The Difference That Makes A Difference That Peirce Makes • 22

Posted on September 19, 2019 by Jon Awbrey

Peirce Society Facebook Page • JC • JA • JA • JA • JC

A discussion — well, more like a series of posts and counterposts — arose last week on the Facebook Page of the Charles S. Peirce Society, and I’ve been going back over it this week because it seemed to invite a useful re-examination of some old but important issues. There appears to be some sort of disagreement, or maybe just failure to communicate, but I’m still having trouble putting my finger on what the source of the issue might be.

One factor seems to be different understandings about the relationship between Peirce’s brand of semiotics and standard first order logic. One thing I’ve noticed before is that people who view Peirce’s work through the filter of first order logic are not likely to see what many of us appreciate in his semiotic approach to logic. There are commentators on Peirce’s logical systems who treat them as nothing more than first order logics in other syntaxes, but I am not one of those. There is something more general and powerful going on with Peirce’s conception of “logic as formal semiotic”, in other words, a normative science of signs.

I still see that factor playing a role in the background of the animadversion but I’m beginning to think there’s probably a much simpler explanation.

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

Posted in Analogy, C.S. Peirce, Communication, Descriptive Science, Fixation of Belief, Formal Systems, Information, Inquiry, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Normative Science, Paradigms, Peirce, Pragmatic Maxim, Pragmatism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Triadicity | Tagged Analogy, C.S. Peirce, Communication, Descriptive Science, Fixation of Belief, Formal Systems, Information, Inquiry, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Normative Science, Paradigms, Peirce, Pragmatic Maxim, Pragmatism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Triadicity | Leave a comment

The Difference That Makes A Difference That Peirce Makes • 21

Posted on September 18, 2019 by Jon Awbrey

Re: Ontolog Forum • John Bottoms
Re: The Difference That Makes A Difference That Peirce Makes : 20

The reflections in my previous blog post developed over several weeks observing various discussions around the web where people seemed to be spending most of their effort talking past each other and hardly ever getting any ideas or information out of one skull and into another. It’s not the first time I’ve noticed belief systems, comfort zones, conceptual silos, paradigms, whatever we call them, acting like immune systems, insulating our mental metabolisms from intellectual antigens.

Anyhow, it’s a working hypothesis to prime future inquiry …

As far as Peirce references go, the choices are legion, so I’ll just link to one place I have in mind at the moment where Peirce sets out a number of truly radical ideas, ones I view as missed opportunities — so far as I know he never fully followed up on them.

Peirce’s 1870 Logic Of Relatives

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

The Difference That Makes A Difference That Peirce Makes • 20

Posted on September 17, 2019 by Jon Awbrey

Cross-paradigm communication, like cross-disciplinary and cross-cultural communication, can be difficult. Sometimes people do not even recognize the existence of other paradigms, disciplines, cultures, long before it comes to the question of their value. Readers of Peirce know he often uses important words in more primordial senses than later came into fashion. Other times his usage embodies a distinct analysis of the concept in question. More than once I’ve found myself remarking how Peirce “anticipates” some strikingly “modern” idea in logic, mathematics, or science, only to find its roots lay deep in the history of thought. Whether he anticipates a future sense or preserves an ancient sense is not always easy to answer.

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

Riffs and Rotes • 4

Posted on September 16, 2019 by Jon Awbrey

Prompted by a recent discussion of prime numbers and complex dynamics on one of the Santa Fe Institute’s FaceBook pages, I posted a link to an old project of mine, going back to a time when I was first learning programming in college and working as an orderly in a hospital x-ray department. Something about the collision of those influences in the medium of my gray matter led me to see curious connections among self-documenting programs, self-indexing data structures, and molecular tagging. Shortly afterwards a couple of Mathematical Games columns by Martin Gardner started me thinking about Gödel numbers and links among graph theory, logic, and number theory.

At any rate, here’s a report on what came of that —

Riffs and Rotes

Posted in Algebra, Combinatorics, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Riffs and Rotes | Tagged Algebra, Combinatorics, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Riffs and Rotes | Leave a comment

Differential Logic and Dynamic Systems • Overview

Posted on September 10, 2019 by Jon Awbrey

In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade‑off between dynamic paradigms and symbolic paradigms. Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system’s state through time. Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system’s description or an agent’s state of information. Logic‑based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus. The work laid out in this report is intended to address that lack.

This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms. The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self‑contained work, to serve as a conceptual frame and a technical background for the network project.

Part 1

Review and Transition

A Functional Conception of Propositional Calculus

Qualitative Logic and Quantitative Analogy

Philosophy of Notation : Formal Terms and Flexible Types

Special Classes of Propositions

Basis Relativity and Type Ambiguity

The Analogy Between Real and Boolean Types

Theory of Control and Control of Theory

Propositions as Types and Higher Order Types

Reality at the Threshold of Logic

Tables of Propositional Forms

A Differential Extension of Propositional Calculus

Differential Propositions : Qualitative Analogues of Differential Equations

An Interlude on the Path

The Extended Universe of Discourse

Intentional Propositions

Life on Easy Street

Part 2

Back to the Beginning : Exemplary Universes

A One-Dimensional Universe

Example 1. A Square Rigging

Back to the Feature

Tacit Extensions

Example 2. Drives and Their Vicissitudes

Part 3

Transformations of Discourse

Foreshadowing Transformations : Extensions and Projections of Discourse

Extension from 1 to 2 Dimensions

Extension from 2 to 4 Dimensions

Thematization of Functions : And a Declaration of Independence for Variables

Thematization : Venn Diagrams

Thematization : Truth Tables

Propositional Transformations

Alias and Alibi Transformations

Transformations of General Type

Analytic Expansions : Operators and Functors

Operators on Propositions and Transformations

Differential Analysis of Propositions and Transformations

The Secant Operator : E

The Radius Operator : e

The Phantom of the Operators : η

The Chord Operator : D

The Tangent Operator : T

Part 4

Transformations of Discourse (cont.)

Transformations of Type B² → B¹

Analytic Expansion of Conjunction

Tacit Extension of Conjunction

Enlargement Map of Conjunction

Digression : Reflection on Use and Mention

Difference Map of Conjunction

Differential of Conjunction

Remainder of Conjunction

Summary of Conjunction

Analytic Series : Coordinate Method

Analytic Series : Recap

Terminological Interlude

End of Perfunctory Chatter : Time to Roll the Clip!

Operator Maps : Areal Views

Operator Maps : Box Views

Operator Diagrams for the Conjunction J = uv

Part 5

Transformations of Discourse (concl.)

Taking Aim at Higher Dimensional Targets

Transformations of Type B² → B²

Logical Transformations

Local Transformations

Difference Operators and Tangent Functors

Epilogue, Enchoiry, Exodus

Appendices

Appendix 1. Propositional Forms and Differential Expansions

Table A1. Propositional Forms on Two Variables

Table A2. Propositional Forms on Two Variables

Table A3. Ef Expanded Over Differential Features

Table A4. Df Expanded Over Differential Features

Table A5. Ef Expanded Over Ordinary Features

Table A6. Df Expanded Over Ordinary Features

Appendix 2. Differential Forms

Table A7. Differential Forms Expanded on a Logical Basis

Table A8. Differential Forms Expanded on an Algebraic Basis

Table A9. Tangent Proposition as Pointwise Linear Approximation

Table A10. Taylor Series Expansion Df = df + d²f

Table A11. Partial Differentials and Relative Differentials

Table A12. Detail of Calculation for the Difference Map

Appendix 3. Computational Details

Operator Maps for the Logical Conjunction f₈(u, v)

Computation of εf₈

Computation of Ef₈

Computation of Df₈

Computation of df₈

Computation of rf₈

Computation Summary for Conjunction

Operator Maps for the Logical Equality f₉(u, v)

Computation of εf₉

Computation of Ef₉

Computation of Df₉

Computation of df₉

Computation of rf₉

Computation Summary for Equality

Operator Maps for the Logical Implication f₁₁(u, v)

Computation of εf₁₁

Computation of Ef₁₁

Computation of Df₁₁

Computation of df₁₁

Computation of rf₁₁

Computation Summary for Implication

Operator Maps for the Logical Disjunction f₁₄(u, v)

Computation of εf₁₄

Computation of Ef₁₄

Computation of Df₁₄

Computation of df₁₄

Computation of rf₁₄

Computation Summary for Disjunction

Appendix 4. Source Materials

Appendix 5. Various Definitions of the Tangent Vector

References

Works Cited

Works Consulted

Incidental Works

Document History

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: Differential Logic | Facebook | Mathstodon • Laws of Form • Ontolog Forum

Posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization | Tagged Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization | 10 Comments

Animated Logical Graphs • 31

Posted on September 1, 2019 by Jon Awbrey

Re: Systems Science • Aleksandar Malečić
Re: Animated Logical Graphs • 21

AM:: Each step on its own, as far as I can follow them, makes sense. You are, if I understand it correctly, trying to figure out something fundamental, the rock bottom reality. When can we expect that results of such a research to become “applicable to more than one of the traditional departments of knowledge”? What kinds of tragedy, disaster, misunderstanding, mismanagement, or failure would/will be preventable by your approach?

The larger questions asked above — interdisciplinary inquiry, the interest in integration, the synthesis of ideas across isolated silos of specialization, and what it might mean for the future — are issues Susan Awbrey and I addressed from a pragmatic semiotic perspective:

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. Abstract. Online.
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. Online.

From that vantage point, what I’m about here is just a subgoal of a subgoal, panning what bits of elemental substrate can be found ever nearer that elusive “rock bottom reality”.

cc: Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

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 • 30

Posted on August 25, 2019 by Jon Awbrey

The duality between Entitative and Existential interpretations of logical graphs is a good example of a mathematical symmetry, in this case a symmetry of order two. Symmetries of this and higher orders give us conceptual handles on excess complexity in the manifold of sensuous impressions, making it well worth the effort to seek them out and grasp them where we find them.

In that vein, here’s a Rosetta Stone to give us a grounding in the relationship between boolean functions and our two readings of logical graphs.

$\text{Boolean Functions on Two Variables}$
$\text{Boolean Function}$	$\text{Entitative Graph}$	$\text{Existential Graph}$
$f_{0}$
$\text{false}$	$\text{false}$	$\text{false}$
$f_{1}$
$\text{neither}~ x ~\text{nor}~ y$	$\lnot (x \lor y)$	$\lnot x \land \lnot y$
$f_{2}$
$y ~\text{and not}~ x$	$\lnot x \land y$	$\lnot x \land y$
$f_{3}$
$\text{not}~ x$	$\lnot x$	$\lnot x$
$f_{4}$
$x ~\text{and not}~ y$	$x \land \lnot y$	$x \land \lnot y$
$f_{5}$
$\text{not}~ y$	$\lnot y$	$\lnot y$
$f_{6}$
$x ~\text{not equal to}~ y$	$x \ne y$	$x \ne y$
$f_{7}$
$\text{not both}~ x ~\text{and}~ y$	$\lnot x \lor \lnot y$	$\lnot (x \land y)$
$f_{8}$
$x ~\text{and}~ y$	$x \land y$	$x \land y$
$f_{9}$
$x ~\text{equal to}~ y$	$x = y$	$x = y$
$f_{10}$
$y$	$y$	$y$
$f_{11}$
$\text{if}~ x ~\text{then}~ y$	$x \Rightarrow y$	$x \Rightarrow y$
$f_{12}$
$x$	$x$	$x$
$f_{13}$
$\text{if}~ y ~\text{then}~ x$	$x \Leftarrow y$	$x \Leftarrow y$
$f_{14}$
$x ~\text{or}~ y$	$x \lor y$	$x \lor y$
$f_{15}$
$\text{true}$	$\text{true}$	$\text{true}$

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

Abduction, Deduction, Induction, Analogy, Inquiry • 27

Posted on August 20, 2019 by Jon Awbrey

Re: Gil Kalai • Avi Wigderson : Integrating Computational Modeling, Algorithms, and Complexity into Theories of Nature Marks a New Scientific Revolution!

I took a look at Avi’s paper “On the Nature of the Theory of Computation” (OtNotToC). There is naturally a good dose of TOC but little on the type of World-Objective Contact (WOC) it takes to connect with empirical science. Just on that sample it reminds me of projects like Wolfram’s “New Kind Of Science”. They all do a good job of convincing us to use computational media as virtual laboratories for conducting experimental mathematics, but they leave us hanging when it comes to analyzing the relation between what goes on inside the box of computation and the natural world outside the box.

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

Posted in Abduction, Analogy, Aristotle, Artificial Intelligence, C.S. Peirce, Deduction, Induction, Inquiry, Inquiry Driven Systems, Intelligent Systems Engineering, Logic, Mental Models, Peirce, Scientific Method, Semiotics, Systems | Tagged Abduction, Analogy, Aristotle, Artificial Intelligence, C.S. Peirce, Deduction, Induction, Inquiry, Inquiry Driven Systems, Intelligent Systems Engineering, Logic, Mental Models, Peirce, Scientific Method, Semiotics, Systems | 6 Comments

Animated Logical Graphs • 29

Posted on August 11, 2019 by Jon Awbrey

Re: Animated Logical Graphs • 21
Re: Ontolog Forum • Joseph Simpson

JS:: I tend to view equivalence and distinction as relationships as opposed to operations. I do not know if this makes any significant difference in this context.

Dear Joe,

I invoked the general concepts of equivalence and distinction at this point in order to keep the wider backdrop of ideas in mind but since we’ve been focusing on boolean functions to coordinate the semantics of propositional calculi we can get a sense of the links between operations and relations by looking at their relationship in a boolean frame of reference.

Let $\mathbb{B} = \{ 0, 1 \}$ and $k$ a positive integer. Then $\mathbb{B}^k$ is the set of $k$ -tuples of elements of $\mathbb{B}.$

A $k$ -variable boolean function is a mapping $\mathbb{B}^k \to \mathbb{B}.$
A $k$ -place boolean relation is a subset of $\mathbb{B}^k.$

The correspondence between boolean functions and boolean relations may be articulated as follows.

Any $k$ -place relation $L,$ as a subset of $\mathbb{B}^k,$ has a corresponding indicator function (or characteristic function) $f_L : \mathbb{B}^k \to \mathbb{B}$ defined by the rule that $f_L (x) = 1$ if $x$ is in $L$ and $f_L (x) = 0$ if $x$ is not in $L.$
Any $k$ -variable function $f : \mathbb{B}^k \to \mathbb{B}$ is the indicator function of a $k$ -place relation $L_f$ consisting of all the $x$ in $\mathbb{B}^k$ where $f(x) = 1.$ The set $L_f$ is called the fiber of $1$ or the pre-image of $1$ in $\mathbb{B}^k$ and is commonly notated as $f^{-1}(1).$

Resources

Logic Syllabus • Relation Theory
Logical Equality • Logical Inequality
Boolean Function • Boolean-Valued Function

cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: FB | Logical Graphs • Laws of Form

Abduction, Deduction, Induction, Analogy, Inquiry • 26

Posted on August 6, 2019 by Jon Awbrey

Re: Gil Kalai • Avi Wigderson : Integrating Computational Modeling, Algorithms, and Complexity into Theories of Nature Marks a New Scientific Revolution!

Projects giving a central place to computation in scientific inquiry go back to Hobbes and Leibniz, at least, and then came Babbage and Peirce. One of the first issues determining their subsequent development is the degree to which they identify computation with deduction. The next question concerns how many types of reasoning they count as contributing to the logic of empirical science:

Is deduction alone sufficient?
Are deduction and induction irreducible to each other and sufficient in tandem?
Are there in fact three irreducible types of inference: abduction, deduction, induction?

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

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Inquiry Into Inquiry

The Difference That Makes A Difference That Peirce Makes • 22

The Difference That Makes A Difference That Peirce Makes • 21

The Difference That Makes A Difference That Peirce Makes • 20

Animated Logical Graphs • 31

Animated Logical Graphs • 30

Abduction, Deduction, Induction, Analogy, Inquiry • 27

Animated Logical Graphs • 29

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Abduction, Deduction, Induction, Analogy, Inquiry • 26

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