Inquiry Into Inquiry

Animated Logical Graphs • 55

Posted on January 30, 2021 by Jon Awbrey

WB:: Kauffman’s 2001 piece on “Peirce” (title is “The Mathematics of Charles Sanders Peirce” [PDF]) is IMO fundamental to this discussion.; Here’s a brief excerpt from a piece I did in 2005:
“Boundary Logic and Alpha Existential Graphs (AEG)”; 4.3 LoF and Alpha Graphs Compared

AEG applies the diagrammatic structure of enclosure specifically to logic. The representations of LoF and AEG are isomorphic, while the systems of transformation rules are remarkably close to being the same. …

Dear William,

Many thanks for your excerpt. It highlights many of the most critical points in comparing the systems of Peirce and Spencer Brown so I’ll take up the topic of duality first. Over the years I’ve always found that to be one of the stickier wickets in the whole field. I’ll discuss it in my Animated Logical Graphs series as that’s where I’ve recently redoubled my efforts to explain the issue and why it’s important.

Regards,

Jon

cc: Cybernetics (1) (2) • Laws of Form • FB | Logical Graphs • Ontolog Forum (1) (2)
• Structural Modeling (1) (2) • Systems Science (1) (2)

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | 9 Comments

Animated Logical Graphs • 54

Posted on January 27, 2021 by Jon Awbrey

Re: Peter Cameron • Doing Research
Re: Gil Kalai • Chomskian Linguistics

Oneirocritical Interlude

Speaking of dreams, the night before last I had a dream where I was listening to a lecturer and something he said brought to mind a logical formula having the form $``\text{if if if } a, b, c, d",$ which I visualized as the Peircean logical graph shown below.

I knew I had seen something the day before prompting that fragment and a search through my browser history turned up Gil Kalai’s post on Chomskian Linguistics where I’d read the phrase “anti anti anti missile missile missile missile”.

cc: Cybernetics (1) (2) • Laws of Form • FB | Logical Graphs • Ontolog Forum (1) (2)
• Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) • Structural Modeling (1) (2)
• Systems Science (1) (2)

Charles Sanders Peirce, George Spencer Brown, and Me • 13

Posted on January 26, 2021 by Jon Awbrey

Re: Laws of Form • Dirk Baecker

DB:: I guess you know Fernando Zalamea’s work on Peirce. He thinks that all of GSB’s important ideas are already in Peirce’s Existential Graphs.; I think he may be right, but then there is the issue of elegance, beauty, and clarity, and here, GSB leads the field.

Dear Dirk,

As you may have gleaned from the bio-graphical narrative I’ve been salvaging from the old LoF group, I started down the intertwining Logical Graph / Laws of Form road sometime in the late 60s, all of which took me pretty far along my own eigenvectors before I hit on the work of Zalamea and others of that persuasion in the present millennium.

In most of the things I’ve written in the past about the relative contributions of Peirce and those who came after, Spencer Brown in particular, I tended to give Peirce a lot of credit for anticipating the developments others clarified or brought to fruition. More lately I’ve observed just how bewildered the untutored reader can become when faced with Peirce’s writings on logical graphs and logic generally, so I’ve been rethinking my apportionment of credit. At any rate, I commented on what I thought was added by whom all through my bio essay and I will continue doing that as I go.

Regards,

Jon

cc: Cybernetics • Laws of Form • Ontolog • Peirce • Structural Modeling • Systems Science

Posted in Abstraction, Amphecks, Analogy, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Differential Logic, Duality, Form, Graph Theory, Inquiry, Inquiry Driven Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Peirce, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Sign Relations, Spencer Brown, Theorem Proving, Time, Topology | Tagged Abstraction, Amphecks, Analogy, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Differential Logic, Duality, Form, Graph Theory, Inquiry, Inquiry Driven Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Peirce, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Sign Relations, Spencer Brown, Theorem Proving, Time, Topology | 6 Comments

Charles Sanders Peirce, George Spencer Brown, and Me • 12

Posted on January 25, 2021 by Jon Awbrey

Re: Laws of Form • Dirk Baecker

DB:: Thanks to James Bowery for inviting me to this group. Maybe it is not exactly what I am looking for, since I am interested in a sociological reading of LoF. Which means that I am searching for a mathematics more akin to semantics than to physics. I am still not sure whether in this respect LoF is a wonderful metaphor to understand basic features of an oscillating communication or whether chapter 11 opens up possibilities to compute semantic values starting with imaginary states.

Dear Dirk,

Thanks for this comment and all the links. I’ve been working along the lines of Peirce’s logical graphs and Spencer Brown’s calculus of indications in parallel since those heady early days of the late 60s and Peirce’s semiotics or theory of triadic sign relations is very much a piece of how I understand both. In that light I’ll feel encouraged to share bits of that along with my work on LoF — and now everyone knows who to blame! 😉

Regards,

Jon

cc: Cybernetics • Laws of Form • Ontolog • Peirce • Structural Modeling • Systems Science

Animated Logical Graphs • 53

Posted on January 22, 2021 by Jon Awbrey

Much of the work I do on the C.S. Peirce/Spencer Brown approach to “mathematical hypostases underlying logic” now goes under the heading of Animated Logical Graphs. There’s a Survey of related resources I update from time to time at the following location.

Survey of Animated Logical Graphs

Resources

Charles Sanders Peirce, George Spencer Brown, and Me • 11

Posted on January 18, 2021 by Jon Awbrey

There’s a new Laws of Form group in town. James Bowery et al. have just revived the earlier group on a new platform and everything looks pretty handy so far. There’s an honest-to-goodness 60s vibe about it for me since it’s well-known to those in the know how Spencer Brown’s work builds on Peirce’s, not just because his calculus of indications resurrects aspects of Peirce’s alpha level logical graphs but because the broader scope of his interests touched on inductive reasoning and the whole welter of knotty tangles in the pragmatics of communication, computation, concept formation which Arbib, Ashby, Bateson, Korzybski, R.D. Laing, McCulloch, Peirce, Polanyi, Watzlawick, and others probed in the matrix of quasi-paradoxes and games people play with symbols.

All of which inspires me to revise and extend the series of posts I shared with the old group a few years back, fixing in passing the large number of now broken links.

cc: Cybernetics • Laws of Form • Ontolog • Peirce • Structural Modeling • Systems Science

Sign Relations, Triadic Relations, Relation Theory • Discussion 5

Posted on January 16, 2021 by Jon Awbrey

Re: Cybernetics • Cliff Joslyn
Re: Sign Relations, Triadic Relations, Relation Theory • Discussion (3) (4)

Dear Cliff,

I’m still collecting my wits from the mind-numbing events of the past two weeks so I’ll copy your last remarks here and work through them step by step.

CJ:: I think what you have is sound, and can be described in a number of ways. In years past in seeking ways to both qualify and quantify variety in systems I characterized this distinction as between “dimensional variety” and “cardinal variety”. Thankfully, this seems straightforward from a mathematical perspective, namely in a standard relational system $S = \times_{i=1}^k X_i,$ where the $X_i$ are dimensions (something that can vary), typically cast as sets, so that $\times$ here is Cartesian product.

Relational systems are just the context we need. It is usual to begin at a moderate level of generality by considering a space $X$ of the following form.

$X ~ = ~ \times_{i=1}^k X_i ~ = ~ X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k.$

(I’ll use $X$ instead of $S$ here because I want to save the letter $``S"$ for sign domains when we come to the special case of sign relational systems.)

We can now define a relation $L$ as a subset of a cartesian product.

$L \subseteq X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k.$

There are two common ways of understanding the subset symbol $``\subseteq"$ in this context. Using language from computer science I’ll call them the weak typing and strong typing interpretations.

Under weak typing conventions $L$ is just a set which happens to be a subset of the cartesian product $X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k$ but which could just as easily be cast as a subset of any other qualified superset. The mention of a particular cartesian product is accessory but not necessary to the definition of the relation itself.
Under strong typing conventions the cartesian product $X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k$ in the type-casting $L \subseteq X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k$ is an essential part of the definition of $L.$ Employing a conventional mathematical idiom, a $k$ -adic relation over the nonempty sets $X_1, X_2, \ldots, X_{k-1}, X_k$ is defined as a $(k+1)$ -tuple $(L, X_1, X_2, \ldots, X_{k-1}, X_k)$ where $L$ is a subset of $X_1 \times X_2 \times \ldots \times X_{k-1} \times X_k.$

We have at this point opened two fronts of interest in cybernetics, namely, the generation of variety and the recognition of constraint. There’s more detail on this brand of relation theory in the resource article linked below. I’ll be taking the strong typing approach to relations from this point on, largely because it comports more naturally with category theory and thus enjoys ready applications to systems and their transformations.

But my eye-brain system is going fuzzy on me now, so I’ll break here and continue later …

Resources

cc: Cybernetics • Ontolog • Peirce List (1) (2) (3) • Structural Modeling • Systems Science

Posted in C.S. Peirce, Icon Index Symbol, Information, Inquiry Driven Systems, Logic, Logic of Relatives, Mathematics, Peirce, Pragmatism, Relation Theory, Semiosis, Semiotics, Sign Relations, Triadic Relations, Triadicity, Visualization | Tagged C.S. Peirce, Icon Index Symbol, Information, Inquiry Driven Systems, Logic, Logic of Relatives, Mathematics, Peirce, Pragmatism, Relation Theory, Semiosis, Semiotics, Sign Relations, Triadic Relations, Triadicity, Visualization | 7 Comments

Icon Index Symbol • 20

Posted on January 11, 2021 by Jon Awbrey

Questions Concerning Certain Faculties Claimed For Signs

Re: FB | Semiotics, Books, Links, News • Jon Awbrey • Dalibor Lošťák

JA:: Icon, Index, Symbol and all other classifications are ideal types abstracted from concrete signs and there are no pure types in actual existence. However, it is a consequence of triadic relation irreducibility that symbols are in a genuine sense the generic type while icons and indices are specializations or so-called “degenerate” cases.

DL:: I think the first sentence answers the question brilliantly. However, I disagree with your assertion about the “degenerate cases”. It is my understanding that iconicity is the aspect of a sign that represents its Firstness, which is incapable of degeneracy. This also leads me to the notion that fully degenerate Thirdness, as applied to a Symbol, is not an Icon. I would be very interested to read your thoughts on this.

The way I see Categories applying to Peirce’s logic and semiotics may be gleaned from the following Survey page.

Survey of Precursors Of Category Theory

The series beginning with the following post might be a good place to start.

Peirce’s Categories • 1

cc: Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in Abduction, Algorithms, Animata, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognition, Computation, Data Structures, Deduction, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Objective Frameworks, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged Abduction, Algorithms, Animata, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognition, Computation, Data Structures, Deduction, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Objective Frameworks, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | 7 Comments

Icon Index Symbol • 19

Posted on January 10, 2021 by Jon Awbrey

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List • Steven Skaggs

SS:: As far as visual semiotics is concerned, it is helpful to think of a “terrain” or map of a semantic mode territory in which of icon-index-symbol form the points of the triangular map, or gamut.; Visual entities (and we can extend this beyond that sense modality but that is what I do most of my work in) can be plotted on this gamut. For instance, an extremely legible typographic word would be in the upper right corner, while an illegible scribble would appear down near the bottom. But expressive calligraphy, or perhaps a graffiti tag, might well occupy somewhere in between. The late works of Paul Klee would mostly be placed in the center. A passport photograph the upper left corner, etc.

cc: Peirce List (1) (2) (3) (4) (5) (6) (7)

Icon Index Symbol • 18

Posted on January 8, 2021 by Jon Awbrey

Questions Concerning Certain Faculties Claimed For Signs

Re: FB | Semiotics, Books, Links, News • Muntadher Almahdawi

Another one of those recurring questions just came up in a Facebook group devoted to Semiotics and I thought it would be useful to try my hand at a fresh attempt to answer it — or at least promote further discussion.

MA: Can index become symbol? Why or why not?

Icon, Index, Symbol and all other classifications are ideal types abstracted from concrete signs and there are no pure types in actual existence. However, it is a consequence of triadic relation irreducibility that symbols are in a genuine sense the generic type while icons and indices are specializations or so-called degenerate cases.

cc: Peirce List (1) (2) (3) (4) (5) (6) (7)

	Objects, Models, The… on Survey of Abduction, Deduction…
	Objects, Models, The… on Objects, Models, Theories •…
	Objects, Models, The… on Objects, Models, Theories •…
	Objects, Models, The… on Objects, Models, Theories •…
	Survey of Abduction,… on Objects, Models, Theories •…
	Abou Youssef on Objects, Models, Theories •…
	Objects, Models, The… on Survey of Abduction, Deduction…
	Objects, Models, The… on Objects, Models, Theories •…
	Objects, Models, The… on Objects, Models, Theories •…
	Objects, Models, The… on Objects, Models, Theories •…

Inquiry Into Inquiry

Animated Logical Graphs • 55

4.3 LoF and Alpha Graphs Compared

Animated Logical Graphs • 54

Charles Sanders Peirce, George Spencer Brown, and Me • 13

Charles Sanders Peirce, George Spencer Brown, and Me • 12

Animated Logical Graphs • 53

Resources

Charles Sanders Peirce, George Spencer Brown, and Me • 11

Sign Relations, Triadic Relations, Relation Theory • Discussion 5

Resources

Icon Index Symbol • 20

Questions Concerning Certain Faculties Claimed For Signs

Icon Index Symbol • 19

Questions Concerning Certain Faculties Claimed For Signs

Icon Index Symbol • 18

Questions Concerning Certain Faculties Claimed For Signs

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