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Differential Propositional Calculus • Appendices
Author: Jon Awbrey
• Overview • Part 1 • Part 2 • Appendices • References •
Contents
- 1 Appendices
- 1.1 Appendix 1. Propositional Forms and Differential Expansions
- 1.1.1 Table A1. Propositional Forms on Two Variables
- 1.1.2 Table A2. Propositional Forms on Two Variables
- 1.1.3 Table A3. Ef Expanded Over Differential Features
- 1.1.4 Table A4. Df Expanded Over Differential Features
- 1.1.5 Table A5. Ef Expanded Over Ordinary Features
- 1.1.6 Table A6. Df Expanded Over Ordinary Features
- 1.2 Appendix 2. Differential Forms
- 1.2.1 Table A7. Differential Forms Expanded on a Logical Basis
- 1.2.2 Table A8. Differential Forms Expanded on an Algebraic Basis
- 1.2.3 Table A9. Tangent Proposition as Pointwise Linear Approximation
- 1.2.4 Table A10. Taylor Series Expansion Df = df + d²f
- 1.2.5 Table A11. Partial Differentials and Relative Differentials
- 1.2.6 Table A12. Detail of Calculation for the Difference Map
- 1.3 Appendix 3. Computational Details
- 1.4 Appendix 4. Source Materials
- 1.5 Appendix 5. Various Definitions of the Tangent Vector
- 1.1 Appendix 1. Propositional Forms and Differential Expansions
Appendices
Appendix 1. Propositional Forms and Differential Expansions
Table A1. Propositional Forms on Two Variables
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Table A2. Propositional Forms on Two Variables
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Table A3. Ef Expanded Over Differential Features
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Table A4. Df Expanded Over Differential Features
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Table A5. Ef Expanded Over Ordinary Features
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Table A6. Df Expanded Over Ordinary Features
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Appendix 2. Differential Forms
The actions of the difference operator and the tangent operator on the 16 bivariate propositions are shown in Tables A7 and A8.
Table A7 expands the differential forms that result over a logical basis:
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This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive cells of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:
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Table A8 expands the differential forms that result over an algebraic basis:
This set consists of the positive propositions in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the positive differential basis.
Table A7. Differential Forms Expanded on a Logical Basis
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Table A8. Differential Forms Expanded on an Algebraic Basis
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Table A9. Tangent Proposition as Pointwise Linear Approximation
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Table A10. Taylor Series Expansion Df = df + d²f
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Table A11. Partial Differentials and Relative Differentials
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Table A12. Detail of Calculation for the Difference Map
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Appendix 3. Computational Details
Operator Maps for the Logical Conjunction f8⟨u, v⟩
Computation of εf8
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Computation of Ef8
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Computation of Df8
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Computation of df8
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Computation of rf8
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Computation Summary for Conjunction
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Operator Maps for the Logical Equality f9⟨u, v⟩
Computation of εf9
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Computation of Ef9
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Computation of Df9
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Computation of df9
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Computation of rf9
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Computation Summary for Equality
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Operator Maps for the Logical Implication f11⟨u, v⟩
Computation of εf11
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Computation of Ef11
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Computation of Df11
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Computation of df11
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Computation of rf11
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Computation Summary for Implication
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Operator Maps for the Logical Disjunction f14⟨u, v⟩
Computation of εf14
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Computation of Ef14
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Computation of Df14
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Computation of df14
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Computation of rf14
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Computation Summary for Disjunction
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Appendix 4. Source Materials
Appendix 5. Various Definitions of the Tangent Vector
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