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# Differential Propositional Calculus • Overview

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**Author: Jon Awbrey**

• Overview • Part 1 • Part 2 • Appendices • References •

A **differential propositional calculus** is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.

## Casual Introduction

## Cactus Calculus

## Formal Development

### Elementary Notions

### Special Classes of Propositions

#### Linear Propositions

#### Positive Propositions

#### Singular Propositions

### Differential Extensions

## Appendices

### Appendix 1. Propositional Forms and Differential Expansions

#### Table A1. Propositional Forms on Two Variables

#### Table A2. Propositional Forms on Two Variables

#### Table A3. E*f* Expanded Over Differential Features

#### Table A4. D*f* Expanded Over Differential Features

#### Table A5. E*f* Expanded Over Ordinary Features

#### Table A6. D*f* 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 D*f* = d*f* + d²*f*

#### 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 ε*f*_{8}

##### Computation of E*f*_{8}

##### Computation of D*f*_{8}

##### Computation of d*f*_{8}

##### Computation of r*f*_{8}

##### Computation Summary for Conjunction

#### Operator Maps for the Logical Equality *f*_{9}(*u*, *v*)

##### Computation of ε*f*_{9}

##### Computation of E*f*_{9}

##### Computation of D*f*_{9}

##### Computation of d*f*_{9}

##### Computation of r*f*_{9}

##### Computation Summary for Equality

#### Operator Maps for the Logical Implication *f*_{11}(*u*, *v*)

##### Computation of ε*f*_{11}

##### Computation of E*f*_{11}

##### Computation of D*f*_{11}

##### Computation of d*f*_{11}

##### Computation of r*f*_{11}

##### Computation Summary for Implication

#### Operator Maps for the Logical Disjunction *f*_{14}(*u*, *v*)

##### Computation of ε*f*_{14}

##### Computation of E*f*_{14}

##### Computation of D*f*_{14}

##### Computation of d*f*_{14}

##### Computation of r*f*_{14}

##### Computation Summary for Disjunction

### Appendix 4. Source Materials

### Appendix 5. Various Definitions of the Tangent Vector

## References

• Overview • Part 1 • Part 2 • Appendices • References •

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