Tag Archives: Graph Theory

Unknown's avatar Jon Awbrey 12:48 pm on February 3, 2026
Tags: , , , , , , , , , , , , , , , Graph Theory, , Indicator Functions, , , , , , , , Time, Topology,   

Differential Logic • Overview

A reader once told me “venn diagrams are obsolete” and of course we all know how unwieldy they become as our universes of discourse expand beyond four or five dimensions.  Indeed, one of the first lessons I learned when I set about implementing Peirce’s graphs and Spencer Brown’s forms on the computer is that 2‑dimensional representations of logic quickly become death traps on numerous conceptual and computational counts.

Still, venn diagrams do us good service at the outset in visualizing the relationships among extensional, functional, and intensional aspects of logic.  A facility with those connections is critical to the computational applications and statistical generalizations of propositional logic commonly used in mathematical and empirical practice.

All things considered, then, it is useful to make the links between various styles of imagery in logical representation as visible as possible.  The first few steps in that direction are set out in the sketch of Differential Logic to follow.

Part 1

Introduction

Cactus Language for Propositional Logic

Differential Expansions of Propositions

Bird’s Eye View

Worm’s Eye View

Panoptic View • Difference Maps

Panoptic View • Enlargement Maps

Part 2

Propositional Forms on Two Variables

Transforms Expanded over Ordinary and Differential Variables

Enlargement Map Expanded over Ordinary Variables

Enlargement Map Expanded over Differential Variables

Difference Map Expanded over Ordinary Variables

Difference Map Expanded over Differential Variables

Operational Representation

Part 3

Field Picture

Differential Fields

Propositions and Tacit Extensions

Enlargement and Difference Maps

Tangent and Remainder Maps

Least Action Operators

Goal-Oriented Systems

Further Reading

Document History

Document History

Differential Logic • Ontology List 2002

Dynamics And Logic • Inquiry List 2004

Dynamics And Logic • NKS Forum 2004

Resources

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

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #category-theory, #change, #cybernetics, #differential-analytic-turing-automata, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #graph-theory, #hologrammautomaton, #indicator-functions, #inquiry-driven-systems, #leibniz, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #topology, #visualization

Unknown's avatar Jon Awbrey 5:28 pm on January 5, 2026
Tags: , Arithmetic, , Computation, Graph Theory, , , , Number Theory, Recursion, Representation, Riffs and Rotes, ,   

Riffs and Rotes • Happy New Year 2026

\text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}.

\begin{array}{llcl} \text{Then} & 2026 & = & 2 \cdot 1013 \\ && = & p_1 p_{170} \\ && = & p_1 p_{2 \cdot 5 \cdot 17} \\ && = & p_1 p_{p_1 p_3 p_7} \\ && = & p_1 p_{p_1 p_{p_2} p_{p_4}} \\ && = & p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}} \end{array}

No information is lost by dropping the terminal 1s.  Thus we may write the following form.

2026 = p p_{p p_{p_p} p_{p_{p^p}}}

The article linked below tells how forms of that order correspond to a family of digraphs called riffs and a family of graphs called rotes.  The riff and rote for 2026 are shown in the next two Figures.

Riff 2026

Riff 2026

Rote 2026

Rote 2026

Reference

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

#algebra, #arithmetic, #combinatorics, #computation, #graph-theory, #group-theory, #logic, #mathematics, #number-theory, #recursion, #representation, #riffs-and-rotes, #semiotics, #visualization

Unknown's avatar Jon Awbrey 7:16 am on November 20, 2025
Tags: Algorithms, , , , , , , , Data Structures, , , , Graph Theory, , , , , , , , , , , ,   

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

Documentation

Blog Series

Blog Dialogs

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 • Mathstodon • 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

Unknown's avatar Jon Awbrey 11:30 am on November 15, 2025
Tags: , , , , , , Computational Complexity, , , , Graph Theory, , , , , , , , , , Proof Theory, , , , Theorem Proving,   

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

Elements

Examples

Blog Series

  • Logical Graphs • Interpretive Duality • (1) • (2) • (3) • (4)
  • Logical Graphs, Iconicity, Interpretation • (1) • (2)
  • Genus, Species, Pie Charts, Radio Buttons • (1)

Excursions

Applications

Anamnesis

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

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #computational-complexity, #constraint-satisfaction-problems, #differential-logic, #equational-inference, #graph-theory, #group-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

Unknown's avatar Jon Awbrey 11:48 am on November 9, 2025
Tags: , Algebra of Logic, , , , Discrete Mathematics, Duality, Dyadic Relations, , , Graph Theory, , , , , , , , , Set Theory, , , ,   

Survey of Relation Theory • 9

In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications.  This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Elements

Relational Concepts

Relation Composition Relation Construction Relation Reduction
Relative Term Sign Relation Triadic Relation
Logic of Relatives Hypostatic Abstraction Continuous Predicate

Illustrations

Information‑Theoretic Perspective

  • Mathematical Demonstration and the Doctrine of Individuals • (1) • (2)

Blog Series

Peirce’s 1870 “Logic of Relatives”

Peirce’s 1880 “Algebra of Logic” Chapter 3

Peirce’s 1885 “Algebra of Logic”

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

Resources

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

#algebra, #algebra-of-logic, #c-s-peirce, #category-theory, #combinatorics, #discrete-mathematics, #duality, #dyadic-relations, #formal-languages, #foundations-of-mathematics, #graph-theory, #group-theory, #logic, #logic-of-relatives, #logical-graphs, #mathematics, #model-theory, #relation-theory, #semiotics, #set-theory, #sign-relational-manifolds, #sign-relations, #triadic-relations, #type-theory

Unknown's avatar Jon Awbrey 11:20 am on November 7, 2025
Tags: , , , , , , , , , , , , , , Frankl Conjecture, , Gradient Descent, Graph Theory, , , , , , , , ,   

Survey of Differential Logic • 8

This is a Survey of work in progress on Differential Logic, resources under development toward a more systematic treatment.

Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.  To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a differential logical calculus — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Elements

Blog Series

Architectonics

Applications

Blog Dialogs

Explorations

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

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #category-theory, #change, #cybernetics, #differential-analytic-turing-automata, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #frankl-conjecture, #functional-logic, #gradient-descent, #graph-theory, #hologrammautomaton, #inquiry-driven-systems, #leibniz, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #visualization