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Differential Logic and Dynamic Systems • Overview

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Author: Jon Awbrey



Tangent Functor Ferris Wheel.jpg
Stand and unfold yourself. Hamlet: Francsico—1.1.2

This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems.  The work pursued here is coordinated with a parallel application that focuses 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.

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

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

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

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