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Higher Order Sign Relations • Discussion 1

Re: FB | Charles S. Peirce SocietyJohn Corcoran

Questions about the proper treatment of use and mention from the standpoint of Peirce’s theory of signs came up recently in discussions on Facebook.  In pragmatic semiotics the trade‑off between “signs-of-objects” and “signs-as-objects” opens up the wider space of higher order sign relations.  In previous work on Inquiry Driven Systems I introduced the subject in the following way.

When interpreters reflect on their use of signs they require an appropriate technical language in which to pursue their reflections.  They need signs referring to sign relations, signs referring to elements and components of sign relations, and signs referring to properties and classes of sign relations.  The orders of signs developing as reflection evolves can be organized under the heading of “higher order signs” and the reflective sign relations involving them can be referred to as “higher order sign relations”.

References

Resources

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Higher Order Sign Relations • 1

Higher Order Sign Relations • Introduction

When interpreters reflect on their use of signs they require an appropriate technical language in which to pursue their reflections.  They need signs referring to sign relations, signs referring to elements and components of sign relations, and signs referring to properties and classes of sign relations.  The orders of signs developing as reflection evolves can be organized under the heading of “higher order signs” and the reflective sign relations involving them can be referred to as “higher order sign relations”.

Some years ago I was formatting my old dissertation proposal on Inquiry Driven Systems for the web when the subject of “signs about signs” arose on the Peirce List.  It called to mind the part of my document on Higher Order Sign Relations, on which basis Reflective Interpretive Frameworks are constructed, and the introduction to which begins as above.

Resources

cc: Academia.edu • BlueSky • Laws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

oeis.orgInquiry Driven Systems • Overview - OeisWiki

Signs Of Signs • 4

Re: Michael HarrisLanguage About Language

But then inevitably I find myself wondering whether a proof assistant, or even a formal system, can make the distinction between “technical” and “fundamental” questions.  There seems to be no logical distinction.  The formalist answer might involve algorithmic complexity, but I don’t think that sheds any useful light on the question.  The materialist answer (often? usually?) amounts to just‑so stories involving Darwin, and lions on the savannah, and maybe an elephant, or at least a mammoth.  I don’t find these very satisfying either and would prefer to find something in between, and I would feel vindicated if it could be proved (in I don’t know what formal system) that the capacity to make such a distinction entails appreciation of music.

Peirce proposed a distinction between corollarial and theorematic reasoning in mathematics which strikes me as similar to the distinction Michael Harris seeks between technical and fundamental questions.

I can’t say I have a lot of insight into how the distinction might be drawn but I recall a number of traditions pointing to the etymology of theorem as having to do with the observation of objects and practices whose depth of detail always escapes full accounting by any number of partial views.

On the subject of music, all I have is the following incidental —

🙞 Riffs and Rotes

Perhaps it takes a number theorist to appreciate it …

Resources

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cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

Mathematics without Apologies, by Michael Harris · About the authorMichael Harris is professor of mathematics at the Université Paris-Diderot and Columbia University.  He is the author or coauthor of more than seventy mathematical books and articles, and has recei…

Signs Of Signs • 3

Re: Michael HarrisLanguage About Language

And if we don’t [keep our stories straight], who puts us away?

One’s answer, or at least one’s initial response to that question will turn on how one feels about formal realities.  As I understand it, reality is that which persists in thumping us on the head until we get what it’s trying to tell us.  Are there formal realities, forms which drive us in that way?

Discussions like those tend to begin by supposing we can form a distinction between external and internal.  That is a formal hypothesis, not yet born out as a formal reality.  Are there formal realities which drive us to recognize them, to pick them out of a crowd of formal possibilities?

Resources

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cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

Mathematics without Apologies, by Michael HarrisMathematics without Apologies, by Michael HarrisAn unapologetic guided tour of the mathematical life

Signs Of Signs • 2

Re: Michael HarrisLanguage About Language

I compared mathematics to a “consensual hallucination”, like virtual reality, and I continue to believe that the aim is to get (consensually) to the point where that hallucination is a second nature.

I think that’s called coherentism, normally contrasted with or complementary to objectivism.  It’s the philosophy of a gang of co‑conspirators who think, “We’ll get off scot‑free so long as we all keep our stories straight.”

Resources

cc: Academia.edu • BlueSky • Laws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

Mathematics without Apologies, by Michael HarrisMathematics without Apologies, by Michael HarrisAn unapologetic guided tour of the mathematical life

Signs Of Signs • 1

Re: Michael HarrisLanguage About Language

There is a language and a corresponding literature treating logic and mathematics as related species of communication and information gathering, namely, the pragmatic‑semiotic tradition transmitted through the lifelong efforts of C.S. Peirce.  It is by no means a dead language but it continues to fly beneath the radar of many trackers in logic and math today.  Nevertheless, the resource remains for those who wish to look into it.

Resources

cc: Academia.edu • BlueSky • Laws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

Mathematics without Apologies, by Michael HarrisMathematics without Apologies, by Michael HarrisAn unapologetic guided tour of the mathematical life

Interpreter and Interpretant • Selection 10

Transfer

Returning to the scene of Dewey’s “Sign of Rain” example, let’s continue examining how the transfer of knowledge through the analogy of experience works in that case.

By way of a recap, we began by considering a fragment of the reasoner’s knowledge base which is logically equivalent to a conjunction of two rules.

may be thought of as a piece of knowledge or item of information allowing for the possibility of certain conditions, expressed in the form of a logical constraint on the present universe of discourse.

Next we found it convenient to express all logical statements in terms of their models, that is, in terms of the primitive circumstances or elements of experience over which they hold true.

  • Let be the chosen set of experiences, or the circumstances in mind under “past experience”.
  • Let be the collective set of experiences, or the prospective total of possible circumstances.
  • Let be the current experience, or the circumstances immediately present to the reasoner.

If we think of the knowledge base as referring to the “regime of experience” over which it is valid, then the sets of models involved in the analogy may be ordered according to the relationships of set inclusion or logical implication existing among them.

Figure 4 shows the subsumption relations involved in the analogy of experience.


In logical terms, the analogy of experience proceeds by inducing a Rule about the validity of a current knowledge base and then by deducing a Fact, the applicability of that knowledge base to a current experience.

  • Step 1 is Inductive, abstracting a Rule from a Case and a Fact.

  • Step 2 is Deductive, admitting a Case to a Rule and arriving at a Fact.

References

  • 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.  ArchiveJournal.  Online (doc) (pdf).
  • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.

Resources

cc: Academia.eduBlueSkyLaws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

Interpreter and Interpretant • Selection 9

Transfer

Let’s examine how the transfer of knowledge through the analogy of experience works in the case of Dewey’s “Sign of Rain” example.

For concreteness, consider a fragment of the reasoner’s knowledge base which is logically equivalent to a conjunction of two rules.

may be thought of as a piece of knowledge or item of information allowing for the possibility of certain conditions, expressed in the form of a logical constraint on the present universe of discourse.

It is convenient to have the option of expressing all logical statements in terms of their models, that is, in terms of the primitive circumstances or elements of experience over which they hold true.

  • Let be the chosen set of experiences, or the circumstances in mind under “past experience”.
  • Let be the collective set of experiences, or the prospective total of possible circumstances.
  • Let be the current experience, or the circumstances immediately present to the reasoner.

If we think of the knowledge base as referring to the “regime of experience” over which it is valid, then the sets of models involved in the analogy may be ordered according to the relationships of set inclusion or logical implication existing among them.

In logical terms, the analogy of experience proceeds by inducing a Rule about the validity of a current knowledge base and then by deducing a Fact, the applicability of that knowledge base to a current experience.

  • Step 1 is Inductive, abstracting a Rule from a Case and a Fact.

  • Step 2 is Deductive, admitting a Case to a Rule and arriving at a Fact.

References

  • 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.  ArchiveJournal.  Online (doc) (pdf).
  • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.

Resources

cc: Academia.eduBlueSkyLaws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

web.archive.orgInterpretation as Action: The Risk of Inquiry

Interpreter and Interpretant • Selection 8

Transfer

What exactly gives the acquisition of a knowledge base its distinctively inductive character?  It is evidently the “analogy of experience” involved in applying what we’ve learned in the past to what confronts us in the present.

Whenever we find ourselves approaching a problem with the thought, If past experience is any guide … we can be sure the analogy of experience has come into play.  We are seeking to find analogies between past experience as a totality and present experience as a point of application.

From a statistical point of view what we mean is this — “If past experience is a fair sample of possible experience then knowledge gained from past experience may usefully apply to present experience”.  It is that mechanism which allows a knowledge base to be carried across gulfs of experience which remain indifferent to the effective contents of its rules.

Next we’ll examine how the transfer of knowledge through the analogy of experience works out in the case of Dewey’s “Sign of Rain” example.

References

  • 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.  ArchiveJournal.  Online (doc) (pdf).
  • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.

Resources

cc: Academia.eduBlueSkyLaws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

web.archive.orgInterpretation as Action: The Risk of Inquiry

Interpreter and Interpretant • Selection 7
inquiryintoinquiry.com/2025/01

Learning —

Rules in a knowledge base, as far as their effective content goes, can be obtained by any mode of inference. For example, consider a proposition of the following form.

• B ⇒ A, Just Before it rains, the Air is cool.

Such a proposition is usually induced from a consideration of many past events. The inductive inference may be observed to fit the following pattern.

• Case : C ⇒ B, In Certain events, it is just Before it rains.
• Fact : C ⇒ A, In Certain events, the Air is cool.
────────────────────────────────────
• Rule : B ⇒ A, Just Before it rains, the Air is cool.

However, the same proposition could also be abduced as an explanation of a singular occurrence or deduced as a conclusion of a prior theory.

References —

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.
pdcnet.org/inquiryct/content/i
academia.edu/57812482/Interpre

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY.
gutenberg.org/files/37423/3742

Resources —

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
inquiryintoinquiry.com/2024/02

Survey of Semiotics, Semiosis, Sign Relations
inquiryintoinquiry.com/2024/01




Inquiry Into Inquiry · Interpreter and Interpretant • Selection 7
More from Inquiry Into Inquiry

Interpreter and Interpretant • Selection 7

Learning

Rules in a knowledge base, as far as their effective content goes, can be obtained by any mode of inference.  For example, consider a proposition of the following form.

Such a proposition is usually induced from a consideration of many past events.  The inductive inference may be observed to fit the following pattern.

However, the same proposition could also be abduced as an explanation of a singular occurrence or deduced as a conclusion of a prior theory.

References

  • 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.  ArchiveJournal.  Online (doc) (pdf).
  • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.

Resources

cc: Academia.eduBlueSkyLaws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

web.archive.orgInterpretation as Action: The Risk of Inquiry

Interpreter and Interpretant • Selection 6
inquiryintoinquiry.com/2025/01

Inquiry and Induction —

To understand the bearing of inductive reasoning on the closing phases of inquiry there are a couple of observations we should make.

• Smaller inquiries are typically woven into larger inquiries, whether the whole pattern of inquiry is carried on by a single agent or by a complex community.

• There are several ways particular instances of inquiry are related to ongoing inquiries at larger scales. Three modes of interaction between component inquiries and compound inquiries may be described under the headings of Learning, Transfer, and Testing of Rules.




Inquiry Into Inquiry · Interpreter and Interpretant • Selection 6
More from Inquiry Into Inquiry

Interpreter and Interpretant • Selection 6

Inquiry and Induction

To understand the bearing of inductive reasoning on the closing phases of inquiry there are a couple of observations we should make.

  • Smaller inquiries are typically woven into larger inquiries, whether the whole pattern of inquiry is carried on by a single agent or by a complex community.
  • There are several ways particular instances of inquiry are related to ongoing inquiries at larger scales.  Three modes of interaction between component inquiries and compound inquiries may be described under the headings of Learning, Transfer, and Testing of Rules.

Throughout the course of inquiry the reasoner makes use of rules which have to be transported across intervals of experience, from masses of experience where they are learned to moments of experience where they are applied.  Inductive reasoning is involved in the learning and transfer of those rules, both in accumulating a knowledge base and in carrying it through the times between acquisition and application.

Learning The main way induction contributes to an ongoing inquiry is through the learning of rules, that is, by creating each rule added to a knowledge base, or any rule used along the way. Transfer The next way induction contributes to an ongoing inquiry is through the operation of analogy, a two‑step combination of induction and deduction which serves to transfer rules from one context to another. Testing Finally, every inquiry making use of a knowledge base amounts to a “field test” of its rules.  If the knowledge base fails to serve any live inquiry in a satisfactory way then there is reason to reconsider and possibly amend its rules.

Next time we’ll examine how the principles of learning, transfer, and testing apply to Dewey’s “Sign of Rain” example.

References

  • 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.  ArchiveJournal.  Online (doc) (pdf).
  • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.

Resources

cc: Academia.eduBlueSkyLaws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

web.archive.orgInterpretation as Action: The Risk of Inquiry

Interpreter and Interpretant • Selection 5
inquiryintoinquiry.com/2025/01

Inquiry and Inference —

If we follow Dewey's “Sign of Rain” story far enough to consider the import of thought for action, we realize the subsequent conduct of the interpreter, progressing up through the natural conclusion of the episode — the quickening steps, the seeking of shelter in time to escape the rain — all those acts amount to a series of further interpretants for the initially recognized signs of rain and the first impressions of the actual case. Just as critical reflection develops the positive and negative signs which gather about an idea, pragmatic interpretation explores the consequential and contrasting actions which give effective and testable meaning to a person's belief in it.




Interpreter and Interpretant • Selection 5

Inquiry and Inference

If we follow Dewey’s “Sign of Rain” story far enough to consider the import of thought for action, we realize the subsequent conduct of the interpreter, progressing up through the natural conclusion of the episode — the quickening steps, the seeking of shelter in time to escape the rain — all those acts amount to a series of further interpretants for the initially recognized signs of rain and the first impressions of the actual case.  Just as critical reflection develops the positive and negative signs which gather about an idea, pragmatic interpretation explores the consequential and contrasting actions which give effective and testable meaning to a person’s belief in it.

Figure 3 charts the progress of inquiry in Dewey’s narrative according to the stages of reasoning identified in Peirce’s theory of inquiry, focusing on the compound pattern of inference formed by the first two steps.


  • Step 1 is Abductive, abstracting a Case from the consideration of a Fact and a Rule.
    •     In the Current situation the Air is cool.
    •     Just Before it rains, the Air is cool.
    •     The Current situation is just Before it rains.
  • Step 2 is Deductive, admitting the Case to another Rule and arriving at a novel Fact.
    •     The Current situation is just Before it rains.
    •     Just Before it rains, a Dark cloud will appear.
    •     In the Current situation, a Dark cloud will appear.

What precedes is nowhere near a complete analysis of Dewey’s example, even so far as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the inquiry process, but perhaps it will do for a start.

References

  • 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.  ArchiveJournal.  Online (doc) (pdf).
  • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.

Resources

cc: Academia.eduBlueSkyLaws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

Interpreter and Interpretant • Selection 4
inquiryintoinquiry.com/2025/01

Interpretation and Inquiry —

To illustrate the role of sign relations in inquiry we begin with Dewey's elegant and simple example of reflective thinking in everyday life.

❝A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something suggested. The pedestrian feels the cold; he thinks of clouds and a coming shower.❞ (John Dewey, How We Think, 6–7).




Interpreter and Interpretant • Selection 4

Interpretation and Inquiry

To illustrate the role of sign relations in inquiry we begin with Dewey’s elegant and simple example of reflective thinking in everyday life.

A man is walking on a warm day.  The sky was clear the last time he observed it;  but presently he notes, while occupied primarily with other things, that the air is cooler.  It occurs to him that it is probably going to rain;  looking up, he sees a dark cloud between him and the sun, and he then quickens his steps.  What, if anything, in such a situation can be called thought?  Neither the act of walking nor the noting of the cold is a thought.  Walking is one direction of activity;  looking and noting are other modes of activity.  The likelihood that it will rain is, however, something suggested.  The pedestrian feels the cold;  he thinks of clouds and a coming shower.

(John Dewey, How We Think, 6–7)

In Dewey’s narrative we can identify the characters of the sign relation as follows.  Coolness is a Sign of the Object rain, and the Interpretant is the thought of the rain’s likelihood.  In his description of reflective thinking Dewey distinguishes two phases, “a state of perplexity, hesitation, doubt” and “an act of search or investigation” (p. 9), comprehensive stages which are further refined in his later model of inquiry.

Reflection is the action the interpreter takes to establish a fund of connections between the sensory shock of coolness and the objective danger of rain by way of the impression rain is likely.  But reflection is more than irresponsible speculation.  In reflection the interpreter acts to charge or defuse the thought of rain by seeking other signs the thought implies and evaluating the thought according to the results of that search.

Figure 2 shows the semiotic relationships involved in Dewey’s story, tracing the structure and function of the sign relation as it informs the activity of inquiry, including both the movements of surprise explanation and intentional action.  The labels on the outer edges of the sign‑relational triple suggest the significance of signs for eventual occurrences and the correspondence of ideas with external orientations.  But there is nothing essential about the dyadic role distinctions they imply, as it is only in special or degenerate cases that such projections preserve enough information to determine the original sign relation.


References

  • 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.  ArchiveJournal.  Online (doc) (pdf).
  • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.

Resources

cc: Academia.eduBlueSkyLaws of FormMathstodonResearch Gate
cc: Conceptual GraphsCyberneticsStructural ModelingSystems Science

Continued thread

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources —

Hypostatic Abstraction
inquiryintoinquiry.com/2008/08

Survey of Semiotics, Semiosis, Sign Relations
inquiryintoinquiry.com/2024/01



Hypostatic Abstraction Figure 1
Inquiry Into Inquiry · Hypostatic Abstraction
More from Inquiry Into Inquiry
Continued thread

❝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. I will not undertake to analyze, this evening, this conception of standing for something — but, it is sufficiently plain that it involves the standing to something 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 1866, Chronological Edition 1, pp. 466–467).