Differential Logic and Dynamic Systems • Overview

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



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



Works Cited

Works Consulted

Incidental Works

Document History

Document History

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10 Responses to Differential Logic and Dynamic Systems • Overview

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  10. Poor Richard says:

    Enchoiry is an interesting word. Any relation to entelechy?

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