#triadicrelations — Public Fediverse posts

Reflective Interpretive Frameworks • Incident 2
• https://inquiryintoinquiry.com/2026/08/27/reflective-interpretive-frameworks-incident-2/

Re: Terence Tao • Modular Arithmetic Challenge
• https://terrytao.wordpress.com/2026/06/08/modular-arithmetic-challenge/
• https://competition.sair.foundation/competitions/modular-arithmetic-challenge/overview

The Modular Arithmetic Challenge asks a simple question:

• Can a neural network learn to do modular multiplication efficiently?

Incidental Reflection 1 —

There are alternative models of neural networks which do not depend on threshold neurons and endlessly fiddling with weights.

Incidental Reflection 2 —

The series of three blog posts linked below present a case study comparing two ways of handling a classic example from the Parallel Distributed Processing paradigm, namely, the “Jets and Sharks” database problem, first taking up the original treatment by McClelland and Rumelhart and then proceeding according to a program I developed for propositional logic modeling. The latter method makes use of ideas from Grossberg's competition‑cooperation and winner‑take‑all dynamics, but is purely propositional‑logic based, involving no extraneous weights.

Theme One Program • Jets and Sharks
(1) https://inquiryintoinquiry.com/2024/06/20/theme-one-program-jets-and-sharks-1-a/
(2) https://inquiryintoinquiry.com/2024/06/22/theme-one-program-jets-and-sharks-2-a/
(3) https://inquiryintoinquiry.com/2024/06/23/theme-one-program-jets-and-sharks-3-a/

Resources —

Survey of Theme One Program
• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Differential Analytic Turing Automata
• https://oeis.org/wiki/Differential_Analytic_Turing_Automata_%E2%80%A2_Overview

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations
#PropositionalModels #MinimalNegationOperators #DifferentialAnalyticTuringAutomata
#CactusGraphs #CactusLanguage #DifferentialLogic #NeuralNetworks #SequenceLearning

Reflection On Recursion • Discussion 1

Re: Reflection On Recursion • 1
Re: Laws of Form • John Mingers

Also the work of Maturana and Varela on autopoiesis and the neurophysiology of cognition which also has recursion at its heart.

Thanks, John.  Yes, we certainly find the whole array of self concepts coming into play here — selfhood, autopoiesis or self creation, self reference and self transformation, just to name a few.  But one thing I need to emphasize from the start is how radically different such concepts appear when viewed under x‑rays of Peirce’s pragmatic semiotics.

I forget where I first heard it, but it’s fairly common observation that the persistence of a recurring problem is a symptom of how unlikely it is to be solved in the paradigm where it keeps occurring.

After a while, it simply becomes time to change the paradigm …

Just by way of a first example, take the very idea of “self‑reference”.  The moment we place it in the medium of triadic sign relations we realize signs do not refer to anything at all except insofar as an interpreter refers them.  And when we think to ask, “What is this that we call an interpreter?”, the pragmatic theory of signs tells us we do not know when we turn out the light but under the x‑ray of the pragmatic maxim the sum of its effects is effectively modeled by an extended triadic sign relation.

Everything I’ll be working at here will be done within a framework like that.

Regards,
Jon

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 4
• https://inquiryintoinquiry.com/2026/04/18/reflection-on-recursion-4/

A feature worth noting in the recursion diagram is the function traversing the square from one triadic node to the other. It preserves an image of the object n all the while its precedent p(n) is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.

Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.png

Resources —

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Reflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1

The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_Reflection

Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Reflection On Recursion • 4

A feature of special note in the recursion diagram is the function traversing the square from one triadic node to the other.  It preserves an image of the object all the while its precedent is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 3
• https://inquiryintoinquiry.com/2026/04/13/reflection-on-recursion-3/

One other feature of syntactic recursion deserves to be brought into higher relief. Evidence of it can be found in the recursion diagram by examining the places where three paths meet. On the descending side there is the point where three paths diverge. On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.

Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.png

The arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.

• The three arrows from the initial node represent a function F : N → N×N×N such that F(n) = (p(n), n, f(n)).

• The three arrows at the penultimate node represent a function m : N×N → N such that m(j, k) = jk.

For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.

Resources —

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Reflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1

The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_Reflection

Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Reflection On Recursion • 3

One other feature of syntactic recursion deserves to be brought into higher relief.  Evidence of it can be found in the recursion diagram by examining the places where three paths meet.  On the descending side there is the point where three paths diverge.  On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.

The arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.

• The three arrows from the initial node represent a function such that
• The three arrows at the penultimate node represent a function such that

For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 2
• https://inquiryintoinquiry.com/2026/04/09/reflection-on-recursion-2/

Turning to the form of a simple recursive function f(n) = m(n, f(p(n))), the clause we used to define it earns the title of “syntactic recursion” due to the way the function name “f” occurring in the defined phrase “f(n)” re‑occurs in the defining phrase “m(n, f(p(n)))”.

Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.png

It needs to be clear there is no circle in the definition — each instance of the type f is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat. Instead of a circle then we have two gyres, the gyre down via the precedent function p and the gyre up via the modifier function m.

cc: https://www.academia.edu/community/L24rvm
cc: https://www.academia.edu/community/LE2mrr
cc: https://www.researchgate.net/post/Reflection_On_Recursion

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Reflection On Recursion • 2

Turning to the form of a simple recursive function the clause we used to define it earns the title of “syntactic recursion” due to the way the function name occurring in the defined phrase re‑occurs in the defining phrase

It needs to be clear there is no circle in the definition — each instance of the type is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat.  Instead of a circle then we have two gyres, the gyre down via the predecessor function and the gyre up via the modifier function

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2) (3)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 1.3
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/

Comment 5 —

Recursion is rife in mathematics and computation, typically sporting its recursive character on its sleeve in the fashion of syntax sketched above.

But mathematics and computation are overlearned subjects and practices, enjoying long histories of being gone over with an eye to articulating every last detail of any way they might be conceived and conducted.

So it's fair to ask whether all that artifice truly tutors nature or only creates a rationalized reconstruction of it. Then again, even if that's all it does, is there anything of use to be learned from it?

Comment 6 —

The prevalence of recursion in mathematics arises from the architecture of mathematical systems.

Mathematical systems grow from a fourfold root.

• “Primitives” are taken as initial terms.

• “Definitions” expound ever more complex terms in relation to the primitives.

• “Axioms” are taken as initial truths.

• “Theorems” follow from the axioms by way of inference rules.

Recursive definitions of mathematical objects and inductive proofs of the corresponding theorems follow closely parallel patterns. And again, in computation, recursive programs follow the same patterns in action.

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Reflection On Recursion • 1.2
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/

Comment 3 —

If we discard from the idea of recursion what is not of its essence, we find recursion occurs when our understanding of one situation has recourse to our understanding of other situations.

Very typically, the object situation presents itself as complex, difficult, or unfamiliar while the resource situations are regarded as being better understood.

It must be appreciated, however, that any ranking of situations by level of understanding is contingent on the circumstances in view and may vary radically in alternate settings.

Comment 4 —

Recursion occurs more markedly in “syntactic recursion”, where the recursive process shows its character as such in the symbols of its syntactic expression.

A sense of the difference can be gained by looking at a case of “ostensible syntactic recursion”. (How much substance backs the ostentation is a subject we'll take up, maybe at length, but later …)

Consider the following diagram for the computation of a simple recursive function.

Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.png

For example, the factorial function f(n) = n! has a definition in terms of the predecessor function p(n) = n-1 and the multiplier function m(j, k) = j∙k.

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Reflection On Recursion • 1.1
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/

Ongoing conversations with Dan Everett on Facebook have me backtracking to recurring questions about the relationship between formal language theory (as I once learned it) and the properties of natural languages as they are found occurring in the field.

A point of particular interest is the role of recursion in formal and natural languages, along with collateral questions about its role in the cognitive sciences at large.

It has taken me quite a while to bring my reflections up to the threshold of minimal coherence — and the inquiry remains ongoing — but it may catalyze the thinking process if I simply share what I've thought so far …

Comment 1 —

Recursion is where you find it — so, myself not being a natural language researcher, when someone who is says they don't find it in a given corpus I just take them at their word …

Comment 2 —

The question to which I keep returning has to do with the relationship between two ways we find recursion occurring.

One way I'd call “pragmatic recursion” — if I wanted to be precise and cover its full scope — since so many of its operations occur without conscious direction, but for now I'll defer to more familiar language, calling it “cognitive” or “conceptual” recursion.

Resources —

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Reflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1

The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_Reflection

Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Reflective Interpretive Frameworks • Incident 1
• https://inquiryintoinquiry.com/2026/03/26/reflective-interpretive-frameworks-incident-1/

Re: William Waites • The Agent That Doesn't Know Itself
• https://johncarlosbaez.wordpress.com/2026/03/20/the-agent-that-doesnt-know-itself/

WW: ❝Why Has Nobody Done This?❞

People who study C.S. Peirce would say reflective reasoning requires triadic relations at core and there is work being done on that. One of the challenges is clarifying the role of triadic relations in category theory and raising them into higher relief as fundamental operations.

Note. I was looking for a word to describe a random encounter with something that jogs one's memory of a recurring theme — “incident” plays into the “reflection” theme and looked worth trying for now.

Resources —

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Reflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_Frameworks

The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_Reflection

Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

Notes On Categories
• https://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/
• https://inquiryintoinquiry.com/2021/07/31/notes-on-categories-2/

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Sign Relations • Signs and Inquiry

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many congruences and parallels it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (Dewey, 38).

Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation.  Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.

References

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

Resources

• Sign Relation • OEIS • MyWikiBiz • Wikiversity
• Survey of Semiotics, Semiosis, Sign Relations
• Survey of Inquiry Driven Systems

cc: Academia.edu • Laws of Form • Research Gate • Syscoi
cc: Cybernetics • Structural Modeling • Systems Science

#CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

Sign Relations • 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.

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, New Elements of Mathematics, vol. 4, 20–21

In the general discussion of diverse theories of signs, the question 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 may be said to have only within a particular context of relationships.

Peirce’s definition of a sign defines it in relation to its objects and its interpretant signs, and thus defines signhood in relative terms, by means of a predicate with three places.  In that definition, signhood is a role in a triadic relation, a role a thing bears or plays in a determinate context of relationships — it is not an absolute or non‑relative property of a thing‑in‑itself, one it possesses 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 the term throughout his work, it is clear he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself.  In particular, his use of the term should not be taken to imply a dyadic correspondence, like the kinds 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 directions than the sense of the word referring to strictly deterministic causal‑temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called a formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called 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 have occasion to view the same data, as logic is a normative science where psychology is a descriptive science, and so they have very different aims, methods, and rationales.

Reference

• 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.  Online.

Resources

• Sign Relation • OEIS • MyWikiBiz • Wikiversity
• Survey of Semiotics, Semiosis, Sign Relations

cc: Academia.edu • Laws of Form • Research Gate • Syscoi
cc: Cybernetics • Structural Modeling • Systems Science

#CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

Survey of Semiotics, Semiosis, Sign Relations • 6

C.S. Peirce defines logic as “formal semiotic”, using formal to highlight the place of logic as a normative science, over and above the descriptive study of signs and their role in wider fields of play.  Understanding logic as Peirce understands it thus requires a companion study of semiotics, semiosis, and sign relations.

What follows is a Survey of blog and wiki resources on the theory of signs, variously known as semeiotic or semiotics, and the actions referred to as semiosis which transform signs among themselves in relation to their objects, all as based on C.S. Peirce’s concept of triadic sign relations.

Elements

• Semeiotic
• Sign Relations

Blog Series

• Semeiotic

• Sign Relations • Anthesis • Definition • Signs and Inquiry • Examples • Dyadic Aspects • Denotation • Connotation • Ennotation • Semiotic Equivalence Relations (1) (2)
• Comments • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)
• Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14)

• Peircean Semiotics and Triadic Sign Relations • (1) • (2) • (3)

• Zeroth Law Of Semiotics
• Comments • (1) • (2) • (3) • (4) • (5) • (6) • (7)
• Discussions • (1) • (2) • (3) • (4)

• All Liar, No Paradox
• Comments • (1)
• Discussions • (1) • (2)

• Sign Relational Manifolds • (1) • (2) • (3) • (4) • (5)
• Discussions • (1) • (2) • (3)

• Semiotics, Semiosis, Sign Relations • (1) • (2) • (3)
• Comments • (1) • (2) • (3) • (4)
• Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19)

• Sign Relations, Triadic Relations, Relation Theory • (1) • (2) • (3) • (4)
• Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)

• Higher Order Sign Relations • (1)
• Discussions • (1) • (2) • (3) • (4) • (5)

Blog Dialogs

• Signs Of Signs • (1) • (2) • (3) • (4)

• Icon Index Symbol • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20)

• Sign Relations, Triadic Relations, Relations • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11)

• Systems of Interpretation • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)
• Discussions • (1)

Sources

• C.S. Peirce • On the Definition of Logic
• C.S. Peirce • Logic as Semiotic
• C.S. Peirce • Objective Logic

• C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • (1) • (2)
• C.S. Peirce • Algebra of Logic 1885 • Selections • (1) • (2) • (3) • (4)

Topics

• Interpreter and Interpretant
• Selections • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10)
• Discussions • (1) • (2) • (3) • (4)

• Tone, Token, Type
• Discussions • (1)

Excursions

• Semiositis • (1)
• Signspiel • (1)
• Skiourosemiosis • (1)

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, 269–284.  Abstract.  Online.
• Awbrey, S.M., and Awbrey, J.L. (September 1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA.  Online.
• 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.  Archive.  Journal.  Online (doc) (pdf).
• Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.

cc: FB | Semeiotics • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#CSPeirce #IconIndexSymbol #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #Semiotics #SignRelations #TriadicRelations #Triadicity #Visualization

Interpreter and Interpretant • Selection 7
• https://inquiryintoinquiry.com/2025/01/15/interpreter-and-interpretant-selection-7-a/

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.
• https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052
• https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY.
• https://www.gutenberg.org/files/37423/37423-h/37423-h.htm

Resources —

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
• https://inquiryintoinquiry.com/2024/02/27/survey-of-abduction-deduction-induction-analogy-inquiry-4/

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

#Peirce #Logic #Semiotics #SignRelations #TriadicRelations
#Interpretation #Interpreter #Interpretant #Hermeneutics
#JohnDewey #Inquiry #Abduction #Deduction #Induction
#Abstraction #HypostaticAbstraction #Reflection

Interpreter and Interpretant • Selection 6
• https://inquiryintoinquiry.com/2025/01/13/interpreter-and-interpretant-selection-6-a/

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.

#Peirce #Logic #Semiotics #SignRelations #TriadicRelations
#Interpretation #Interpreter #Interpretant #Hermeneutics
#JohnDewey #Inquiry #Abduction #Deduction #Induction
#Abstraction #HypostaticAbstraction #Reflection

Interpreter and Interpretant • Selection 5
• https://inquiryintoinquiry.com/2025/01/11/interpreter-and-interpretant-selection-5-a/

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.

#Peirce #Logic #Semiotics #SignRelations #TriadicRelations
#Interpretation #Interpreter #Interpretant #Hermeneutics
#JohnDewey #Inquiry #Abduction #Deduction #Induction
#Abstraction #HypostaticAbstraction #Reflection

References —

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY.
• https://www.gutenberg.org/files/37423/37423-h/37423-h.htm

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.
• https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html
• https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052
• https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry
• https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry

Resources —

Survey of Inquiry Driven Systems
• https://inquiryintoinquiry.com/2024/02/28/survey-of-inquiry-driven-systems-6/

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

#Peirce #Logic #Semiotics #Semiosis #SignRelations #TriadicRelations
#Cybersemiotics #Interpreter #Interpretant #Hermeneutics #Hermenaut
#JohnDewey #HowWeThink #Inquiry #Abduction #Deduction #Induction
#Abstraction #HypostaticAbstraction #Reflection #Interpretation

Interpreter and Interpretant • Selection 4
• https://inquiryintoinquiry.com/2025/01/09/interpreter-and-interpretant-selection-4-a/

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).

#Peirce #Logic #Semiotics #Semiosis #SignRelations #TriadicRelations
#Cybersemiotics #Interpreter #Interpretant #Hermeneutics #Hermenaut
#JohnDewey #HowWeThink #Inquiry #Abduction #Deduction #Induction
#Abstraction #HypostaticAbstraction #Reflection #Interpretation

Peircean Semiotics and Triadic Sign Relations • 3
• https://inquiryintoinquiry.com/2024/08/23/peircean-semiotics-and-triadic-sign-relations-3-a/

Having labored mightily to bring out a new edition of my primer on sign relations, including material on the pivotal concept of semiotic equivalence relations, I thought it worth the candle to post a notice of the new version here.

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

Semiotic Equivalence Relations
• https://oeis.org/wiki/Sign_relation#Semiotic_equivalence_relations

#Peirce #Inquiry #Logic #LogicOfRelatives #RelationTheory
#Semiotics #Semiosis #SignRelations #TriadicRelations

Peircean Semiotics and Triadic Sign Relations • 2
• https://inquiryintoinquiry.com/2024/08/22/peircean-semiotics-and-triadic-sign-relations-2-a/

When I returned to graduate school for the third time around, this time in systems engineering, I had in mind integrating my long‑standing projects investigating the dynamics of information, inquiry, learning, and reasoning, viewing each as a process whose trajectory evolves over time through the medium which gives it concrete embodiment, namely, a triadic sign relation.

Up until that time I don't believe I'd ever given much thought to sign relations that had anything smaller than infinite domains of objects, signs, and interpretant signs. Countably infinite domains are what come natural in logic, since that is the norm for the formal languages it uses. Continuous domains come first to mind when turning to physical systems, despite the fact that systems with a discrete or quantized character often enter the fray.

So it came as a bit of a novelty to me when my advisor, following the motto of engineers the world over to “Keep It Simple, Stupid!” — affectionately known by the acronym KISS — asked me to construct the simplest non‑trivial finite example of a sign relation I could possibly come up with. The outcome of that exercise I wrote up in the following primer on sign relations.

Inquiry Driven Systems • Sign Relations : A Primer
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_1#Sign_Relations_:_A_Primer

Inquiry Driven Systems • Semiotic Equivalence Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_1#Semiotic_Equivalence_Relations

#Peirce #Inquiry #Logic #LogicOfRelatives #RelationTheory
#Semiotics #Semiosis #SignRelations #TriadicRelations

Peircean Semiotics and Triadic Sign Relations • 1
• https://inquiryintoinquiry.com/2024/08/20/peircean-semiotics-and-triadic-sign-relations-1-a/

As a “guide for the perplexed”, at least when it comes to semiotics, I'll use this thread to collect a budget of resources I think have served to clarify the topic in the past.

By way of a first offering, let me recommend the following most excellent paper, which I can say with all due modesty in light of the fact all its excellence is due to my most excellent co‑author.

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
• https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry

#Peirce #Inquiry #Logic #LogicOfRelatives #RelationTheory
#Semiotics #Semiosis #SignRelations #TriadicRelations

Sign Relations, Triadic Relations, Relation Theory • Discussion 12
• https://inquiryintoinquiry.com/2024/02/24/sign-relations-triadic-relations-relation-theory-discussion-12/

Re: Sign Relations, Triadic Relations, Relation Theory • 1
• https://inquiryintoinquiry.com/2024/02/18/sign-relations-triadic-relations-relation-theory-1-a/

A note from a longtime correspondent points out a search of the available texts turns up no use of the plural form “semiotics” by Peirce and just one place where he uses the plural form “Semeiotics”. That prompts me to make the following excuse for my use or abuse of Peirce's terms, as the case may be.

Peirce has always been one of my chief resources in the quest to understand how logic and math and science work. There is much to be gained by getting his distinctive ideas across to active practitioners in those fields. In doing that I find it better to tweak the words a bit, if that's what it takes to preserve the idea, than to hallow the words at the risk of losing the idea.

As far as semiotics by any name goes, what seems to work best without too much clanging in modern ears is parsing “semiotics” in line with words like “mathematics” and “cybernetics”, plus we can now use the singular form as the adjective “semiotic”.

#Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
#LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

Sign Relations, Triadic Relations, Relation Theory • 4.2
• https://inquiryintoinquiry.com/2024/02/22/sign-relations-triadic-relations-relation-theory-4/

Here's an earlier draft version of Peirce's 1902 definition of a sign, which he gives in the process of defining logic.

Selections from C.S. Peirce, “Carnegie Application” (1902)

❝No. 12. On the Definition of Logic❞ [Earlier Draft]

❝Logic is formal semiotic. A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C. This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time. It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident. The word “formal” in the definition is also defined.❞ (NEM 4, 54).

Reference —

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.
• Online ( https://cspeirce.com/menu/library/bycsp/l75/l75.htm )

#Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
#LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

Sign Relations, Triadic Relations, Relation Theory • 4.1
• https://inquiryintoinquiry.com/2024/02/22/sign-relations-triadic-relations-relation-theory-4/

For ease of reference, here are two variants of Peirce's 1902 definition of a sign, which he gives in the process of defining logic.

Selections from C.S. Peirce, “Carnegie Application” (1902)

❝No. 12. On the Definition of Logic❞

❝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.❞ (NEM 4, 20–21).

Reference —

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.
• Online ( https://cspeirce.com/menu/library/bycsp/l75/l75.htm )

#Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
#LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

Sign Relations, Triadic Relations, Relation Theory • 3
• https://inquiryintoinquiry.com/2024/02/20/sign-relations-triadic-relations-relation-theory-3/

The middle ground between relations in general and the sign relations informing logic, inquiry, and communication is occupied by triadic relations, also called ternary or 3‑place relations.

Triadic relations are some of the most pervasive in mathematics, over and above the importance of sign relations for logic.

For a primer on triadic relations, with examples from mathematics and semiotics, see the article linked below.

Triadic Relations
• OEIS Wiki ( https://oeis.org/wiki/Triadic_relation )
• Wikiversity ( https://en.wikiversity.org/wiki/Triadic_relation )

#Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
#LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

Sign Relations, Triadic Relations, Relation Theory • 2
• https://inquiryintoinquiry.com/2024/02/19/sign-relations-triadic-relations-relation-theory-2/

I always have trouble deciding whether to start with the genus and drive down to the species or begin with concrete examples and accompany Sisyphus up Mt. Abstraction.

To start at the wide end of the funnel, the following article takes up relations in general, focusing on the discrete mathematical varieties we find most useful in applications, for example, as background for empirical data sets and relational data bases.

Relation Theory
• OEIS Wiki ( https://oeis.org/wiki/Relation_theory )
• Wikiversity ( https://en.wikiversity.org/wiki/Relation_theory )

#Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
#LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

Sign Relations, Triadic Relations, Relation Theory • 1
• https://inquiryintoinquiry.com/2024/02/18/sign-relations-triadic-relations-relation-theory-1-a/

To understand how signs work in Peirce's theory of triadic sign relations, or “semiotics”, we have to understand, in order of increasing generality, sign relations, triadic relations, and relations in general, each as conceived in Peirce's logic of relative terms and the corresponding mathematics of relations.

Toward that understanding, here are the current versions of articles I long ago contributed to Wikipedia and Wikiversity and continue to develop at a number of other places.

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

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

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

#Peirce #Logic #Mathematics #ScientificMethod #Semiotics #Semiosis
#LogicOfRelatives #RelationTheory #TriadicRelations #SignRelations

#Semiotics, #Semiosis, #SignRelations • #SurveyPage
• https://inquiryintoinquiry.com/2022/06/26/survey-of-semiotics-semiosis-sign-relations-3/

This is a Survey of blog and wiki resources on the #TheoryOfSigns, also known as #Semiotic or #Semeiotics, and the actions referred to as #Semiosis which transform #Signs among themselves in relation to their #Objects, all as based on C.S. #Peirce's concept of #TriadicSignRelations.

#Logic #RelationTheory #LogicOfRelatives
#TriadicRelationIrreducibility #Semiositis
#Relations #TriadicRelations #TernaryRelations