#inquirydrivensystems — Public Fediverse posts

Objects, Models, Theories • 4
• https://inquiryintoinquiry.com/2026/09/18/objects-models-theories-4-a/

What are objects, models, theories, and how do they relate to one another?

Recurring questions about the relationship between objects, models, and theories take on a different aspect when viewed from the perspective of Peirce's pragmatic semiotic, in other words, when cast within a framework of triadic sign relations.

In contemplating the array of questions which come to mind I always find it helpful to ruminate on the diagram shown below — I might even call it a “mandala” for its wealth of symbolic features and its aid in organizing the pro‑&‑con‑fusion of mental impressions.

Aristotle's “Paradigm”
• https://inquiryintoinquiry.com/wp-content/uploads/2013/11/aristotles-paradigm.jpg

Here is the corresponding text from Aristotle and the context that leads on to Peirce's viewpoint.

Aristotle's “Paradigm” • Reasoning by Analogy or Example
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Aristotle's_Paradigm

Resources —

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
• https://inquiryintoinquiry.com/2025/05/01/survey-of-abduction-deduction-induction-analogy-inquiry-5/

#Peirce #Logic #Mathematics #Semiotics #Inference #Inquiry #InformationTheory
#Abduction #Deduction #Induction #MathematicalModels #ModelTheory #MentalModels
#Analogy #AdaptiveSystems #InquiryDrivenSystems #LearningTheory #TriadicRelations

Objects, Models, Theories • 2
• https://inquiryintoinquiry.com/2026/09/12/objects-models-theories-2-a/

Re: Gödel's Lost Letter • The Graph Of Math
• https://rjlipton.com/2013/11/15/the-graph-of-math/

GLL:
❝Kurt Gödel is said to have been a latecomer to appreciating the power of Model Theory. He was of course the greatest architect of Proof Theory, which stands in contrast to Model Theory. Model Theory concerns itself with what could be true, while Proof Theory deals with what can be proved. The latter sounds more definite, but they are supplementary: a statement is capable of being true somewhere precisely when its negation cannot be proved. The question is, where is that somewhere? And when?❞

What — if anything — is the common sense that connects the different senses of the word “model”, as it has been used over the years in logic, mathematics, and the special sciences? It's a problem I've been running into for several decades now and I think I can trace the roots of it going back as far as Aristotle’s treatment of analogy.

Resources —

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
• https://inquiryintoinquiry.com/2025/05/01/survey-of-abduction-deduction-induction-analogy-inquiry-5/

#Peirce #Logic #Mathematics #Semiotics #Inference #Inquiry #InformationTheory
#Abduction #Deduction #Induction #MathematicalModels #ModelTheory #MentalModels
#Analogy #AdaptiveSystems #InquiryDrivenSystems #LearningTheory #TriadicRelations

Objects, Models, Theories • 1
• https://inquiryintoinquiry.com/2026/09/10/objects-models-theories-1-a/

Happy Birthday, Charles Sanders Peirce❢ — September 10, 1839

I return once more to a recurring subject.

Re: Artem Kaznatcheev • Three Types of Mathematical Models
• https://egtheory.wordpress.com/2013/09/08/mathematical-models/

In speaking of models one tends to find denizens of different disciplines talking at cross purposes to one another. Logicians use the word to describe what may be distinguished as “logical models”, saying a model is whatever satisfies a theory — anything a theory holds true of — and that is the sense used in the logical subject of “model theory”.

Almost everyone else uses the word to describe what may be called “analogical models”, analogues being things holding enough properties in common with other things that learning about Thing 2 (the analogue system) can teach us about Thing 1 (the object system).

It is actually quite easy to integrate those two senses of the word “model” into a coherent picture of the whole situation, namely, the triadic relationship among objects, analogues, and theories. We'll get into that further as the discussion proceeds.

Resources —

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
• https://inquiryintoinquiry.com/2025/05/01/survey-of-abduction-deduction-induction-analogy-inquiry-5/

cc: https://www.academia.edu/community/VDDKwZ
cc: https://www.researchgate.net/post/Objects_Models_Theories

#Peirce #Logic #Mathematics #Semiotics #Inference #Inquiry #InformationTheory
#Abduction #Deduction #Induction #MathematicalModels #ModelTheory #MentalModels
#Analogy #AdaptiveSystems #InquiryDrivenSystems #LearningTheory #TriadicRelations

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

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

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

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

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.

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.

For example, the factorial function has a definition in terms of the predecessor function and the multiplier function

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.

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

Differential Logic • 18

Tangent and Remainder Maps

If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition in the following way.

The next venn diagram shows the differential proposition we get by extracting the linear approximation to the difference map at each cell or point of the universe   What results is the logical analogue of what would ordinarily be called the differential of but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for whenever it’s necessary to single it out.

To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.

To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.

Capping the analysis of the proposition in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map which happens to be linear in pairs of variables.

Reading the arrows off the map produces the following data.

In short, is a constant field, having the value at each cell.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

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

Differential Logic • 17

Enlargement and Difference Maps

Continuing with the example the following venn diagram shows the enlargement or shift map in the same style of field picture we drew for the tacit extension

A very important conceptual transition has just occurred here, almost tacitly, as it were.  Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields and both of the type is very useful, because it allows us to consider those fields as integral mathematical objects which can be operated on and combined in the ways we usually associate with algebras.

In the present case one notices the tacit extension and the enlargement are in a sense dual to each other.  The tacit extension indicates all the arrows out of the region where is true and the enlargement indicates all the arrows into the region where is true.  The only arc they have in common is the no‑change loop at   If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of shown in the following venn diagram.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

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

Differential Logic • 15

Differential Fields

The structure of a differential field may be described as follows.  With each point of there is associated an object of the following type:  a proposition about changes in that is, a proposition   In that frame of reference, if is the universe generated by the set of coordinate propositions then is the differential universe generated by the set of differential propositions   The differential propositions and may thus be interpreted as indicating and respectively.

A differential operator of the first order type we are currently considering, takes a proposition and gives back a differential proposition   In the field view of the scene, we see the proposition as a scalar field and we see the differential proposition as a vector field, specifically, a field of propositions about contemplated changes in

The field of changes produced by on is shown in the following venn diagram.

The differential field specifies the changes which need to be made from each point of in order to reach one of the models of the proposition that is, in order to satisfy the proposition

The field of changes produced by on is shown in the following venn diagram.

The differential field specifies the changes which need to be made from each point of in order to feel a change in the felt value of the field

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

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

Differential Logic • 14

Field Picture

Let us summarize the outlook on differential logic we’ve reached so far.  We’ve been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse to considering a larger universe of discourse   An operator of that general type, namely, acts on each proposition of the source universe to produce a proposition of the target universe

The operators we’ve examined so far are the enlargement or shift operator and the difference operator   The operators and act on propositions in that is, propositions of the form which amount to propositions about the subject matter of and they produce propositions of the form which amount to propositions about specified collections of changes conceivably occurring in

At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and help us keep our wits about us as we venture into ever more rarefied airs of abstraction.

One good picture comes to us by way of the field concept.  Given a space a field of a specified type over is formed by associating with each point of an object of type   If that sounds like the same thing as a function from to the space of things of type — it is nothing but — and yet it does seem helpful to vary the mental images and take advantage of the figures of speech most naturally springing to mind under the emblem of the field idea.

In the field picture a proposition becomes a scalar field, that is, a field of values in

For example, consider the logical conjunction shown in the following venn diagram.

Each of the operators takes us from considering propositions here viewed as scalar fields over to considering the corresponding differential fields over analogous to what in real analysis are usually called vector fields over

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

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

Differential Logic • 13

Transforms Expanded over Ordinary and Differential Variables

Two views of how the difference operator acts on the set of sixteen functions are shown below.  Table A5 shows the expansion of over the set of ordinary variables and Table A6 shows the expansion of over the set of differential variables.

Difference Map Expanded over Ordinary Variables

Difference Map Expanded over Differential Variables

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

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

Survey of Pragmatic Semiotic Information • 9

This is a Survey of blog and wiki posts on a theory of information which grows out of pragmatic semiotic ideas.  All my projects are exploratory in character but this line of inquiry is more open‑ended than most.  The question is —

What is information and how does it impact the spectrum of activities answering to the name of inquiry?

Setting out on what would become his lifelong quest to explore and explain the “Logic of Science”, C.S. Peirce pierced the veil of historical confusions obscuring the issue and fixed on what he called the “laws of information” as the key to solving the puzzle.

The first hints of the Information Revolution in our understanding of scientific inquiry may be traced to Peirce’s lectures of 1865–1866 at Harvard University and the Lowell Institute.  There Peirce took up “the puzzle of the validity of scientific inference” and claimed it was “entirely removed by a consideration of the laws of information”.

Fast forward to the present and I see the Big Question as follows.  Having gone through the exercise of comparing and contrasting Peirce’s theory of information, however much it yet remains in a rough‑hewn state, with Shannon’s paradigm so pervasively informing the ongoing revolution in our understanding and use of information, I have reason to believe Peirce’s idea is root and branch more general and has the potential, with due development, to resolve many mysteries still bedeviling our grasp of inference, information, and inquiry.

Inference, Information, Inquiry

• OEIS Wiki • Information = Comprehension × Extension

• Blog Series 2026 • Information = Comprehension × Extension • Preamble

• Blog Series 2024 • Information = Comprehension × Extension • Preamble
• Selections • (1) • (2) • (3) • (4) • (5) • (6)
• Comments • (1) • (2) • (3) • (4) • (5) • (6) • (7)

• Blog Series 2019 • { Information = Comprehension × Extension } • Revisited
• Selections • (1) • (2) • (3) • (4) • (5) • (6)
• Comments • (1) • (2) • (3) • (4) • (5)
• Discussions • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17)

• Blog Series 2016 • { Information = Comprehension × Extension }
• Selections • (1) • (2) • (3) • (4) • (5) • (6)
• Comments • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)
• Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7)

Pragmatic Semiotic Information

• Pragmatic Semiotic Information • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)
• Comments • (1) • (2) • (3)
• Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21)

Semiotics, Semiosis, Sign Relations

• Semeiotic • OEIS Wiki • Wikiversity
• Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

Sign Relations, Triadic Relations, Relation Theory

• Blog Series • (1)
• Discusssions • (1) • (2)

Excursions

• Pragmatic Semiotic Information
• Peirce’s Logic Of Information

Blog Dialogs

• Semiotic Theory Of Information • (1) • (2) • (3) • (4) • (5) • (6)
• Icon Index Symbol • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17)
• Sign Relations • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)
• Sign Relations, Triadic Relations, Relations • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11)

References

• Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”.  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).

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

Higher Order Sign Relations • Discussion 1
• https://inquiryintoinquiry.com/2025/03/27/higher-order-sign-relations-discussion-1-a/

Re: FB | Charles S. Peirce Society • John Corcoran
• https://www.facebook.com/groups/peircesociety/posts/1768975423238442/

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 —

John Corcoran
• https://johncorcoran.academia.edu/

Schemata : The Concept of Schema in the History of Logic
• https://www.academia.edu/12691868/SCHEMATA_THE_CONCEPT_OF_SCHEMA_IN_THE_HISTORY_OF_LOGIC

Use And Mention, Use Without Mention, Mention Without Use
• https://www.academia.edu/s/ea64a3484e/schemata#comment_525151

Resources —

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

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 #Inquiry #Logic #Mathematics #Reflection
#Semiotics #SignRelations #HigherOrderSignRelations
#InquiryDrivenSystems #ReflectiveInterpretiveFrameworks
#Arithmetization #GödelNumbers #Quotation #UseAndMention

Higher Order Sign Relations • 1
• https://inquiryintoinquiry.com/2025/03/22/higher-order-sign-relations-1-a/

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 —

Inquiry Driven Systems
• 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

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

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 #Inquiry #Logic #Mathematics #Reflection
#Semiotics #SignRelations #HigherOrderSignRelations
#InquiryDrivenSystems #ReflectiveInterpretiveFrameworks

Theory and Therapy of Representations • 5
• https://inquiryintoinquiry.com/2025/02/25/theory-and-therapy-of-representations-5-b/

Re: R.J. Lipton and K.W. Regan • Legal Complexity
• https://rjlipton.com/2022/09/04/legal-complexity/

❝I do not pretend to understand the moral universe;
the arc is a long one, my eye reaches but little ways;
I cannot calculate the curve and complete the figure by
the experience of sight; I can divine it by conscience.
And from what I see I am sure it bends towards justice.❞

🙞 Theodore Parker
• https://web.archive.org/web/20200302045624/https://books.google.com/books?id=eHgYAAAAYAAJ&pg=PA48#v=onepage&q&f=false

The arc of the moral universe may bend toward justice — there's hope it will.
For the logic of laws to converge on justice may take some doing on our part.

Resources —

Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7/

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

#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention

Theory and Therapy of Representations • 4
• https://inquiryintoinquiry.com/2025/02/24/theory-and-therapy-of-representations-4-b/

Re: Ontolog Forum • Paola Di Maio
• https://groups.google.com/g/ontolog-forum/c/Ek_7cCCyFkQ/m/qI0kQv4UAgAJ

JA: What are the forces distorting our representations of what's observed, what's expected, and what's intended?

PDM: The short answer is — the force behind all distortions is our own unenlightened mind, and all the shortfalls this comes with.

I think that's true, we have to keep reflecting on the state of our personal enlightenments. If we can do that without losing our heads and our systems thinking caps, there will be much we can do to promote the general Enlightenment of the State.

On both personal and general grounds we have a stake in the projects of self‑governing systems — whether it is possible for them to exist and what it takes for them to thrive in given environments. Systems on that order have of course been studied from many points of view and at many levels of organization. Whether we address them under the names of adaptive, cybernetic, error-correcting, intelligent, or optimal control systems they all must be capable to some degree of learning, reasoning, and self‑guidance.

Resources —

Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7/

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

#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention

Theory and Therapy of Representations • 3.2
• https://inquiryintoinquiry.com/2025/02/23/theory-and-therapy-of-representations-3-b/

Scene 2. Theory and Therapy of Representations • 1
• https://inquiryintoinquiry.com/2025/02/21/theory-and-therapy-of-representations-1-b/

Statistics were originally the data a ship of state needed for stationkeeping and staying on course. The Founders of the United States, like the Cybernauts of the Enlightenment they were, engineered a ship of state with checks and balances and error-controlled feedbacks for the sake of representing both reality and the will of the people. In that connection Max Weber saw how a state's accounting systems are intended as representations of realities its crew and passengers must observe or perish.

That brings us to Question 2 —

• What are the forces distorting our representations of what's observed, what's expected, and what's intended?

Resources ─

Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/

Pragmatic Theory Of Truth
• https://oeis.org/wiki/Pragmatic_Theory_Of_Truth

#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention

Theory and Therapy of Representations • 3.1
• https://inquiryintoinquiry.com/2025/02/23/theory-and-therapy-of-representations-3-b/

Representation is a concept we find at the intersection of cybernetics, epistemology, logic, mathematics, psychology, and sociology. In my studies it led me from math to psych and back again, with sidelong glances at the history of democratic governance. Its time come round again, I find myself returning to the scenes of two recurring questions.

Scene 1. Pragmatic Truth • Discussion 18
• https://inquiryintoinquiry.com/2019/11/14/pragmatic-truth-discussion-18/

We do not live in axiom systems. We do not live encased in languages, formal or natural. There is no reason to think we will ever have exact and exhaustive theories of what's out there, and the truth, as we know, is “out there”. Peirce understood there are more truths in mathematics than are dreamt of in logic — and Gödel’s realism should have put the last nail in the coffin of logicism — but some ways of thinking just never get a clue.

That brings us to Question 1 —

• What are formalisms and all their embodiments in brains and computers good for?

Resources ─

Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/

Pragmatic Theory Of Truth
• https://oeis.org/wiki/Pragmatic_Theory_Of_Truth

#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention

Theory and Therapy of Representations • 2
• https://inquiryintoinquiry.com/2025/02/22/theory-and-therapy-of-representations-2-b/

In a complex society, people making decisions and taking actions at places remote from you have the power to affect your life in significant ways. Those people govern your life, they are your government, no matter what spheres of influence they inhabit, private or public. The only way you get a choice in that governance is if there are paths of feedback permitting you to affect the life of those decision makers and action takers in significant ways. That is what accountability, response-ability, and representative government are all about.

Naturally, some people are against that.

In the United States there has been a concerted campaign for as long as I can remember — but even more concerted since the Reagan Regime — to get the People to abdicate their hold on The Powers That Be and just let some anonymous corporate entity send us the bill after the fact. They keep trying to con the People into thinking they can starve the beast, to limit government, when what they are really doing is feeding the beast of corporate control, weakening their own power over the forces that govern their lives.

That is the road to perdition as far as responsible government goes. There is not much of anything one leader or one administration can do unsupported if the People do not constantly demand a government of, by, and for the People.

Resource ─

Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/

#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention

Theory and Therapy of Representations • 1
• https://inquiryintoinquiry.com/2025/02/21/theory-and-therapy-of-representations-1-b/

❝Again, in a ship, if a man were at liberty to do what he chose, but were devoid of mind and excellence in navigation (αρετης κυβερνητικης), do you perceive what must happen to him and his fellow sailors?❞

─ Plato • Alcibiades 135 A

Statistics were originally the data a ship of state needed for stationkeeping and staying on course. The Founders of the United States, like the Cybernauts of the Enlightenment they were, engineered a ship of state with checks and balances and error‑controlled feedbacks for the sake of representing both reality and the will of the people. In that connection Max Weber saw how a state's accounting systems are intended as representations of realities its crew and passengers must observe or perish.

The question for our time is —

• What are the forces distorting our representations of what's observed, what's expected, and what's intended?

Repercussions ─

The Place Where Three Wars Meet
• https://inquiryintoinquiry.com/2012/06/21/the-place-where-three-wars-meet/

Resource ─

Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/

#AdaptiveSystems #Cybernetics #SystemsTheory #Governance #Democracy
#Plato #Peirce #MaxWeber #Accountability #Representation #Statistics
#Inquiry #InquiryDrivenSystems #Observation #Expectation #Intention

Basal Ingredients Of Society • ℞
• https://inquiryintoinquiry.com/2025/02/20/basal-ingredients-of-society-%e2%84%9e-a/

❝THE SOCIAL COMPACT❞

❝If then we discard from the social compact what is not of its essence, we shall find that it reduces itself to the following terms:

❝“Each of us puts his person and all his power in common under the supreme direction of the general will, and, in our corporate capacity, we receive each member as an indivisible part of the whole.”❞

Reference —

Jean Jacques Rousseau, “The Social Contract”, G.D.H. Cole (trans.), Great Books of The Western World, Volume 38.

#AdaptiveSystems #Cybernetics #Democracy #Governance
#Inquiry #InquiryDrivenSystems #LearningOrganizations
#Reciprocity #Rousseau #SocialCompact #Sustainability

Basal Ingredients Of Society • Prologue
• https://inquiryintoinquiry.com/2025/02/18/basal-ingredients-of-society-prologue-a/

I settled on the acronym BIOS to suggest the vital elements of life in society, a life in association with others, and not just any association but one whose flickers of life are sustained for more than a few vicissitudes of history. Sustainability in that life requires democracy, a society based on a distinctive form of social compact.

• https://www.academia.edu/community/54MZbO
• https://bsky.app/profile/inquiryintoinquiry.bsky.social/post/3lihriaerw224
• https://www.researchgate.net/post/Basal_Ingredients_Of_Society

#AdaptiveSystems #Cybernetics #Democracy #Governance
#Inquiry #InquiryDrivenSystems #LearningOrganizations
#Reciprocity #Rousseau #SocialCompact #Sustainability

Differential Propositional Calculus • 10

Special Classes of Propositions (cont.)

Let’s pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general.  We can do this by recruiting our visual imaginations and drawing up a sufficient budget of venn diagrams for each family of propositions.  The case for 3 variables is exemplary enough for a start.

Linear Propositions

The linear propositions, may be written as sums:

One thing to keep in mind about these sums is that the values in are added “modulo 2”, that is, in such a way that

In a universe of discourse based on three boolean variables, the linear propositions take the shapes shown in Figure 8.

At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following three forms.

Next are the venn diagrams for the three linear propositions of rank 2, which may be expressed by the following three forms, respectively.

Next are the three linear propositions of rank 1, which are none other than the three basic propositions,

At the bottom is the linear proposition of rank 0, the everywhere false proposition or the constant function, which may be expressed by the form or by a simple

Resources

• Logic Syllabus
• Survey of Differential Logic

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

#Amphecks #Animata #BooleanAlgebra #BooleanFunctions #CSPeirce #CactusGraphs #CategoryTheory #Change #Cybernetics #DifferentialAnalyticTuringAutomata #DifferentialCalculus #DifferentialLogic #DiscreteDynamics #EquationalInference #FunctionalLogic #GraphTheory #Hologrammautomaton #IndicatorFunctions #InquiryDrivenSystems #Leibniz #Logic #LogicalGraphs #Mathematics #MinimalNegationOperators #PropositionalCalculus #Time #Topology #Visualization

Pragmatic Truth • 3.2
• https://inquiryintoinquiry.com/2024/07/12/pragmatic-truth-3/

Truth Predicates —

One of the first questions to be asked in this setting concerns the relationship between the significant performance and its reflective critique. If one expresses oneself in a particular fashion, and someone says “that’s true”, is there anything useful at all to be said in general terms about the relationship between those two acts? For instance, does the critique add value to the expression criticized, does it say something significant in its own right, or is it but an insubstantial echo of the original sign?

Resources —

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

Pragmatic Maxim
• https://inquiryintoinquiry.com/2023/08/07/pragmatic-maxim-a/

Truth Theory
• https://oeis.org/wiki/Truth_theory

Pragmatic Theory Of Truth • Document History
• https://oeis.org/wiki/Pragmatic_Theory_Of_Truth
• https://oeis.org/wiki/Pragmatic_Theory_Of_Truth#Document_history

Correspondence Theory Of Truth
• https://oeis.org/wiki/Correspondence_Theory_Of_Truth

#Peirce #Logic #Inquiry #Truth #PragmaticTruth #TruthTheory #Semiotics #Meaning
#JohnDewey #WilliamJames #PragmaticMaxim #Pragmatism #InquiryDrivenSystems

Pragmatic Truth • 3.1
• https://inquiryintoinquiry.com/2024/07/12/pragmatic-truth-3/

Truth Predicates —

An inquiry into the character of truth generally begins with the idea of an informative, meaningful, or significant element, the goodness of whose information, meaning, or significance may be put in question and needs to be evaluated. Depending on context, the element may be called an artefact, expression, image, impression, lyric, mark, performance, picture, sentence, sign, string, symbol, text, thought, token, utterance, word, work, and so on. However that may be, one has the task of judging whether the bearers of information, meaning, or significance are indeed truth‑bearers or not. That judgment is typically expressed in the form of a specific “truth predicate”, whose positive application to a sign, or so on, asserts the truth of the sign.

Considered within the broadest horizon, there is little reason to imagine the process of judging a work, which leads to a predication of false or true, is necessarily amenable to formalization, and that task may always remain what is commonly called a judgment call. But there are many well-circumscribed domains where it is useful to consider disciplined forms of evaluation and the observation of those limits allows for the institution of what is called a “method” of judging truth and falsity.

Resources —

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

Pragmatic Maxim
• https://inquiryintoinquiry.com/2023/08/07/pragmatic-maxim-a/

Truth Theory
• https://oeis.org/wiki/Truth_theory

Pragmatic Theory Of Truth
• https://oeis.org/wiki/Pragmatic_Theory_Of_Truth

Correspondence Theory Of Truth
• https://oeis.org/wiki/Correspondence_Theory_Of_Truth

#Peirce #Logic #Inquiry #Truth #PragmaticTruth #TruthTheory
#InquiryDrivenSystems #PragmaticMaxim #Pragmatism #Semiotics

Pragmatic Truth • 2
• https://inquiryintoinquiry.com/2024/07/10/pragmatic-truth-2/

Truth as the Good of Logic —

Pragmatic theories of truth enter on a stage set by the philosophies of former ages, with special reference to the Ancient Greeks, the Scholastics, and Immanuel Kant. Recalling a few elements of that background can provide valuable insight into the play of ideas as they have developed up through our time. Because pragmatic ideas about truth are often confused with a number of quite distinct notions it is useful say a few words about those other theories and to highlight the points of significant contrast.

In one classical formulation, truth is defined as the good of logic, where logic is classed as a normative science, in other words, an inquiry into a good or value which seeks to arrive at knowledge of it and the means to achieve it. In that view, truth cannot be discussed to much effect outside the context of inquiry, knowledge, and logic, all very broadly conceived.

Resources —

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

Pragmatic Maxim
• https://inquiryintoinquiry.com/2023/08/07/pragmatic-maxim-a/

Truth Theory
• https://oeis.org/wiki/Truth_theory

Pragmatic Theory Of Truth • Document History
• https://oeis.org/wiki/Pragmatic_Theory_Of_Truth
• https://oeis.org/wiki/Pragmatic_Theory_Of_Truth#Document_history

Correspondence Theory Of Truth
• https://oeis.org/wiki/Correspondence_Theory_Of_Truth

#Peirce #Logic #Inquiry #Truth #PragmaticTruth #TruthTheory #Semiotics #Meaning
#JohnDewey #WilliamJames #PragmaticMaxim #Pragmatism #InquiryDrivenSystems

Constraints and Indications • 2
• https://inquiryintoinquiry.com/2024/07/04/constraints-and-indications-2-a/

Coping with collaboration, communication, context, integration, interoperability, perspective, purpose, and the reality of the information dimension demands a transition from conceptual environments bounded by dyadic relations to those informed by triadic relations, especially the variety of triadic sign relations employed by pragmatic semiotics.

Along the lines of my first post on this topic I am presently concerned with the logical and mathematical requirements of dealing with constraints but when it comes to the constraints involved in communicating across cultural and disciplinary barriers I could recommend a paper Susan Awbrey and I wrote for a conference devoted to those very issues.

Conference Presentation —

Awbrey, S.M., and Awbrey, J.L. (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.
• https://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm

Published Paper —

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.
• https://journals.sagepub.com/doi/abs/10.1177/1350508401082013
• https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities

#Peirce #Logic #Semiotics #Indication #Inference #Information #Inquiry
#Ashby #Cybernetics #Constraint #Control #Regulation #RequisiteVariety
#AdaptiveSystems #IntelligentSystems #InquiryDrivenSystems #Pragmatics

Constraints and Indications • 1.2
• https://inquiryintoinquiry.com/2024/07/02/constraints-and-indications-1-a/

Re: Peirce List • Kaina Stoicheia and the Symbol Grounding Problem
• https://web.archive.org/web/20121231182129/http://comments.gmane.org/gmane.science.philosophy.peirce/9165

Reference —

• Ashby, W.R. (1956), Introduction to Cybernetics, Methuen, London, UK.

Resources —

Survey of Cybernetics
• https://inquiryintoinquiry.com/2024/01/25/survey-of-cybernetics-4/

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

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

#Peirce #Ashby #Cybernetics #InquiryDrivenSystems #PragmaticSemioticInformation
#Constraint #ControlSystems #Indication #Information #Inquiry #IntelligentSystems
#AdaptiveSystems #ErrorControlledRegulation #Feedback #SystemsTheory #Uncertainty
#Determination #Intentionality #Learning #Reasoning #SemioticInformation #Semiotics

Constraints and Indications • 1.1
• https://inquiryintoinquiry.com/2024/07/02/constraints-and-indications-1-a/

The system‑theoretic concept of “constraint” is one that unifies a manifold of other notions — definition, determination, habit, information, law, predicate, regularity, and so on. Indeed, it is often the best way to understand the entire complex of concepts.

Entwined with the concept of “constraint” is the concept of “information”, the power signs bear to reduce uncertainty and advance inquiry. Asking what consequences those ideas have for Peirce’s theory of triadic sign relations led me some years ago to the thoughts recorded on the following page.

Pragmatic Semiotic Information
• https://oeis.org/wiki/Pragmatic_Semiotic_Information

Here I am thinking of the concept of constraint that constitutes one of the fundamental ideas of classical cybernetics and mathematical systems theory.

For example, here is how W. Ross Ashby introduces the concept of constraint in his Introduction to Cybernetics (1956).

❝A most important concept, with which we shall be much concerned later, is that of “constraint”. It is a relation between two sets, and occurs when the variety that exists under one condition is less than the variety that exists under another. Thus, the variety of the human sexes is 1 bit; if a certain school takes only boys, the variety in the sexes within the school is zero; so as 0 is less than 1, constraint exists.❞ (1964 ed., p. 127).

At its simplest, then, constraint is an aspect of the subset relation.

The objective of an agent, organism, or similar regulator is to keep within its viable region, a particular subset of its possible state space. That is the constraint of primary interest to the agent.

#Peirce #Ashby #Cybernetics #InquiryDrivenSystems

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

This is a Survey of work in progress on Inquiry Driven Systems, material I plan to refine toward a more systematic treatment of the subject.

An “inquiry driven system” is a system having among its state variables some representing its state of information with respect to various questions of interest, for example, its own state and the states of potential object systems. Thus it has a component of state tracing a trajectory though an “information state space”.

Please follow the above link for the full set of resources.
Articles treating the more central ideas are linked below.

Elements —

Prospects for Inquiry Driven Systems
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems

Introduction to Inquiry Driven Systems
• https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems

Background —

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

Functional Logic • Quantification Theory
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Quantification_Theory

Functional Logic • Higher Order Propositions
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Higher_Order_Propositions

Developments —

Inquiry Driven Systems • Inquiry Fields
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Inquiry_Fields

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

Applications —

Conceptual Barriers to Creating Integrative Universities
• https://www.academia.edu/1266492/Conceptual_Barriers_to_Creating_Integrative_Universities

Interpretation as Action • The Risk of Inquiry
• https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry

An Architecture for Inquiry • Building Computer Platforms for Discovery
• https://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery

Exploring Research Data Interactively • Theme One : A Program of Inquiry
• https://www.academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry

#Peirce #Inquiry #InquiryDrivenSystems #InformationStateSpace
#Semiotics #SignRelations #Interpretation #SystemsEngineering
#Abduction #Deduction #Induction #Analogy #DynamicsOfInquiry

Icon Index Symbol • 15

Questions Concerning Certain Faculties Claimed For Signs

I put down the cup and turn to my mind.  It is up to my mind to find the truth.  But how?  What grave uncertainty, whenever the mind feels overtaken by itself;  when it, the seeker, is also the obscure country where it must seek and where all its baggage will be nothing to it.  Seek?  Not only that:  create.  It is face to face with something that does not yet exist and that only it can accomplish, and bring into its light.

— Proust • In Search of Lost Time

Re: Peirce List (1) (2) • Tom Gollier • Jerry Rhee • John Sowa

That passage from Proust epitomizes for me one of the most distinctive features of the inquiry process, the fact that its object is a state of information that the inquisiturus, the agent of inquiry, may never have known before.

I have thought of inquiry and intelligence in terms of cybernetic or system-theoretic processes ever since my first encounters with the works of Arbib, Ashby, Bateson, McCulloch, Wiener, Young, and others during my undergrad years.  In the early 90s I returned to grad school in a Systems Engineering program with the idea that I might be able to string together many loose threads of unfinished business that continued to tug at my brain.  Here’s a bit I wrote at the outset of that project, that comes to mind in this context:

Prospects for Inquiry Driven Systems • Architecture of Inquiry

It is important to remember that knowledge is a different sort of goal from the run-of-the-mill setpoints a system might have.  The typical goal is a state that a system has actually experienced many times before, like normal body temperature for a human being.  But a particular state of knowledge an intelligent system moves toward may be a state it has never been through before.  The fundamental equivocation on this point expressed in Plato’s Meno, whether learning is functionally equivalent to remembering, was discussed above.  In spite of this quibble, it still seems necessary to regard states of knowledge as a distinctive class.  The reasons for this may lie in the fact that a useful definition of inquiry for human beings necessarily involves a whole community of inquiry.

Resources

• Prospects for Inquiry Driven Systems
• Introduction to Inquiry Driven Systems
• Inquiry Driven Systems • Inquiry Into Inquiry

cc: Peirce List (1) (2)

Icon Index Symbol • 14

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) • Helmut Raulien
Re: Icon Index Symbol • (10) (11) (12) (13)

Let me sum up the main points of the above exchange before moving on.

Mathematics is useful in our present endeavor because it treats relations in general.  In addition — and multiplication, too — mathematics is chock full of well‑studied examples of triadic relations.  When it comes to the job of analyzing sign relations and teasing out their structures we can save ourselves a lot of trouble and trial and error by examining this record of prior art and adapting its methods to handle sign relations.

On the other hand, there are hints in Peirce’s work of how triadic relations extend across a threshold of complexity in such a way that relations of all higher adicities can be analyzed in terms of 1‑adic, 2‑adic, and 3‑adic relations.  This is where the analogy between Peirce’s category theory and mathematical category theory both forms and breaks.  The utility of categories in mathematical theory derives from the ubiquity of functions in mathematical practice, and functions are dyadic relations.  Still, triadic relations pervade the background of mathematical category theory, being visible in the triadic composition relation and in the concept of “natural transformations”, the clarification of which is one of the original motivations for the subject.  Bringing the triadic roots of category theory into higher relief is one of my motives for bringing about an encounter with Peirce’s categories, an effort to which I have given not a few years of thought.

That brings us to the case of sign relations proper.  These types of triadic relations form our first stepping stones and also our first stumbling blocks in the inquiry into inquiry and I gave some indications why that might be true.  I don’t know if I can do better than that at this time, but I’ll think on it further after that all‑essential secondness of caffeination.

Resources

• Sign Relations
• Triadic Relations
• Relation Theory
• Peirce’s Logic Of Information
• Precursors Of Category Theory
• Information = Comprehension × Extension
• Peirce’s 1870 Logic Of Relatives • Overview
• Peirce’s 1870 Logic Of Relatives • Preliminaries

cc: Peirce List (1) (2) (3)

Readings On Determination

This is an anchor post for a set of readings on determination, taking a census of the senses Peirce used in his pragmatic approach to information, inquiry, science, and signs with a pinch of readings from other writers for historical and contemporary context.

I will start with a set of readings I collected some twenty years ago.  Once that groundwork is laid down, there is a set of ideas coming from the relational programming paradigm in computer science I think would mesh nicely with Peirce’s theory of information and could be extended to cover the triadic relations of his semiotics.

Resource

• Collection Of Source Materials

cc: Peirce List