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Differential Logic and Dynamic Systems • Overview
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Author: Jon Awbrey
• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Appendices • References • Document History •
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 f_{8}(u, v)
Computation of εf_{8}
Computation of Ef_{8}
Computation of Df_{8}
Computation of df_{8}
Computation of rf_{8}
Computation Summary for Conjunction
Operator Maps for the Logical Equality f_{9}(u, v)
Computation of εf_{9}
Computation of Ef_{9}
Computation of Df_{9}
Computation of df_{9}
Computation of rf_{9}
Computation Summary for Equality
Operator Maps for the Logical Implication f_{11}(u, v)
Computation of εf_{11}
Computation of Ef_{11}
Computation of Df_{11}
Computation of df_{11}
Computation of rf_{11}
Computation Summary for Implication
Operator Maps for the Logical Disjunction f_{14}(u, v)
Computation of εf_{14}
Computation of Ef_{14}
Computation of Df_{14}
Computation of df_{14}
Computation of rf_{14}
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
• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Appendices • References • Document History •
Categories:
- Adaptive systems
- Artificial intelligence
- Boolean algebra
- Boolean functions
- Category theory
- Combinatorics
- Computation theory
- Cybernetics
- Differential logic
- Discrete systems
- Dynamical systems
- Formal languages
- Formal sciences
- Formal systems
- Functional logic
- Graph theory
- Group theory
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- Neural networks
- Peirce, Charles Sanders
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