon Awbrey

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Jun 11, 2020, 1:54:31 PM6/11/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Signs and Inquiry
At: http://inquiryintoinquiry.com/2020/06/11/sign-relations-%e2%80%a2-signs-and-inquiry/

All,

Here's a paragraph on an issue I've explored in more depth
both before and after writing this bit of ice-breaker to it,
but since it's a topic I get back to every other summer or so
I'll leave this much as a segue and a reminder of the season.

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 that 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). Thus, there is an active and intricate form of
cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper
character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of
sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view.

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.
ttps://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

Jon Awbrey

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Jun 13, 2020, 2:16:03 PM6/13/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 3
At: http://inquiryintoinquiry.com/2020/06/13/sign-relations-%e2%80%a2-discussion-3/

Re: Sign Relations • Definition
https://inquiryintoinquiry.com/2020/06/01/sign-relations-%e2%80%a2-definition/
Re: Ontolog Forum
https://groups.google.com/d/topic/ontolog-forum/cpgB6B6UjRs/overview
::: Alex Shkotin
https://groups.google.com/d/msg/ontolog-forum/cpgB6B6UjRs/MY7-3dYGBQAJ

Regarding Peirce's definition of a sign linked above, Alex Shkotin writes:

<QUOTE>

A Sign is unusually active in Peirce's definition:

A (a sign) brings B (interpretant sign) into correspondence with C (object of sign).

Moreover, A determines B or even creates B.

It would be nice to get an example of such an active sign, its interpretant sign, and an object. My point is to make
the Peirce definition as clear as to be formalized.

</QUOTE>

Dear Alex,

Thanks for your comment. It points to a problem lurking in the wings through all these discussions, so let's nudge it
on stage and throw a better light on it.

I remember my first formal logic prof in college being rather adamant about the difference between a logical formula,
which supposedly bore its “logical form” on its sleeve — I recall the very figure he used — and any of its diverse and
sundry natural language paraphrases. As time wore on I would reconfigure many of the lessons impressed on me in those
days, but that one has stuck, I'm guessing because it goes without saying in mathematical and scientific practice.

This treble clef, to vary the figure — forms as objects, formulas as signs, and paraphrases as interpretant signs — is
the key to a fundamental theme.

A very wide field of discussion opens up at this point. To begin we have the logical jump from forms to formulas and
the semiotic drift from formulas to paraphrases. Further on we'll encounter a range of tensions between formal and
informal contexts of inquiry.

Susan Awbrey and I discussed a related set of issues in our “Conceptual Barriers” paper. Here is how we set up our
treatment of three problematics.

<QUOTE>

• Problematic 1 is the tension that arises along a dimension of increasing formalization in our mental models of the
world, between what we may call the ‘informal context’ of real-world practice and the ‘formal context’ of specialized study.

• Problematic 2 is the difficulty in communication that is created by differing mental models of the world, in other
words, by the tendency among groups of specialists to form internally coherent but externally disparate systems of
mental images.

• Problematic 3 is a special type of communication difficulty that commonly arises between the ‘Two Cultures’ of the
scientific and the humanistic disciplines. A significant part of the problem derives from the differential emphasis
that each group places on its use of symbolic and conceptual systems, limiting itself to either the denotative or the
connotative planes of variation, but seldom integrating the two.

</QUOTE>

Please excuse the sweeping preamble. It wasn't meant to sweep your observations under the rug — it's just so many
discussions here and there on the web in recent days are reminding me of the larger designs beyond my more mundane focus
on brass tacks matters. I'll bring this all back to bear on the everyday life of signs the next chance I get.

Regards,

Jon

References
==========

• Awbrey, S.M., and Awbrey, J.L. (2001),
“Conceptual Barriers to Creating Integrative Universities”,
Organization : The Interdisciplinary Journal of Organization,
Theory, and Society 8(2), Sage Publications, London, UK, pp. 269–284.
Abstract:
http://org.sagepub.com/cgi/content/abstract/8/2/269
Online:
https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities

• 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://arisbe.sitehost.iu.edu/menu/library/bycsp/L75/l75.htm

Jon Awbrey

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Jun 15, 2020, 2:15:20 PM6/15/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations ??? Discussion 4
At: http://inquiryintoinquiry.com/2020/06/15/sign-relations-%e2%80%a2-discussion-4/

Re: Peirce List
At: https://list.iupui.edu/sympa/arc/peirce-l/2020-06/thrd2.html#00026
Re: Edwina Taborsky
At: https://list.iupui.edu/sympa/arc/peirce-l/2020-06/msg00101.html

All,

A note on a couple of recurring themes may be useful at this point.

1. Peirce's ???metaphorical argument??? for transforming discussion of interpretive agents, whether individuals or
communities, to discussion of interpretant signs is as follows.

<QUOTE>

I think we need to reflect upon the circumstance that every word implies some proposition or, what is the same thing,
every word, concept, symbol has an equivalent term ??? or one which has become identified with it, ??? in short, has an
interpretant.

Consider, what a word or symbol is; it is a sort of representation. Now a representation is something which stands for
something. ??? A thing cannot stand for something without standing to something for that something. Now, what is this
that a word stands to? Is it a person?

We usually say that the word homme stands to a Frenchman for man. It would be a little more precise to say that it
stands to the Frenchman's mind ??? to his memory. It is still more accurate to say that it addresses a particular
remembrance or image in that memory. And what image, what remembrance? Plainly, the one which is the mental equivalent
of the word homme ??? in short, its interpretant. Whatever a word addresses then or stands to, is its interpretant or
identified symbol. ???

The interpretant of a term, then, and that which it stands to are identical. Hence, since it is of the very essence of
a symbol that it should stand to something, every symbol ??? every word and every conception ??? must have an interpretant ???
or what is the same thing, must have information or implication. (Peirce, CE 1, 466???467).

</QUOTE>

There's additional discussion of this passage at the following locations.

??? Inquiry Driven Systems
https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview
??? C'est Moi
https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_1#C.27est_Moi

??? Information = Comprehension ?? Extension
https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension
??? Selection 18
https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension#Selection_18

2. When we employ mathematical models to describe any domain of phenomena, we are always proceeding hypothetically, and
the modality of all mathematics, in its own right, is the possible. That is because mathematical existence is existence
in the moderate sense of ???what's not inconsistent???. In the conventional idiom, ???it's would be's all the way down???, and
the usual scales of modality are flattened down to one mode, to wit, Be ???. It is not until we take the risk of acting
on our abduced model that we encounter genuine brute force Secondness.

References
==========

??? 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://arisbe.sitehost.iu.edu/menu/library/bycsp/L75/l75.htm

??? Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition,
Peirce Edition Project (eds.), Indiana University Press, Bloomington
and Indianapolis, IN, 1981???. Cited as (CE volume, page).

Regards,

Jon

Jon Awbrey

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Jun 15, 2020, 5:25:19 PM6/15/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
[Something along the line is screwing up unicode again. I'll try this way.]
[As always, there's a fully formatted version on my blog at the link below.]
A note on a couple of recurring themes may be useful at this point.

1. Peirce's "metaphorical argument" for transforming discussion
of interpretive agents, whether individuals or communities,
to discussion of interpretant signs is as follows.

<QUOTE>

I think we need to reflect upon the circumstance that every word implies some proposition or, what is the same thing,
every word, concept, symbol has an equivalent term -- or one which has become identified with it, -- in short, has an
"interpretant".

Consider, what a word or symbol is; it is a sort of representation. Now a representation is something which stands for
something. ... A thing cannot stand for something without standing "to" something "for" that something. Now, what is
this that a word stands "to"? Is it a person?

We usually say that the word "homme" stands to a Frenchman for "man". It would be a little more precise to say that it
stands to the Frenchman's mind -- to his memory. It is still more accurate to say that it addresses a particular
remembrance or image in that memory. And what "image", what remembrance? Plainly, the one which is the mental
equivalent of the word "homme" -- in short, its interpretant. Whatever a word addresses then or "stands to", is its
interpretant or identified symbol. ...

The interpretant of a term, then, and that which it stands to are identical. Hence, since it is of the very essence of
a symbol that it should stand "to" something, every symbol -- every "word" and every "conception" -- must have an
interpretant -- or what is the same thing, must have information or implication. (Peirce, CE 1, 466–467).

</QUOTE>

There's additional discussion of this passage at the following locations.

Inquiry Driven Systems : C'est Moi
https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview
https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_1#C.27est_Moi

Information = Comprehension × Extension : Selection 18
https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension
https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension#Selection_18

2. When we employ mathematical models to describe any domain of phenomena,
we are always proceeding hypothetically and tentatively, and the modality
of all mathematics, in its own right, is "the possible". That is because
mathematical existence is existence in the moderate sense of "whatever's
not inconsistent". In the idiom, "It's would-be's all the way down."
In effect the usual scales of modality are flattened down to one mode,
to wit, Be flat. It is not until we take the risk of acting on our
abduced model that we encounter genuine brute force Secondness.

References
==========

* 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.

* Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition,
Peirce Edition Project (eds.), Indiana University Press, Bloomington
and Indianapolis, IN, 1981-. Cited as (CE volume, page).

Jon Awbrey

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Jun 16, 2020, 8:12:37 AM6/16/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 5
At: http://inquiryintoinquiry.com/2020/06/16/sign-relations-%e2%80%a2-discussion-5/

Re: Sign Relations • Discussion 4
At: https://inquiryintoinquiry.com/2020/06/15/sign-relations-%e2%80%a2-discussion-4/

The transformative idea here is not the convertibility of term logic, propositional logic, and monadic predicate logic
-- which has been a commonplace of logic since Aristotle, if not in those words, and only got forgot during the false
subtleties of the Frege-Russell tradition, though even Quine was woke enough in time to write a nice essay on it -- but
rather the transformation from interpreter models to interpretant models. The latter are what Peirce and all in his
train need for constructing abstract formal theories neutral on psychologism, materialism, biologism, and various other
all too stolid -isms.

There's more discussion of Peirce's passage to the interpretant at the following locations.

* Inquiry Driven Systems
( https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview )
* C'est Moi
( https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_1#C.27est_Moi )

* Information = Comprehension × Extension
( https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension )
* Selection 18
( https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension#Selection_18 )

Regards,

Jon

Jon Awbrey

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Jun 16, 2020, 2:56:29 PM6/16/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Examples
At: http://inquiryintoinquiry.com/2020/06/16/sign-relations-%e2%80%a2-examples/

Because the examples to follow have been artificially constructed to be as simple as possible, their detailed
elaboration can run the risk of trivializing the whole theory of sign relations. Despite their simplicity, however,
these examples have subtleties of their own, and their careful treatment will serve to illustrate many 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" (SOI) 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 x S x 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 as a subset of O x S x 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 as a subset of S. In that case it becomes
convenient to lump signs and interpretants together into 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 uncluttered 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 as subsets of O x S x I.

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

These 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 names an object and
two co-referent signs, making up an ordered triple of the form (o, s, i) called an "elementary relation", that is, one
element of the relation's set-theoretic extension.

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

Regards,

Jon
Sign Relation Twin Tables LA & LB.png

Jon Awbrey

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Jun 21, 2020, 9:33:02 AM6/21/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Dyadic Aspects
At: http://inquiryintoinquiry.com/2020/06/21/sign-relations-%e2%80%a2-dyadic-aspects/

We take up the dyadic aspects of triadic relations ...

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

Table 2. Dyadic Aspects of Triadic Relations
https://inquiryintoinquiry.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 interpretant i in the interpretant domain 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.

References
==========

• 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://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
https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052
https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry

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
Dyadic Projections of Triadic Relations.png

Jon Awbrey

unread,
Jun 22, 2020, 11:20:18 AM6/22/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 6
At: http://inquiryintoinquiry.com/2020/06/22/sign-relations-%e2%80%a2-discussion-6/
Re: Alex Shkotin
https://groups.google.com/d/msg/ontolog-forum/cpgB6B6UjRs/kpi3Zf7uBQAJ

Dear Alex,

We all love natural languages, our native tongues, but each one has a mind of its own and a habit of saying both more
and less and something other than the meanings we intend at the moment of utterance. So maybe it's a love-hate
relationship, or at least a Liebeskampf.

Whether we are endowed with an inborn faculty for language, even a genetic blueprint for selected species of languages
on a par with our naturally evolved motor and sense organs, or whether we acquire our initial languages from scratch,
every natural language worth its salt preserves a rich heritage of biological and cultural meanings its users will
assimilate, consciously or otherwise. I would not say “resistance is futile” but habits of thought built into our first
and second natures demand persistent habits of critical reflection to break.

We do use natural language paraphrases to “express the meaning of [a logical formula] using different words, especially
to achieve greater clarity” and up to a point they serve that end. But there's a catch. If a natural language
paraphrase could express the precise meaning of a logical formula with greater clarity, what would be the use of the
formula?

Well, that's the beginning of a post I started on the spectrum of formality from form to formal object to formula to
paraphrase. But I decided to let it simmer for another day. Now that we have a workbench stocked with concrete
examples of triadic relations and sign relations we might as well use them to illustrate the abstractions while keeping
our feet on more solid ground.

I'll turn to that task next.

Regards,

Jon

Jon Awbrey

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Jun 23, 2020, 10:15:45 AM6/23/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Denotation
http://inquiryintoinquiry.com/2020/06/23/sign-relations-%e2%80%a2-denotation/

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, alternatively 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 × S.

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

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

Regards,

Jon

Sign Relation Twin Tables Den LA & Den LB.png

Jon Awbrey

unread,
Jun 24, 2020, 11:15:15 AM6/24/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Connotation
http://inquiryintoinquiry.com/2020/06/24/sign-relations-%e2%80%a2-connotation/

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

References
==========

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

Jon Awbrey

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Jun 26, 2020, 5:28:35 PM6/26/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations ??? Discussion 7
http://inquiryintoinquiry.com/2020/06/26/sign-relations-%e2%80%a2-discussion-7/

Re: Sign Relations ??? Definition
https://inquiryintoinquiry.com/2020/06/01/sign-relations-%e2%80%a2-definition/
https://groups.google.com/d/msg/ontolog-forum/cpgB6B6UjRs/1pPLJheLAQAJ

Dear Alex,

Please forgive the long and winding dissertation. I've been through many discussions of Peirce's definition of ???logic
as formal semiotic??? but I keep discovering new ways of reading what I once thought a fairly straightforward proposition.
That's all useful information but it makes me anxious to avoid all the missteps of exposition I may have made in the
past. At any rate, I think I've set enough background and context ??? it will take more, but later ??? to begin addressing
your comments now.

For ease of reference here is Peirce's twofold definition again.

<QUOTE Peirce>

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>

Turning to your first comment ???

https://inquiryintoinquiry.com/2020/06/13/sign-relations-%e2%80%a2-discussion-3/

<QUOTE Alex Shkotin>

A Sign is unusually active in Peirce's definition:

A (a sign) brings B (interpretant sign) into correspondence with C (object of sign).

Moreover, A determines B or even creates B.

It would be nice to get an example of such an active sign, its interpretant sign, and an object. My point is to make
the Peirce definition as clear as to be formalized.

</QUOTE>

Several issues stand out. There are questions about paraphrases,
the active character of signs, and the nature of what is being defined.

??? The problem of paraphrases arises at this point because it affects
how literally we ought to take the words in a natural language proxy
for a logical or mathematical formula.

For example, a conventional idiom in describing a mathematical function f : X ??? Y is to say f ???maps??? or ???sends??? an
element of X to an element of Y. A concrete verb may quicken the intuition but the downside is its power to evoke
excess meanings beyond the abstract intention. It is only as we become more familiar with the formal subject matter of
sign relations that we can decide what kind of ???bringing??? and ???creating??? and ???determining??? is really going on in all
that sign, object, interpretant relating, whether at the abstract level or in a given application.

??? There is the question of a sign's active character.
Where's the dynamic function in all this static structure?
Klaus Krippendorff raised the same question in regard to the
Parable of the Sunflower back at the beginning of this discussion.

https://inquiryintoinquiry.com/2020/05/28/sign-relations-%e2%80%a2-discussion-1/

<QUOTE Klaus Krippendorff>

[Peirce's] triadic explanations do not cover the dynamics of the sunflower???s behavior. It favors static descriptions
which cybernetics is fundamentally opposed to, moreover including the cybernetician as enactor of his or her conceptual
system.

</QUOTE>

I have not forgotten this question. Indeed, it's the
question at the heart of my work on Inquiry Driven Systems

https://inquiryintoinquiry.com/2020/06/26/survey-of-inquiry-driven-systems-%e2%80%a2-2/

which led me back to grad school in Systems Engineering
???to develop mutual applications of systems theory and
artificial intelligence to each other???.

https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems

Anything approaching an adequate answer to that question is going to be one of those things requiring more background
and context, all in good time, but there are a few hints we can take from Peirce's text about the way forward.

<QUOTE Peirce>

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.

</QUOTE>

My reading of that tells me about a division of
labor across three levels of abstraction. There is
a level of psychological experience and social activity,
a level of dynamic process and temporal pattern, and
a level of mathematical form.

To be continued ???

Regards,

Jon

Jon Awbrey

unread,
Jun 27, 2020, 12:40:16 PM6/27/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Alex, All ...

Those are just ordinary unicodes for things like
bullets, ellipses, mdashes, quotes, right arrows,
math symbols, etc.

Every now and then some processor along the way fails to do its duty
and I get those triple question marks. It doesn't seem to be anything
I've done and or can do and it usually goes away after a couple of days.

I've been to the Mozilla help forums and no one seems to know why.
It appears to be an ISP issue, not anything at my source since my
automatic bcc: to myself is okay, nor anything at the listserves.

It started late last year after some changes at Yahoo,
the middleman AT&T imposed on me a few years back.

I'll append a fresh copy below to see if
it's worked its way out of the system.

At any rate, I always include a link my blog copy,
which is much better formatted in LaTeX with Figs
and Tables and all.

Cf: Sign Relations • Discussion 7
http://inquiryintoinquiry.com/2020/06/26/sign-relations-%e2%80%a2-discussion-7

Regards,

Jon

Test
=============
bullet •
ellipses …
mdash —
quotes “ ”
rightarrow →

Cf: Sign Relations • Discussion 7
http://inquiryintoinquiry.com/2020/06/26/sign-relations-%e2%80%a2-discussion-7/

Re: Sign Relations • Definition
Please forgive the long and winding dissertation. I've been through many discussions of Peirce's definition of “logic
as formal semiotic” but I keep discovering new ways of reading what I once thought a fairly straightforward proposition.
That's all useful information but it makes me anxious to avoid all the missteps of exposition I may have made in the
past. At any rate, I think I've set enough background and context — it will take more, but later — to begin addressing
your comments now.

For ease of reference here is Peirce's twofold definition again.

<QUOTE Peirce>

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>

Turning to your first comment —

https://inquiryintoinquiry.com/2020/06/13/sign-relations-%e2%80%a2-discussion-3/

<QUOTE Alex Shkotin>

A Sign is unusually active in Peirce's definition:

A (a sign) brings B (interpretant sign) into correspondence with C (object of sign).

Moreover, A determines B or even creates B.

It would be nice to get an example of such an active sign, its interpretant sign, and an object. My point is to make
the Peirce definition as clear as to be formalized.

</QUOTE>

Several issues stand out. There are questions about paraphrases,
the active character of signs, and the nature of what is being defined.

• The problem of paraphrases arises at this point because it affects
how literally we ought to take the words in a natural language proxy
for a logical or mathematical formula.

For example, a conventional idiom in describing a mathematical function f : X → Y is to say f “maps” or “sends” an
element of X to an element of Y. A concrete verb may quicken the intuition but the downside is its power to evoke
excess meanings beyond the abstract intention. It is only as we become more familiar with the formal subject matter of
sign relations that we can decide what kind of “bringing” and “creating” and “determining” is really going on in all
that sign, object, interpretant relating, whether at the abstract level or in a given application.

• There is the question of a sign's active character.
Where's the dynamic function in all this static structure?
Klaus Krippendorff raised the same question in regard to the
Parable of the Sunflower back at the beginning of this discussion.

https://inquiryintoinquiry.com/2020/05/28/sign-relations-%e2%80%a2-discussion-1/

<QUOTE Klaus Krippendorff>

[Peirce's] triadic explanations do not cover the dynamics of the sunflower’s behavior. It favors static descriptions
which cybernetics is fundamentally opposed to, moreover including the cybernetician as enactor of his or her conceptual
system.

</QUOTE>

I have not forgotten this question. Indeed, it's the
question at the heart of my work on Inquiry Driven Systems

https://inquiryintoinquiry.com/2020/06/26/survey-of-inquiry-driven-systems-%e2%80%a2-2/

which led me back to grad school in Systems Engineering
“to develop mutual applications of systems theory and
artificial intelligence to each other”.

https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems

Anything approaching an adequate answer to that question is going to be one of those things requiring more background
and context, all in good time, but there are a few hints we can take from Peirce's text about the way forward.

<QUOTE Peirce>

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.

</QUOTE>

My reading of that tells me about a division of
labor across three levels of abstraction. There is
a level of psychological experience and social activity,
a level of dynamic process and temporal pattern, and
a level of mathematical form.

To be continued …

Regards,

Jon

On 6/27/2020 3:56 AM, Alex Shkotin wrote:> Dear Jon,
>
> Just a clarification question: what does it mean "???" in your message
> above? Is that something you put in the message by your own hand? What does
> the sign "???" mean in this case?
> I study a phenomenon of definition and a way to formalize them. Peirce
> definition of sign is a challenging example.
> The point number one is that any scientific, technological and law term
> definition may be written in Simple English with variables introduced by
> Stagirite.
>
> Regards,
>
> Alex
>

Jon Awbrey

unread,
Jun 29, 2020, 1:45:54 PM6/29/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Ennotation
http://inquiryintoinquiry.com/2020/06/29/sign-relations-%e2%80%a2-ennotation/

A third aspect of a sign's complete meaning concerns the reference its objects have to 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, alternatively written in
any of forms, proj_OI L, L_OI, proj_13 L, and L_13, is defined as follows.

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

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

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

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

Regards,

Jon

Sign Relation Twin Tables Enn LA & Enn LB.png

Jon Awbrey

unread,
Jul 1, 2020, 3:00:42 AM7/1/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Semiotic Equivalence Relations 1
http://inquiryintoinquiry.com/2020/07/01/sign-relations-%e2%80%a2-semiotic-equivalence-relations-1/

A “semiotic equivalence relation” (SER) is a special type of equivalence relation arising in the analysis of sign
relations. Generally speaking, any equivalence relation is associated with a family of equivalence classes which
partition the underlying set of elements, known as the “domain” or “space” of the relation. 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 these 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 these 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 makes for 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”}. But it needs to be observed that the semiotic partitions for
interpreters A and B are not identical, indeed, they are orthogonal to each other. This allows us to regard the “form”
of these 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 these Tables serves to explain what is meant by saying the SEPs for A and B
are orthogonal to each other.

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

Regards,

Jon

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

Jon Awbrey

unread,
Jul 2, 2020, 11:50:21 AM7/2/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 8
http://inquiryintoinquiry.com/2020/07/02/sign-relations-%e2%80%a2-discussion-8/

Re: Sign Relations • Ennotation
https://inquiryintoinquiry.com/2020/06/29/sign-relations-%e2%80%a2-ennotation/

Re: Peirce List ( https://list.iupui.edu/sympa/arc/peirce-l/2020-07/thrd1.html#00000 )
::: Helmut Raulien ( https://list.iupui.edu/sympa/arc/peirce-l/2020-07/msg00002.html )

Dear Helmut,

The important thing now is to extend our perspective beyond one sign at a time and one object, sign, interpretant at a
time to comprehending a sign relation as a specified set of object, sign, interpretant triples embedded in the set of
all possible triples in a specified context.

In my mind's eye, no doubt influenced by my early interest in Gestalt Psychology, I always picture a sign relation as a
gestalt composed of figure and ground. The triples in the sign relation form a figure set in relief against the
background of all possible triples and the triples left over form the ground of the gestalt.

From a mathematical point of view, the set of possible triples is a “cartesian product” of the following form.

• O × S × I = {(o, s, i) : o ∈ O ∧ s ∈ S ∧ i ∈ I}.

Here, O is “object domain”, the set of objects under discussion, S is the “sign domain”, the specified set of signs, and
I is the “interpretant domain”, the specified set of interpretants.

On this canvass, in this frame, any number of sign relations might be set as figures and each of them would be delimited
as a salient subset of the cartesian product in view. Letting L be any such sign relation, mathematical convention
provides the following description of its relation to the set of possible triples.

• L ⊆ O × S × I.

It's important to note at this point that the specified cartesian product and the specified subset of it are equally
critical parts of the sign relation's definition.

Well, it took a lot longer to set the scene than I thought it would when I got up this morning, so I'll break here and
get back to your specific comments when I next get a chance.

Regards,

Jon

Jon Awbrey

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Jul 3, 2020, 10:20:16 AM7/3/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Semiotic Equivalence Relations 2
http://inquiryintoinquiry.com/2020/07/03/sign-relations-%e2%80%a2-semiotic-equivalence-relations-2/

A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in
particular. (NB. The notations will be much more readable on the blog page linked above.)

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

Thus we have the following definitions.

• [x]_E = {y ∈ X : (x, y) ∈ E}

• x =_E y ⇔ (x, y) ∈ E

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. That is, let [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

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. These modifications are 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.

• [“A”]_{L_A} = [“i”]_{L_A}

• [“B”]_{L_A} = [“u”]_{L_A}

or

• “A” =_{L_A} “i”

• “B” =_{L_A} “u”

Thus it induces the semiotic partition:

• {{“A”, “i”}, {“B”, “u”}}

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

• [“A”]_{L_B} = [“u”]_{L_B}

• [“B”]_{L_B} = [“i”]_{L_B}

or

• “A” =_{L_B} “u”

• “B” =_{L_B} “i”

Thus it induces the semiotic partition:

• {{“A”, “u”}, {“B”, “i”}}

Jon Awbrey

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Jul 5, 2020, 12:00:15 PM7/5/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 9
http://inquiryintoinquiry.com/2020/07/05/sign-relations-%e2%80%a2-discussion-9/
Thanks for your comments. They prompt me to say a little more about the mathematical character of the sign relational
models I'm using.

Peirce without mathematics is like science without mathematics. In every direction of research he pioneered or
prospected, information, inquiry, logic, semiotics, we trace his advances only so far, barely scratch the surface before
we need to bring in mathematical models adequate to the complexity of the phenomena under investigation.

In recent years there has been a tendency in certain quarters to ignore the mathematical substrate of Peirce's pragmatic
thought, even a refusal to use the mathematical tools he crafted to the task of sharpening our understanding. I do not
recall that attitude being prevalent when I began my studies of Peirce's work some fifty years ago. The issue in the
“reception of Peirce” over most of that time has largely been the tendency of people imbued in the traditions of
“analytic philosophy” to dismiss Peirce out of hand. But that school of thought had no problem with using mathematics,
aside from the short-sighted attempts to reduce mathematics to logic and all relations to dyadic ones.

Maybe this late resistance to Peirce's mathematical groundwork has come about through an overly selective viewing of his
entire spectrum of work or maybe it's just a matter of taste. Whatever this case, it's critical for people who are
looking for adequate models of the complex phenomena involved in belief systems, communication, intelligent systems,
knowledge representation, scientific inquiry, and so on to recognize that all the resources we need for working with
relations in general as sets of ordered tuples and sign relations in particular as sets of ordered triples are already
available in Peirce's technical works from 1870 on.

Okay, it looks like I've used up my morning again with more preliminary matters but it seemed important to clear up a
few things about the overall mathematical approach.

Regards,

Jon

Jon Awbrey

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Jul 9, 2020, 5:30:27 PM7/9/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Sign Relations • Discussion 10
http://inquiryintoinquiry.com/2020/07/09/sign-relations-%e2%80%a2-discussion-10/

Re: Cybernetics • Klaus Krippendorff • Bernard Scott
Re: Ontolog • Mihai Nadin • John Sowa • Alex Shkotin
Re: Peirce List • Helmut Raulien • Edwina Taborsky

Dear Colleagues,

While engaged in a number of real and imaginary dialogues
with people I continue to owe full replies, I thought it
might be a good time to stand back and take in the view
from this vantage point. I summed up the desired outlook
a few days ago in the following way.

“The important thing now is to extend our perspective beyond
one sign at a time and one object, sign, interpretant at a time
to comprehending a sign relation as a specified set of object,
sign, interpretant triples embedded in the set of all possible
triples in a specified context.”

If we now comprehend each sign relation L as an extended
collection of triples (o, s, i), where each object o belongs
to a set O of objects, each sign s belongs to a set S of signs,
each interpretant i belongs to a set I of interpretants, and the
whole sign relation L is embedded as a subset in the product space
O × S × I, then our level of description ascends to the point where
we take whole sign relations of this sort as the principal subjects
of classification and structural analysis.

Once we adopt a whole systems perspective on sign relations
we begin to see many commonplace topics in a fresh light.

Agency
======

That Peirce remodels his theory of semiosis from speaking
of interpretive agents to speaking of interpretant signs is
a familiar theme by now. By way of reminder, we discussed
this transformation just recently in Discussion 4 and
Discussion 5 of this series.

But we have to wonder: Why does Peirce make this shift,
this change of basis from interpreters to interpretants?
He does it because the idea of an interpreter stands in
need of clarification and the way he advises to clarify
ideas is to apply the pragmatic maxim. The result is an
operational definition of an interpreter in terms of its
effects on signs in relation to their objects.

It would seem we have replaced an interpreter with a sign relation.
More precisely, we are taking a sign relation as our effective model
for the interpreter in question. Now we must not take this the wrong
way. There is no suggestion of reducing the hypostatic agent to a
particular sign relation. It falls within our capacity merely to
clarify our concept of the agent to a moderate degree, to construct
a model or a representation capturing aspects of the agent's activity
bearing on a particular application.

With that I’ve run out of time for today.
The topic for next time will be Context ...

References
==========

Pragmatic Maxim
https://inquiryintoinquiry.com/2008/08/07/pragmatic-maxim/

Sign Relations • Discussion 4
https://inquiryintoinquiry.com/2020/06/15/sign-relations-%e2%80%a2-discussion-4/

Sign Relations • Discussion 5
https://inquiryintoinquiry.com/2020/06/16/sign-relations-%e2%80%a2-discussion-5/

Sign Relations • Discussion 8
https://inquiryintoinquiry.com/2020/07/02/sign-relations-%e2%80%a2-discussion-8/

Regards,

Jon
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