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 report is intended to address that lack.

This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms. The work pursued here is coordinated with a parallel application focusing 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.

# 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

# Part 2

## 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 3

## 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 4

## Transformations of Discourse (cont.)

### 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 5

## Transformations of Discourse (concl.)

### 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. 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

## References

### Works Cited

### Works Consulted

### Incidental Works

# Document History

## Document History

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