Inquiry Into Inquiry

Survey of Precursors Of Category Theory • 6

Posted on May 5, 2025 by Jon Awbrey

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. A Survey of resources on the topic is given below, still very rough and incomplete, but perhaps a few will find it of use.

Background

Blog Series

Notes On Categories • (1)
Precursors Of Category Theory • (1) • (2) • (3) • (4) • (5) • (6)

Discussions • (1) • (2) • (3)

Categories à la Peirce

C.S. Peirce • A Guess at the Riddle

Peirce’s Categories • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21)

C.S. Peirce and Category Theory • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)

cc: FB | Peirce Matters • Laws of Form • Ma^thstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Abstraction, Ackermann, Analogy, Aristotle, C.S. Peirce, Carnap, Category Theory, Foundations of Mathematics, Hilbert, Hypostatic Abstraction, Kant, Logic, Mathematics, Propositions As Types Analogy, Relation Theory, Saunders Mac Lane, Semiotics, Type Theory, Universals | Tagged Abstraction, Ackermann, Analogy, Aristotle, C.S. Peirce, Carnap, Category Theory, Foundations of Mathematics, Hilbert, Hypostatic Abstraction, Kant, Logic, Mathematics, Propositions As Types Analogy, Relation Theory, Saunders Mac Lane, Semiotics, Type Theory, Universals | 2 Comments

Survey of Pragmatic Semiotic Information • 9

Posted on May 4, 2025 by Jon Awbrey

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 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

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 • Ma^thstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Abduction, C.S. Peirce, Communication, Control, Cybernetics, Deduction, Determination, Discovery, Doubt, Epistemology, Fixation of Belief, Induction, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretation, Invention, Knowledge, Learning Theory, Logic, Logic of Relatives, Logic of Science, Mathematics, Philosophy of Science, Pragmatic Information, Probable Reasoning, Process Thinking, Relation Theory, Scientific Inquiry, Scientific Method, Semeiosis, Semiosis, Semiotic Information, Semiotics, Sign Relational Manifolds, Sign Relations, Surveys, Triadic Relations, Uncertainty, Visualization | Tagged Abduction, C.S. Peirce, Communication, Control, Cybernetics, Deduction, Determination, Discovery, Doubt, Epistemology, Fixation of Belief, Induction, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretation, Invention, Knowledge, Learning Theory, Logic, Logic of Relatives, Logic of Science, Mathematics, Philosophy of Science, Pragmatic Information, Probable Reasoning, Process Thinking, Relation Theory, Scientific Inquiry, Scientific Method, Semeiosis, Semiosis, Semiotic Information, Semiotics, Sign Relational Manifolds, Sign Relations, Surveys, Triadic Relations, Uncertainty, Visualization | 6 Comments

Survey of Inquiry Driven Systems • 7

Posted on May 4, 2025 by Jon Awbrey

This is a Survey of work in progress on Inquiry Driven Systems, material I plan to refine toward a more compact and 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.

Anthem

The object of reasoning is to find out …

Elements

Background

Blog Series

Pragmatic Cosmos • (1)

In the Way of Inquiry

Pragmatic Theory Of Truth • (1) • (2) • (3) • (4) • (5) • (6)

Discussions • (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)

Blog Dialogs

Inquiry Driven Systems

Comments • (1) • (2) • (3) • (4) • (5) • (6)
Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

Inquiry Into Inquiry

On Balance
On Initiative • (1) • (2) • (3) • (4) • (5)
In Medias Res • Flash Back
Understanding • (1) • (2)
Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

Architectonics of Inquiry • (1)

Higher Order Sign Relations • (1)

Discussions • (1) • (2) • (3) • (4) • (5)

Developments

Applications

Conceptual Barriers to Creating Integrative Universities
(Abstract) (Online)
Interpretation as Action • The Risk of Inquiry
(Journal) (doc) (pdf)
An Architecture for Inquiry • Building Computer Platforms for Discovery
(Online)
Exploring Research Data Interactively • Theme One : A Program of Inquiry
(Online)

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

Posted in Abduction, Adaptive Systems, Analogy, Animata, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognitive Science, Cybernetics, Deduction, Educational Systems Design, Educational Technology, Fixation of Belief, Induction, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Interpretation, Logic, Logic of Science, Mathematics, Mental Models, Pragmatic Maxim, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged Abduction, Adaptive Systems, Analogy, Animata, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognitive Science, Cybernetics, Deduction, Educational Systems Design, Educational Technology, Fixation of Belief, Induction, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Interpretation, Logic, Logic of Science, Mathematics, Mental Models, Pragmatic Maxim, Semiotics, Sign Relations, Triadic Relations, Visualization | 8 Comments

Survey of Differential Logic • 8

Posted on May 3, 2025 by Jon Awbrey

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

Differential Propositional Calculus

Part 1 • Part 2 • Appendices • References

Differential Logic • Part 1 • Part 2 • Part 3

Differential Logic and Dynamic Systems

Part 1 • Part 2 • Part 3 • Part 4 • Part 5
Appendices • References

Blog Series

Differential Logic • The Logic of Change and Difference

Differential Logic • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18)

Comments • (1) • (2) • (3) • (4) • (5) • (6) • (7)
Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16)

Differential Propositional Calculus • (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)

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

Differential Logic and Dynamic Systems

Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7)

Differential Logic, Dynamic Systems, Tangent Functors • (1)

Comments • (1) • (2)
Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

Explorations

Frankl Conjecture • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)

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

Posted in 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 | Tagged 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 | Leave a comment

Survey of Definition and Determination • 4

Posted on May 3, 2025 by Jon Awbrey

In the early 1990s, “in the middle of life’s journey” as the saying goes, I returned to grad school in a systems engineering program with the idea of taking a more systems-theoretic approach to my development of Peircean themes, from signs and scientific inquiry to logic and information theory.

Two of the first questions calling for fresh examination were the closely related concepts of definition and determination, not only as Peirce used them in his logic and semiotics but as researchers in areas as diverse as computer science, cybernetics, physics, and systems science would find themselves forced to reconsider the concepts in later years. That led me to collect a sample of texts where Peirce and a few other writers discuss the issues of definition and determination. There are copies of those selections at the following sites.

Collection Of Source Materials

What follows is a Survey of blog and wiki posts on Definition and Determination, with a focus on the part they play in Peirce’s interlinked theories of signs, information, and inquiry. In classical logical traditions the concepts of definition and determination are closely related and their bond acquires all the more force when we view the overarching concept of constraint from an information-theoretic point of view, as Peirce did beginning in the 1860s.

Blog Dialogs

Definition and Determination • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21) • (22)

Readings On Determination • (1)

Discussion • (1) • (2) • (3) • (4)

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

Posted in C.S. Peirce, Comprehension, Constraint, Definition, Determination, Extension, Form, Indication, Information, Information = Comprehension × Extension, Inquiry Driven Systems, Logic, Mathematics, Scientific Method, Semiotics, Sign Relations, Structure, Systems Theory, Visualization | Tagged C.S. Peirce, Comprehension, Constraint, Definition, Determination, Extension, Form, Indication, Information, Information = Comprehension × Extension, Inquiry Driven Systems, Logic, Mathematics, Scientific Method, Semiotics, Sign Relations, Structure, Systems Theory, Visualization | Leave a comment

Survey of Cybernetics • 5

Posted on May 2, 2025 by Jon Awbrey

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

This is a Survey of blog posts relating to Cybernetics. It includes the selections from Ashby’s Introduction and the comment on them I’ve posted so far, plus two series of reflections on the governance of social systems in light of cybernetic and semiotic principles.

Anthem

Ouch❢

Ashby’s Introduction to Cybernetics

Chapter 10 • Regulation In Biological Systems

Selections • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10)

Discussions • (1) • (2) • (3) • (4)

Chapter 11 • Requisite Variety

Selections • (11) • (12) • (13)

Blog Series

Basal Ingredients Of Society (BIOS)

Prologue • Prescription

Comments • (1) • (2) • (3) • (4) • (5) • (6) • (7)

Theory and Therapy of Representations • (1) • (2) • (3) • (4) • (5)

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

Posted in Abduction, C.S. Peirce, Communication, Control, Cybernetics, Deduction, Determination, Discovery, Doubt, Epistemology, Fixation of Belief, Induction, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretation, Invention, Knowledge, Learning Theory, Logic, Logic of Relatives, Logic of Science, Mathematics, Peirce, Philosophy, Philosophy of Science, Pragmatic Information, Probable Reasoning, Process Thinking, Relation Theory, Scientific Inquiry, Scientific Method, Semeiosis, Semiosis, Semiotic Information, Semiotics, Sign Relational Manifolds, Sign Relations, Surveys, Triadic Relations, Uncertainty | Tagged Abduction, C.S. Peirce, Communication, Control, Cybernetics, Deduction, Determination, Discovery, Doubt, Epistemology, Fixation of Belief, Induction, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretation, Invention, Knowledge, Learning Theory, Logic, Logic of Relatives, Logic of Science, Mathematics, Peirce, Philosophy, Philosophy of Science, Pragmatic Information, Probable Reasoning, Process Thinking, Relation Theory, Scientific Inquiry, Scientific Method, Semeiosis, Semiosis, Semiotic Information, Semiotics, Sign Relational Manifolds, Sign Relations, Surveys, Triadic Relations, Uncertainty | 1 Comment

Survey of Animated Logical Graphs • 8

Posted on May 2, 2025 by Jon Awbrey

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)

Logic Syllabus

Discussions • (1) • (2)

Logical Graphs

Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11)

Logical Graphs • Interpretive Duality • (1) • (2) • (3) • (4)

Logical Graphs, Iconicity, Interpretation • (1) • (2)

Logical Graphs, Truth Tables, Venn Diagrams • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

Minimal Negation Operators • (1) • (2) • (3) • (4)

Discussions • (1) • (2)

Genus, Species, Pie Charts, Radio Buttons • (1)

Discussions • (1) • (2) • (3) • (4) • (5)

Excursions

Futures Of Logical Graphs
Operator Variables in Logical Graphs • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)

Discussions • (1) • (2)

Interpretive Duality in Logical Graphs • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)
Mathematical Duality in Logical Graphs • (1)

Discussions • (1) • (2)

Transformations of Logical Graphs • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14)

Discussions • (1)

Applications

Anamnesis

CSP, GSB, & Me • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20)

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

Posted in 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 | Tagged 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 | 77 Comments

Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 5

Posted on May 1, 2025 by Jon Awbrey

This is a Survey of blog and wiki posts on three elementary forms of inference, as recognized by a logical tradition extending from Aristotle through Charles S. Peirce. Particular attention is paid to the way the inferential rudiments combine to form the more complex patterns of analogy and inquiry.

Anthem

Tenacity, Authority, Plausibility, Inquiry

Blog Dialogs

Abduction, Deduction, Induction, Analogy, Inquiry • (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)

Blog Series

Functional Logic • Inquiry and Analogy • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21)

Discussions • (1)

Blog Surveys

OEIS Wiki

Ontolog Forum

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

Posted in Abduction, Aristotle, C.S. Peirce, Deduction, Dewey, Discovery, Doubt, Fixation of Belief, Functional Logic, Icon Index Symbol, Induction, Inference, Information, Inquiry, Invention, Logic, Logic of Science, Mathematics, Morphism, Paradigmata, Paradigms, Pattern Recognition, Peirce, Philosophy, Pragmatic Maxim, Pragmatism, Scientific Inquiry, Scientific Method, Semiotics, Sign Relations, Surveys, Syllogism, Triadic Relations, Visualization | Tagged Abduction, Aristotle, C.S. Peirce, Deduction, Dewey, Discovery, Doubt, Fixation of Belief, Functional Logic, Icon Index Symbol, Induction, Inference, Information, Inquiry, Invention, Logic, Logic of Science, Mathematics, Morphism, Paradigmata, Paradigms, Pattern Recognition, Peirce, Philosophy, Pragmatic Maxim, Pragmatism, Scientific Inquiry, Scientific Method, Semiotics, Sign Relations, Surveys, Syllogism, Triadic Relations, Visualization | Leave a comment

Cactus Language • Preliminaries 9

Posted on April 25, 2025 by Jon Awbrey

We now have the materials in place to formulate a definition of our subject.

The painted cactus language with paints in the set $\mathfrak{P} = \{ p_j : j \in J \}$ is the formal language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*$ defined as follows.

$\begin{array}{ll} \text{PC 1.} & \text{The blank symbol}~ m_1 ~\text{is a sentence.} \\ \text{PC 2.} & \text{The paint}~ p_j ~\text{is a sentence for each}~ j ~\text{in}~ J. \\ \text{PC 3.} & \mathrm{Conc}^0 ~\text{and}~ \mathrm{Surc}^0 ~\text{are sentences.} \\ \text{PC 4.} & \text{For each positive integer}~ n, \\ & \text{if}~ s_1, \ldots, s_n ~\text{are sentences} \\ & \text{then}~ \mathrm{Conc}_{k=1}^n s_k ~\text{is a sentence} \\ & \text{and}~ \mathrm{Surc}_{k=1}^n s_k ~\text{is a sentence.} \end{array}$

In the idiom of formal language theory, a string $s$ is called a sentence of $\mathfrak{L}$ if and only if it belongs to $\mathfrak{L},$ or simply a sentence if the language $\mathfrak{L}$ is understood. A sentence of $\mathfrak{C} (\mathfrak{P})$ is referred to as a painted and rooted cactus expression on the palette $\mathfrak{P},$ or a cactus expression for short.

Resources

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

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | 9 Comments

Cactus Language • Preliminaries 8

Posted on April 22, 2025 by Jon Awbrey

Defining the basic operations of concatenation and surcatenation on arbitrary strings gives them operational meaning for the all‑inclusive language $\mathfrak{L} = \mathfrak{A}^*.$ With that in hand it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.

If $\mathfrak{L}$ is an arbitrary formal language over an alphabet of the type we have been discussing, that is, an alphabet of the form $\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},$ then there are a number of basic structural relations which can be defined on the strings of $\mathfrak{L}.$

Concatenation: $s$ is the concatenation of $s_1$ and $s_2$ in $\mathfrak{L}$
if and only if
$s_1$ is a sentence of $\mathfrak{L},$ $s_2$ is a sentence of $\mathfrak{L},$
and
$s = s_1 \cdot s_2$; $s$ is the concatenation of the $k$ strings $s_1, \ldots, s_k$ in $\mathfrak{L}$
if and only if
$s_j$ is a sentence of $\mathfrak{L}$ for all $j = 1 \ldots k$
and
$s = \mathrm{Conc}_{j=1}^k s_j = s_1 \cdot \ldots \cdot s_k$

Discatenation: $s$ is the discatenation of $s_1$ by $t$
if and only if
$s_1$ is a sentence of $\mathfrak{L},$ $t$ is an element of $\mathfrak{A},$
and
$s_1 = s \cdot t$
in which case we more commonly write
$s = s_1 \cdot t^{-1}$

Subclause: $s$ is a subclause of $\mathfrak{L}$
if and only if
$s$ is a sentence of $\mathfrak{L}$
and
$s$ ends with a $``\text{)}"$

Subcatenation: $s$ is the subcatenation of $s_1$ by $s_2$
if and only if
$s_1$ is a subclause of $\mathfrak{L},$ $s_2$ is a sentence of $\mathfrak{L},$
and
$s = s_1 \cdot (``\text{)}")^{-1} \cdot ``\text{,}" \cdot s_2 \cdot ``\text{)}"$

Surcatenation: $s$ is the surcatenation of the $k$ strings $s_1, \ldots, s_k$ in $\mathfrak{L}$
if and only if
$s_j$ is a sentence of $\mathfrak{L}$ for all ${j = 1 \ldots k}$
and
$s = \mathrm{Surc}_{j=1}^k s_j = ``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot \ldots \cdot ``\text{,}" \cdot s_k \cdot ``\text{)}"$

The converses of the above decomposition relations amount to the corresponding composition operations. As complementary forms of analysis and synthesis they make it possible to articulate the structures of strings and sentences in two directions.

Resources

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

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