Differential Logic and Dynamic Systems • Overview

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 series of posts is intended to address that lack.

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

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 2

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 3

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 4

Taking Aim at Higher Dimensional Targets

Transformations of Type B² → B²

Logical Transformations

Local Transformations

Difference Operators and Tangent Functors

Epilogue, Enchoiry, Exodus

Appendices

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

References

Works Cited

Works Consulted

Incidental Works

Document History

Document History

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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 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Animated Logical Graphs • 31

Re: Systems ScienceAleksandar 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.  AbstractOnline.
  • 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”.

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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 , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Animated Logical Graphs • 30

The duality between Entitative and Existential interpretations of logical graphs is one example of a mathematical symmetry, in this case a symmetry of order 2.  Symmetries of this and higher orders give us conceptual handles on excess complexities in the manifold of sensuous impressions, making it well worth our trouble 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} Cactus Root Cactus Stem
\text{false} \text{false} \text{false}
f_{1} Cactus (xy) Cactus (x)(y)
\text{neither}~ x ~\text{nor}~ y \lnot (x \lor y) \lnot x \land \lnot y
f_{2} Cactus (x(y)) Cactus (x)y
y ~\text{and not}~ x \lnot x \land y \lnot x \land y
f_{3} Cactus (x) Cactus (x)
\text{not}~ x \lnot x \lnot x
f_{4} Cactus ((x)y) Cactus x(y)
x ~\text{and not}~ y x \land \lnot y x \land \lnot y
f_{5} Cactus (y) Cactus (y)
\text{not}~ y \lnot y \lnot y
f_{6} Cactus ((x,y)) Cactus (x,y)
x ~\text{not equal to}~ y x \ne y x \ne y
f_{7} Cactus (x)(y) Cactus (xy)
\text{not both}~ x ~\text{and}~ y \lnot x \lor \lnot y \lnot (x \land y)
f_{8} Cactus ((x)(y)) Cactus xy
x ~\text{and}~ y x \land y x \land y
f_{9} Cactus (x,y) Cactus ((x,y))
x ~\text{equal to}~ y x = y x = y
f_{10} Cactus y Cactus y
y y y
f_{11} Cactus (x)y Cactus (x(y))
\text{if}~ x ~\text{then}~ y x \Rightarrow y x \Rightarrow y
f_{12} Cactus x Cactus x
x x x
f_{13} Cactus x(y) Cactus ((x)y)
\text{if}~ y ~\text{then}~ x x \Leftarrow y x \Leftarrow y
f_{14} Cactus xy Cactus ((x)(y))
x ~\text{or}~ y x \lor y x \lor y
f_{15} Cactus Stem Cactus Root
\text{true} \text{true} \text{true}

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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 , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Abduction, Deduction, Induction, Analogy, Inquiry : 27

Re: Gil KalaiAvi 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.

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Posted in Abduction, Analogy, Aristotle, Artificial Intelligence, C.S. Peirce, Computation, Computational Complexity, Deduction, Induction, Inquiry, Inquiry Driven Systems, Intelligent Systems, Logic, Peirce, Problem Solving, Semiotics | Tagged , , , , , , , , , , , , , , , | 1 Comment

Animated Logical Graphs • 29

Re: Ontolog ForumJoseph Simpson
Re: Animated Logical Graphs • 21

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).

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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 , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Abduction, Deduction, Induction, Analogy, Inquiry : 26

Re: Gil KalaiAvi 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 one identifies computation and deduction.  The next question concerns how many types of reasoning one counts as contributing to the logic of empirical science:

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

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Posted in Abduction, Analogy, Aristotle, Artificial Intelligence, C.S. Peirce, Computation, Computational Complexity, Deduction, Induction, Inquiry, Inquiry Driven Systems, Intelligent Systems, Logic, Peirce, Problem Solving, Semiotics | Tagged , , , , , , , , , , , , , , , | 2 Comments

Animated Logical Graphs • 28

Re: Ontolog ForumJSJA

I will have to focus on other business for a couple of weeks — so just by way of reminding myself what we were talking about at this juncture where logical graphs and differential logic intersect, here’s my comment on R.J. Lipton and K.W. Regan’s blog post about Discrepancy Games and Sensitivity.

Just by way of a general observation, concepts like discrepancy, influence, sensitivity, etc. are differential in character, so I tend to think the proper grounds for approaching them more systematically will come from developing the logical analogue of differential geometry.

I took a few steps in this direction some years ago in connection with an effort to understand a certain class of intelligent systems as dynamical systems.  There’s a motley assortment of links here:

Resources

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