Jon Awbrey 9:44 am on May 3, 2026
Tags: Abstraction ( 3 ), Amphecks ( 25 ), Analogy ( 4 ), Animata ( 27 ), Automated Research Tools ( 4 ), Boolean Algebra ( 25 ), Boolean Functions ( 26 ), C.S. Peirce ( 51 ), Cactus Graphs ( 26 ), Deduction ( 6 ), Differential Logic ( 26 ), Duality ( 3 ), Form ( 3 ), Graph Theory ( 28 ), Iconicity ( 2 ), Laws of Form ( 4 ), Leibniz ( 4 ), logic ( 53 ), Logical Graphs ( 27 ), Mathematics ( 52 ), Minimal Negation Operators ( 26 ), Model Theory ( 4 ), Peirce's Law ( 2 ), Praeclarum Theorema ( 2 ), Proof Theory ( 3 ), Propositional Calculus ( 26 ), Propositional Logic ( 2 ), Semiotics ( 28 ), Spencer Brown, Theorem Proving ( 3 ), Topology ( 3 ), Visualization ( 38 )

Animated Logical Graphs • 2

Re: Peirce List • Jim Willgoose

It’s almost 50 years now since I first encountered the volumes of Peirce’s Collected Papers in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown’s Laws of Form in the Whole Earth Catalog and I sent off for it right away. I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph‑theoretic data structures formed on their ideas to good purpose as the basis of its reasoning engine.

I thought it might contribute to a number of long‑running and ongoing discussions if I could articulate what I think I learned from that experience.

So I’ll try to keep focused on that.

Resources

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

#abstraction, #amphecks, #analogy, #animata, #automated-research-tools, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #deduction, #differential-logic, #duality, #form, #graph-theory, #iconicity, #laws-of-form, #leibniz, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #model-theory, #peirces-law, #praeclarum-theorema, #proof-theory, #propositional-calculus, #propositional-logic, #semiotics, #spencer-brown, #theorem-proving, #topology, #visualization

Jon Awbrey 10:10 am on April 30, 2026
Tags: Abstraction ( 3 ), Amphecks ( 25 ), Analogy ( 4 ), Animata ( 27 ), Automated Research Tools ( 4 ), Boolean Algebra ( 25 ), Boolean Functions ( 26 ), C.S. Peirce ( 51 ), Cactus Graphs ( 26 ), Deduction ( 6 ), Differential Logic ( 26 ), Duality ( 3 ), Form ( 3 ), Graph Theory ( 28 ), Iconicity ( 2 ), Laws of Form ( 4 ), Leibniz ( 4 ), logic ( 53 ), Logical Graphs ( 27 ), Mathematics ( 52 ), Minimal Negation Operators ( 26 ), Model Theory ( 4 ), Peirce's Law ( 2 ), Praeclarum Theorema ( 2 ), Proof Theory ( 3 ), Propositional Calculus ( 26 ), Propositional Logic ( 2 ), Semiotics ( 28 ), Spencer Brown, Theorem Proving ( 3 ), Topology ( 3 ), Visualization ( 38 )

Animated Logical Graphs • 1

For Your Musement …

Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce’s Alpha Graphs for propositional logic.

Proof Animations

Double Negation

Peirce’s Law

Praeclarum Theorema

Two‑Thirds Majority Function

A full discussion of logical graphs can be found in the following article.

Logical Graphs

Additional Resources

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

Jon Awbrey 7:16 am on November 20, 2025
Tags: Algorithms, Animata ( 27 ), Artificial Intelligence ( 2 ), Automated Research Tools ( 4 ), Boolean Functions ( 26 ), C.S. Peirce ( 51 ), Cactus Graphs ( 26 ), Constraint Satisfaction Problems ( 2 ), Data Structures, Differential Logic ( 26 ), Equational Inference ( 24 ), Formal Languages ( 2 ), Graph Theory ( 28 ), Inquiry Driven Systems ( 33 ), Laws of Form ( 4 ), Learning Theory ( 3 ), logic ( 53 ), Logical Graphs ( 27 ), Mathematics ( 52 ), Minimal Negation Operators ( 26 ), Painted Cacti ( 2 ), Propositional Calculus ( 26 ), Propositional Equation Reasoning Systems ( 2 ), Spencer Brown, Visualization ( 38 )

Survey of Theme One Program • 7

This is a Survey of resources relating to the Theme One Program I worked on all through the 1980s. The aim was to develop fundamental algorithms and data structures for integrating empirical learning with logical reasoning. I had earlier developed separate programs for basic components of those tasks, in particular, two‑level formal language learning and propositional constraint satisfaction, the latter using an extension of C.S. Peirce’s logical graphs as a syntax for propositional logic. Thus arose the question of how well it might be possible to get “empiricist” and “rationalist” modes of operation to cooperate. The long‑term vision is the implementation of an Automated Research Tool able to double as a platform for Inquiry Driven Education.

Wiki Hub

Theme One Program • Overview

Documentation

Blog Series

Theme One Program

Motivation • (1) • (2) • (3) • (4) • (5) • (6)
Exposition • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)
Discussion • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10)
Applications, Examples, Exercises

Jets and Sharks • (1) • (2) • (3)

Blog Dialogs

Theme One • A Program Of Inquiry • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20)

Applications

References

Awbrey, S.M., and Awbrey, J.L. (May 1991), “An Architecture for Inquiry • Building Computer Platforms for Discovery”, Proceedings of the Eighth International Conference on Technology and Education, Toronto, Canada, pp. 874–875. Online.
Awbrey, J.L., and Awbrey, S.M. (January 1991), “Exploring Research Data Interactively • Developing a Computer Architecture for Inquiry”, Poster presented at the Annual Sigma Xi Research Forum, University of Texas Medical Branch, Galveston, TX.
Awbrey, J.L., and Awbrey, S.M. (August 1990), “Exploring Research Data Interactively • Theme One : A Program of Inquiry”, Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training, Society for Applied Learning Technology, Washington, DC, pp. 9–15. Online.

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#algorithms, #animata, #artificial-intelligence, #automated-research-tools, #boolean-functions, #c-s-peirce, #cactus-graphs, #constraint-satisfaction-problems, #data-structures, #differential-logic, #equational-inference, #formal-languages, #graph-theory, #inquiry-driven-systems, #laws-of-form, #learning-theory, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #painted-cacti, #propositional-calculus, #propositional-equation-reasoning-systems, #spencer-brown, #visualization

Jon Awbrey 11:30 am on November 15, 2025
Tags: Amphecks ( 25 ), Animata ( 27 ), Boolean Algebra ( 25 ), Boolean Functions ( 26 ), C.S. Peirce ( 51 ), Cactus Graphs ( 26 ), Computational Complexity, Constraint Satisfaction Problems ( 2 ), Differential Logic ( 26 ), Equational Inference ( 24 ), Graph Theory ( 28 ), Group Theory ( 3 ), Laws of Form ( 4 ), logic ( 53 ), Logical Graphs ( 27 ), Mathematics ( 52 ), Minimal Negation Operators ( 26 ), Model Theory ( 4 ), Painted Cacti ( 2 ), Peirce ( 3 ), Proof Theory ( 3 ), Propositional Calculus ( 26 ), Propositional Equation Reasoning Systems ( 2 ), Spencer Brown, Theorem Proving ( 3 ), Visualization ( 38 )

Survey of Animated Logical Graphs • 8

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph‑theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

Beginnings

Logical Graphs • First Impressions

Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14)

Logical Graphs • Formal Development

Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)

Elements

Examples

Double Negation • (1) • (2) • (3)

Peirce’s Law

This Blog • OEIS Wiki
Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7)

Praeclarum Theorema

This Blog • OEIS Wiki
Blog Series • (1) • (2) • (3)

Proof Animations

Blog Series

Alpha Now, Omega Later • (1) • (2) • (3) • (4) • (5) • (6) • (7)

Animated Logical Graphs • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21) • (22) • (23) • (24) • (25) • (26) • (27) • (28) • (29) • (30) • (31) • (32) • (33) • (34) • (35) • (36) • (37) • (38) • (39) • (40) • (41) • (42) • (43) • (44) • (45) • (46) • (47) • (48) • (49) • (50) • (51) • (52) • (53) • (54) • (55) • (56) • (57) • (58) • (59) • (60) • (61) • (62) • (63) • (64) • (65) • (66) • (67) • (68) • (69) • (70) • (71) • (72) • (73) • (74) • (75) • (76) • (77) • (78) • (79) • (80) • (81)

Cactus Language

Overview • (1) • (2) • (3) • (4)
Preliminaries • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17)
Syntax • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)
Generalities About Formal Grammars • (1) • (2) • (3)
Stylistics • (1) • (2) • (3) • (4) • (5) • (6)
Pragmatics • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16)
Mechanics • (1) • (2) • (3) • (4) • (5)
Semantics • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)
Discussions • (1) • (2) • (3)