Systems of Interpretation
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Jon Awbrey
May 5, 2023, 2:00:52 PM5/5/23
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Cf: Systems of Interpretation • 1
https://inquiryintoinquiry.com/2023/05/05/systems-of-interpretation-1-2/
All,
Questions have arisen about the different styles of diagrams
and figures used to represent triadic sign relations in Peircean
semiotics. What do they mean? Which style is best? Among the
most popular pictures some use geometric triangles while others
use the three‑pronged graphs Peirce used in his logical graphs
to represent triadic relations.
Diagrams and figures, like any signs, can serve to communicate
the intended interpretants and thus to coordinate the conduct of
interpreters toward the intended objects — but only in communities of
interpretation where the conventions of interpretation are understood.
Conventions of interpretation are by comparison far more difficult to
communicate.
That brings us to the first question we have to ask about the possibility
of communication in this area, namely, what conventions of interpretation
are needed to make sense of these diagrams, figures, and graphs?
Regards,
Jon
https://inquiryintoinquiry.com/2023/05/05/systems-of-interpretation-1-2/
All,
Questions have arisen about the different styles of diagrams
and figures used to represent triadic sign relations in Peircean
semiotics. What do they mean? Which style is best? Among the
most popular pictures some use geometric triangles while others
use the three‑pronged graphs Peirce used in his logical graphs
to represent triadic relations.
Diagrams and figures, like any signs, can serve to communicate
the intended interpretants and thus to coordinate the conduct of
interpreters toward the intended objects — but only in communities of
interpretation where the conventions of interpretation are understood.
Conventions of interpretation are by comparison far more difficult to
communicate.
That brings us to the first question we have to ask about the possibility
of communication in this area, namely, what conventions of interpretation
are needed to make sense of these diagrams, figures, and graphs?
Regards,
Jon
Jon Awbrey
May 7, 2023, 4:44:48 PM5/7/23
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Cf: Systems of Interpretation • 2
https://inquiryintoinquiry.com/2023/05/07/systems-of-interpretation-2-2/
All,
Let's start as simply as possible. The following Figure
is typical of many I have used to illustrate sign relations
from the time I first began studying Peirce's theory of signs.
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation.png
The above variant comes from a paper Susan Awbrey and I presented at
a conference in 1999, a revised version of which was published in 2001.
As the drafter of that drawing I can speak with authority about the
artist's intentions in drawing it and also about the conventions of
interpretation forming the matrix of its conception and delivery.
Just by way of refreshing my own memory, here is how we set it up —
Figure 2 represents an “elementary sign relation”. It is a single
transaction taking place among 3 entities, the object o, the sign s,
and the interpretant sign i, the association of which is typically
represented by means of the ordered triple (o, s, i).
One of the interpretive conventions implied in that setup is hallowed
by long tradition, going back to the earliest styles of presentation in
mathematics. In it one draws a figure intended as “representative” of
many figures. Regarded as a concrete drawing the figure is naturally
imperfect, individual, peculiar, and special but it's meant to be taken
purely as a representative of its class — generic, ideal, and typical.
That is the main convention of interpretation which goes into giving
diagrams and figures their significant power.
References —
Conceptual Barriers to Creating Integrative Universities
• https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities
Organizations of Learning or Learning Organizations:
The Challenge of Creating Integrative Universities
• https://arisbe.sitehost.iu.edu/menu/library/aboutcsp/awbrey/integrat.htm
Regards,
Jon
https://inquiryintoinquiry.com/2023/05/07/systems-of-interpretation-2-2/
All,
Let's start as simply as possible. The following Figure
is typical of many I have used to illustrate sign relations
from the time I first began studying Peirce's theory of signs.
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation.png
The above variant comes from a paper Susan Awbrey and I presented at
a conference in 1999, a revised version of which was published in 2001.
As the drafter of that drawing I can speak with authority about the
artist's intentions in drawing it and also about the conventions of
interpretation forming the matrix of its conception and delivery.
Just by way of refreshing my own memory, here is how we set it up —
Figure 2 represents an “elementary sign relation”. It is a single
transaction taking place among 3 entities, the object o, the sign s,
and the interpretant sign i, the association of which is typically
represented by means of the ordered triple (o, s, i).
One of the interpretive conventions implied in that setup is hallowed
by long tradition, going back to the earliest styles of presentation in
mathematics. In it one draws a figure intended as “representative” of
many figures. Regarded as a concrete drawing the figure is naturally
imperfect, individual, peculiar, and special but it's meant to be taken
purely as a representative of its class — generic, ideal, and typical.
That is the main convention of interpretation which goes into giving
diagrams and figures their significant power.
References —
Conceptual Barriers to Creating Integrative Universities
• https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities
Organizations of Learning or Learning Organizations:
The Challenge of Creating Integrative Universities
• https://arisbe.sitehost.iu.edu/menu/library/aboutcsp/awbrey/integrat.htm
Regards,
Jon
Jon Awbrey
May 10, 2023, 5:28:22 PM5/10/23
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Cf: Systems of Interpretation • 3
https://inquiryintoinquiry.com/2023/05/10/systems-of-interpretation-3-2/
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation-1.0.png
That “triskelion” stick-figure for an elementary sign relation
or individual triple (o, s, i) is about the simplest possible.
Susan Awbrey and I used a less skeletal figure in an earlier paper,
where our aim was to articulate the commonalities Peirce's concept
of a sign relation shares with its archetype in Aristotle.
Figure 1. The Sign Relation in Aristotle
• https://inquiryintoinquiry.files.wordpress.com/2022/04/awbrey-awbrey-1995-e280a2-figure-1.png
Here is the corresponding passage from “On Interpretation”.
❝Words spoken are symbols or signs (symbola) of affections or
impressions (pathemata) of the soul (psyche); written words
are the signs of words spoken. As writing, so also is speech
not the same for all races of men. But the mental affections
themselves, of which these words are primarily signs (semeia),
are the same for the whole of mankind, as are also the objects
(pragmata) of which those affections are representations or
likenesses, images, copies (homoiomata).❞ (De Interp. i. 16a4).
Reference —
Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”,
Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
Journal ( https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 )
Online (doc) ( https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry )
Online (pdf) ( https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry )
Regards,
Jon
https://inquiryintoinquiry.com/2023/05/10/systems-of-interpretation-3-2/
Figure 2. An Elementary Sign Relation
That “triskelion” stick-figure for an elementary sign relation
or individual triple (o, s, i) is about the simplest possible.
Susan Awbrey and I used a less skeletal figure in an earlier paper,
where our aim was to articulate the commonalities Peirce's concept
of a sign relation shares with its archetype in Aristotle.
Figure 1. The Sign Relation in Aristotle
• https://inquiryintoinquiry.files.wordpress.com/2022/04/awbrey-awbrey-1995-e280a2-figure-1.png
Here is the corresponding passage from “On Interpretation”.
❝Words spoken are symbols or signs (symbola) of affections or
impressions (pathemata) of the soul (psyche); written words
are the signs of words spoken. As writing, so also is speech
not the same for all races of men. But the mental affections
themselves, of which these words are primarily signs (semeia),
are the same for the whole of mankind, as are also the objects
(pragmata) of which those affections are representations or
likenesses, images, copies (homoiomata).❞ (De Interp. i. 16a4).
Reference —
Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”,
Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
Journal ( https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 )
Online (doc) ( https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry )
Online (pdf) ( https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry )
Regards,
Jon
Jon Awbrey
Jun 10, 2023, 9:24:45 AM6/10/23
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Cf: Systems of Interpretation • 4
http://inquiryintoinquiry.com/2023/06/10/systems-of-interpretation-4-2/
Re: Mike Bergman
• https://web.archive.org/web/20160410151041/http://permalink.gmane.org/gmane.science.philosophy.peirce/18534
Re: Valentine Daniel
• https://web.archive.org/web/20160328163000/http://permalink.gmane.org/gmane.science.philosophy.peirce/18540
All,
For its pertinence to the present discussion, here again
is what Peirce wrote about the mathematical way of using
individual or particular cases to make general hypotheses
or suppositions:
Mathematical Demonstration and the Doctrine of Individuals • 1
• https://inquiryintoinquiry.com/2023/05/16/mathematical-demonstration-and-the-doctrine-of-individuals-1-2/
And just so we don't forget Peirce's theory of individuals is not the
run-of-the-mill absolute kind but makes the quality of individuality
relative to the context of discussion — or the frame of reference
as they say in physics — here is what he wrote about that:
Mathematical Demonstration and the Doctrine of Individuals • 2
• https://inquiryintoinquiry.com/2023/05/16/mathematical-demonstration-and-the-doctrine-of-individuals-2-2/
Regards,
Jon (https://mathstodon.xyz/@Inquiry)
http://inquiryintoinquiry.com/2023/06/10/systems-of-interpretation-4-2/
Re: Mike Bergman
• https://web.archive.org/web/20160410151041/http://permalink.gmane.org/gmane.science.philosophy.peirce/18534
Re: Valentine Daniel
• https://web.archive.org/web/20160328163000/http://permalink.gmane.org/gmane.science.philosophy.peirce/18540
All,
For its pertinence to the present discussion, here again
is what Peirce wrote about the mathematical way of using
individual or particular cases to make general hypotheses
or suppositions:
Mathematical Demonstration and the Doctrine of Individuals • 1
• https://inquiryintoinquiry.com/2023/05/16/mathematical-demonstration-and-the-doctrine-of-individuals-1-2/
And just so we don't forget Peirce's theory of individuals is not the
run-of-the-mill absolute kind but makes the quality of individuality
relative to the context of discussion — or the frame of reference
as they say in physics — here is what he wrote about that:
Mathematical Demonstration and the Doctrine of Individuals • 2
• https://inquiryintoinquiry.com/2023/05/16/mathematical-demonstration-and-the-doctrine-of-individuals-2-2/
Regards,
Jon (https://mathstodon.xyz/@Inquiry)
Jon Awbrey
Jun 11, 2023, 2:00:44 PM6/11/23
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Cf: Systems of Interpretation • 5
• https://inquiryintoinquiry.com/2023/06/11/systems-of-interpretation-5-2/
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation-1.0.png
An elementary sign relation is an ordered triple (o, s, i). It is called
“elementary” because it is one element of a sign relation L ⊆ O × S × I,
where O is a set of “objects”, S is a set of “signs”, and I is a set of
“interpretant signs” collectively called the “domains” of the relation.
But what is the significance of that ordering?
In any presentation of subject matter we have to distinguish
the natural order of things from the order of consideration or
presentation in which things are taken up on a given occasion.
The natural order of things comes to light through the discovery
of invariants over a variety of presentations and representations.
That type of order tends to take a considerable effort to reveal.
The order of consideration or presentation is often more arbitrary,
making some aspects of the subject matter more salient than others
depending on the paradigm or perspective one has chosen.
In the case of sign relations, the order in which we take up the
domains O, S, I or the components of a triple (o, s, i) is wholly
arbitrary so long as we maintain the same order throughout the
course of discussion.
Regards,
Jon
• https://inquiryintoinquiry.com/2023/06/11/systems-of-interpretation-5-2/
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation-1.0.png
“elementary” because it is one element of a sign relation L ⊆ O × S × I,
where O is a set of “objects”, S is a set of “signs”, and I is a set of
“interpretant signs” collectively called the “domains” of the relation.
But what is the significance of that ordering?
In any presentation of subject matter we have to distinguish
the natural order of things from the order of consideration or
presentation in which things are taken up on a given occasion.
The natural order of things comes to light through the discovery
of invariants over a variety of presentations and representations.
That type of order tends to take a considerable effort to reveal.
The order of consideration or presentation is often more arbitrary,
making some aspects of the subject matter more salient than others
depending on the paradigm or perspective one has chosen.
In the case of sign relations, the order in which we take up the
domains O, S, I or the components of a triple (o, s, i) is wholly
arbitrary so long as we maintain the same order throughout the
course of discussion.
Regards,
Jon
Jon Awbrey
Jun 14, 2023, 12:04:27 PM6/14/23
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Cf: Systems of Interpretation • 6
• https://inquiryintoinquiry.com/2023/06/14/systems-of-interpretation-6-2/
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation-1.0.png
Re: Peirce List
• https://web.archive.org/web/20160410151020/http://comments.gmane.org/gmane.science.philosophy.peirce/18558
Re: John Collier
• https://web.archive.org/web/20160405154804/http://permalink.gmane.org/gmane.science.philosophy.peirce/18570
<QUOTE JC:>
I strongly agree, Jon. Reading meaning into artefacts of the
representation is not typically transparent. I would say that
the whole symbol represents the sign with its threefold character
and that the node is not some separate signifier. To put it on
this level is, as you suggest, a category error.
</QUOTE>
Precisely. And “artefact” is a very choice word here, with all
the right connotations. It would be unfortunate if this trivial
“triskelion” figure became a caltrop to our thought, blocking the
way of inquiry.
Aside from the ellipses we added to call attention to a couple of
derivative dyadic relations, somewhat loosely called denotative and
connotative in our paper, it is merely typical of the 3-spoke figures
in common use when I was first learning Peirce's theory of signs, often
arising to point out the differences between Saussure's dyadic semiology
and Peirce's triadic semiotics.
But the intervening decades have taught me mostly all the ways diagrams
and figures of that sort can be misinterpreted when the conventions of
interpretation needed to understand them are not up and running. It can
be instructive to carry out post mortems on the various maps of misreading,
though. If one is not up for the morbidity of that then it is probably wiser
to move on to more viable representations.
Regards,
Jon
• https://inquiryintoinquiry.com/2023/06/14/systems-of-interpretation-6-2/
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation-1.0.png
• https://web.archive.org/web/20160410151020/http://comments.gmane.org/gmane.science.philosophy.peirce/18558
Re: John Collier
• https://web.archive.org/web/20160405154804/http://permalink.gmane.org/gmane.science.philosophy.peirce/18570
<QUOTE JC:>
I strongly agree, Jon. Reading meaning into artefacts of the
representation is not typically transparent. I would say that
the whole symbol represents the sign with its threefold character
and that the node is not some separate signifier. To put it on
this level is, as you suggest, a category error.
</QUOTE>
Precisely. And “artefact” is a very choice word here, with all
the right connotations. It would be unfortunate if this trivial
“triskelion” figure became a caltrop to our thought, blocking the
way of inquiry.
Aside from the ellipses we added to call attention to a couple of
derivative dyadic relations, somewhat loosely called denotative and
connotative in our paper, it is merely typical of the 3-spoke figures
in common use when I was first learning Peirce's theory of signs, often
arising to point out the differences between Saussure's dyadic semiology
and Peirce's triadic semiotics.
But the intervening decades have taught me mostly all the ways diagrams
and figures of that sort can be misinterpreted when the conventions of
interpretation needed to understand them are not up and running. It can
be instructive to carry out post mortems on the various maps of misreading,
though. If one is not up for the morbidity of that then it is probably wiser
to move on to more viable representations.
Regards,
Jon
Jon Awbrey
Jun 15, 2023, 11:36:35 AM6/15/23
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Cf: Systems of Interpretation • 7
• https://inquiryintoinquiry.com/2023/06/15/systems-of-interpretation-7-2/
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation-1.0.png
Re: Peirce List
• https://web.archive.org/web/20160410151020/http://comments.gmane.org/gmane.science.philosophy.peirce/18558
Re: Gary Fuhrman
• https://web.archive.org/web/20160405155834/http://permalink.gmane.org/gmane.science.philosophy.peirce/18574
All,
Peirce's existential graphs are a general calculus for expressing the
same subject matter as his logic of relative terms and thus they serve
to represent the structures of many‑place relations. Cast at that level
of generality, there is nothing to prevent existential graphs from being
used to express the special cases of relative terms needed for a theory of
triadic sign relations, for example, terms like “s stands to i for o” or
“__ stands to __ for __” or any number of other forms, depending on the
style one prefers. It may give us pause that we have to use sign relations
in order to mention sign relations but the fact is we do that all the time
whether we are using Peirce's semiotics or not. Peirce's pragmatic analysis
of the process simply provides a clearer account than most other approaches do.
Regards,
Jon
• https://inquiryintoinquiry.com/2023/06/15/systems-of-interpretation-7-2/
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation-1.0.png
Re: Peirce List
• https://web.archive.org/web/20160410151020/http://comments.gmane.org/gmane.science.philosophy.peirce/18558
• https://web.archive.org/web/20160405155834/http://permalink.gmane.org/gmane.science.philosophy.peirce/18574
All,
Peirce's existential graphs are a general calculus for expressing the
same subject matter as his logic of relative terms and thus they serve
to represent the structures of many‑place relations. Cast at that level
of generality, there is nothing to prevent existential graphs from being
used to express the special cases of relative terms needed for a theory of
triadic sign relations, for example, terms like “s stands to i for o” or
“__ stands to __ for __” or any number of other forms, depending on the
style one prefers. It may give us pause that we have to use sign relations
in order to mention sign relations but the fact is we do that all the time
whether we are using Peirce's semiotics or not. Peirce's pragmatic analysis
of the process simply provides a clearer account than most other approaches do.
Regards,
Jon
Jon Awbrey
Jun 16, 2023, 1:45:32 PM6/16/23
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Systems of Interpretation • 8
• http://inquiryintoinquiry.com/2023/06/16/systems-of-interpretation-8-2/
Figure 3. Aspects of a Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/06/aspects-of-a-sign-relation-2.0.png
Re: Peirce List
• https://web.archive.org/web/20160410151020/http://comments.gmane.org/gmane.science.philosophy.peirce/18558
Re: Kirsti Määttänen
• https://web.archive.org/web/20160405155000/http://permalink.gmane.org/gmane.science.philosophy.peirce/18573
All,
One of the chief advantages of Peirce's logical graphs, entitative
and existential, is the way they escape the bounds of 1‑dimensional
syntax and thus make it clear that many constraints of order imposed
by the ordinary lines of linguistic text are not of the essence for
logic but purely rhetorical accidents. That does, of course, leave
open the question of what constraints imposed by the 2‑dimensional
medium of Peirce's logical graphs might also be inessential to logic.
As far as visualizations of sign relations go, without worrying about
their use as a calculus, there is the above 3‑dimensional example from
a paper Susan Awbrey and I presented at conference in 1999 and revised
for publication in 2001.
Resources —
Logical Graphs • Introduction
• https://inquiryintoinquiry.com/2008/07/29/logical-graphs-introduction/
Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2008/09/19/logical-graphs-formal-development/
Logical Graphs • With Animated Proofs
• https://oeis.org/wiki/Logical_Graphs
Regards,
Jon
• http://inquiryintoinquiry.com/2023/06/16/systems-of-interpretation-8-2/
Figure 3. Aspects of a Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/06/aspects-of-a-sign-relation-2.0.png
Re: Peirce List
• https://web.archive.org/web/20160410151020/http://comments.gmane.org/gmane.science.philosophy.peirce/18558
• https://web.archive.org/web/20160405155000/http://permalink.gmane.org/gmane.science.philosophy.peirce/18573
All,
One of the chief advantages of Peirce's logical graphs, entitative
and existential, is the way they escape the bounds of 1‑dimensional
syntax and thus make it clear that many constraints of order imposed
by the ordinary lines of linguistic text are not of the essence for
logic but purely rhetorical accidents. That does, of course, leave
open the question of what constraints imposed by the 2‑dimensional
medium of Peirce's logical graphs might also be inessential to logic.
As far as visualizations of sign relations go, without worrying about
their use as a calculus, there is the above 3‑dimensional example from
a paper Susan Awbrey and I presented at conference in 1999 and revised
for publication in 2001.
Resources —
Logical Graphs • Introduction
• https://inquiryintoinquiry.com/2008/07/29/logical-graphs-introduction/
Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2008/09/19/logical-graphs-formal-development/
Logical Graphs • With Animated Proofs
• https://oeis.org/wiki/Logical_Graphs
Regards,
Jon
Jon Awbrey
Jun 17, 2023, 2:04:57 PM6/17/23
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Cf: Systems of Interpretation • 9
• https://inquiryintoinquiry.com/2023/06/17/systems-of-interpretation-9-2/
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation-1.0.png
Re: Peirce List
• https://web.archive.org/web/20160401033602/http://comments.gmane.org/gmane.science.philosophy.peirce/18534
Jerry Chandler
• https://web.archive.org/web/20160324201000/http://permalink.gmane.org/gmane.science.philosophy.peirce/18552
All,
It is above all important to understand that Peirce's concept
of a sign relation is defined at a higher order of abstraction
than any notion of causal or temporal order.
A sign relation L ⊆ O × S × I is a structure which can “generate” the
temporal sequences of signs making up a semiotic process but there is
no necessary temporal order associated with the relational domains
O, S, I nor with the roles of objects, signs, and interpretant signs
in any triple of the form (o, s, i).
As it happens, generative relationships between a generating structure and
a generated class of structures are very common throughout mathematics and
not unique to semiotics.
Regards,
Jon
• https://inquiryintoinquiry.com/2023/06/17/systems-of-interpretation-9-2/
Figure 2. An Elementary Sign Relation
• https://inquiryintoinquiry.files.wordpress.com/2023/05/elementary-sign-relation-1.0.png
Re: Peirce List
Jerry Chandler
• https://web.archive.org/web/20160324201000/http://permalink.gmane.org/gmane.science.philosophy.peirce/18552
All,
It is above all important to understand that Peirce's concept
of a sign relation is defined at a higher order of abstraction
than any notion of causal or temporal order.
A sign relation L ⊆ O × S × I is a structure which can “generate” the
temporal sequences of signs making up a semiotic process but there is
no necessary temporal order associated with the relational domains
O, S, I nor with the roles of objects, signs, and interpretant signs
in any triple of the form (o, s, i).
As it happens, generative relationships between a generating structure and
a generated class of structures are very common throughout mathematics and
not unique to semiotics.
Regards,
Jon
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