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__NOTOC__'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
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<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
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{| align="center" cellpadding="10"
| [[File:Diff Log Dyn Sys • Tangent Functor Ferris Wheel 2.0.png|448px]]
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{| style="height:36px; width:100%"
| align="left" | ''Stand and unfold yourself.''
| align="right" | Hamlet: Francisco—1.1.2
|}
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.
<h2>[[Differential Logic and Dynamic Systems • Part 1#Review and Transition|Review and Transition]]</h2>
<h2>[[Differential Logic and Dynamic Systems • Part 1#A Functional Conception of Propositional Calculus|A Functional Conception of Propositional Calculus]]</h2>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Qualitative Logic and Quantitative Analogy|Qualitative Logic and Quantitative Analogy]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Philosophy of Notation : Formal Terms and Flexible Types|Philosophy of Notation : Formal Terms and Flexible Types]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Special Classes of Propositions|Special Classes of Propositions]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Basis Relativity and Type Ambiguity|Basis Relativity and Type Ambiguity]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#The Analogy Between Real and Boolean Types|The Analogy Between Real and Boolean Types]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Theory of Control and Control of Theory|Theory of Control and Control of Theory]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Propositions as Types and Higher Order Types|Propositions as Types and Higher Order Types]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Reality at the Threshold of Logic|Reality at the Threshold of Logic]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Tables of Propositional Forms|Tables of Propositional Forms]]</h3>
<h2>[[Differential Logic and Dynamic Systems • Part 1#A Differential Extension of Propositional Calculus|A Differential Extension of Propositional Calculus]]</h2>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Differential Propositions : Qualitative Analogues of Differential Equations|Differential Propositions : Qualitative Analogues of Differential Equations]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#An Interlude on the Path|An Interlude on the Path]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#The Extended Universe of Discourse|The Extended Universe of Discourse]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Intentional Propositions|Intentional Propositions]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 1#Life on Easy Street|Life on Easy Street]]</h3>
<h2>[[Differential Logic and Dynamic Systems • Part 2#Back to the Beginning : Exemplary Universes|Back to the Beginning : Exemplary Universes]]</h2>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#A One-Dimensional Universe|A One-Dimensional Universe]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#Example 1. A Square Rigging|Example 1. A Square Rigging]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#Back to the Feature|Back to the Feature]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#Tacit Extensions|Tacit Extensions]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 2#Example 2. Drives and Their Vicissitudes|Example 2. Drives and Their Vicissitudes]]</h3>
<h2>[[Differential Logic and Dynamic Systems • Part 3#Transformations of Discourse|Transformations of Discourse]]</h2>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 3#Foreshadowing Transformations : Extensions and Projections of Discourse|Foreshadowing Transformations : Extensions and Projections of Discourse]]</h3>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Extension from 1 to 2 Dimensions|Extension from 1 to 2 Dimensions]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Extension from 2 to 4 Dimensions|Extension from 2 to 4 Dimensions]]</h4>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 3#Thematization of Functions : And a Declaration of Independence for Variables|Thematization of Functions : And a Declaration of Independence for Variables]]</h3>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Thematization : Venn Diagrams|Thematization : Venn Diagrams]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Thematization : Truth Tables|Thematization : Truth Tables]]</h4>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 3#Propositional Transformations|Propositional Transformations]]</h3>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Alias and Alibi Transformations|Alias and Alibi Transformations]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Transformations of General Type|Transformations of General Type]]</h4>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 3#Analytic Expansions : Operators and Functors|Analytic Expansions : Operators and Functors]]</h3>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Operators on Propositions and Transformations|Operators on Propositions and Transformations]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 3#Differential Analysis of Propositions and Transformations|Differential Analysis of Propositions and Transformations]]</h4>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Secant Operator : E|The Secant Operator : E]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Radius Operator : e|The Radius Operator : e]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Phantom of the Operators : η|The Phantom of the Operators : η]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Chord Operator : D|The Chord Operator : D]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 3#The Tangent Operator : T|The Tangent Operator : T]]</h5>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 4#Transformations of Type B² → B¹|Transformations of Type B² → B¹]]</h3>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#Analytic Expansion of Conjunction|Analytic Expansion of Conjunction]]</h4>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Tacit Extension of Conjunction|Tacit Extension of Conjunction]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Enlargement Map of Conjunction|Enlargement Map of Conjunction]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Digression : Reflection on Use and Mention|Digression : Reflection on Use and Mention]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Difference Map of Conjunction|Difference Map of Conjunction]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Differential of Conjunction|Differential of Conjunction]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Remainder of Conjunction|Remainder of Conjunction]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Summary of Conjunction|Summary of Conjunction]]</h5>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#Analytic Series : Coordinate Method|Analytic Series : Coordinate Method]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#Analytic Series : Recap|Analytic Series : Recap]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#Terminological Interlude|Terminological Interlude]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 4#End of Perfunctory Chatter : Time to Roll the Clip!|End of Perfunctory Chatter : Time to Roll the Clip!]]</h4>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Operator Maps : Areal Views|Operator Maps : Areal Views]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Operator Maps : Box Views|Operator Maps : Box Views]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Part 4#Operator Diagrams for the Conjunction J = uv|Operator Diagrams for the Conjunction ''J'' = ''uv'']]</h5>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 5#Taking Aim at Higher Dimensional Targets|Taking Aim at Higher Dimensional Targets]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Part 5#Transformations of Type B² → B²|Transformations of Type B² → B²]]</h3>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 5#Logical Transformations|Logical Transformations]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 5#Local Transformations|Local Transformations]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Part 5#Difference Operators and Tangent Functors|Difference Operators and Tangent Functors]]</h4>
<h2>[[Differential Logic and Dynamic Systems • Part 5#Epilogue, Enchoiry, Exodus|Epilogue, Enchoiry, Exodus]]</h2>
<h2>[[Differential Logic and Dynamic Systems • Appendices#Appendices|Appendices]]</h2>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 1. Propositional Forms and Differential Expansions|Appendix 1. Propositional Forms and Differential Expansions]]</h3>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A1. Propositional Forms on Two Variables|Table A1. Propositional Forms on Two Variables]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A2. Propositional Forms on Two Variables|Table A2. Propositional Forms on Two Variables]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A3. Ef Expanded Over Differential Features|Table A3. E''f'' Expanded Over Differential Features]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A4. Df Expanded Over Differential Features|Table A4. D''f'' Expanded Over Differential Features]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A5. Ef Expanded Over Ordinary Features|Table A5. E''f'' Expanded Over Ordinary Features]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A6. Df Expanded Over Ordinary Features|Table A6. D''f'' Expanded Over Ordinary Features]]</h4>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 2. Differential Forms|Appendix 2. Differential Forms]]</h3>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A7. Differential Forms Expanded on a Logical Basis|Table A7. Differential Forms Expanded on a Logical Basis]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A8. Differential Forms Expanded on an Algebraic Basis|Table A8. Differential Forms Expanded on an Algebraic Basis]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A9. Tangent Proposition as Pointwise Linear Approximation|Table A9. Tangent Proposition as Pointwise Linear Approximation]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A10. Taylor Series Expansion Df = df + d²f|Table A10. Taylor Series Expansion D''f'' = d''f'' + d²''f'']]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A11. Partial Differentials and Relative Differentials|Table A11. Partial Differentials and Relative Differentials]]</h4>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Table A12. Detail of Calculation for the Difference Map|Table A12. Detail of Calculation for the Difference Map]]</h4>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 3. Computational Details|Appendix 3. Computational Details]]</h3>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Operator Maps for the Logical Conjunction f8(u, v)|Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(''u'', ''v'')]]</h4>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of εf8|Computation of ε''f''<sub>8</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Ef8|Computation of E''f''<sub>8</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Df8|Computation of D''f''<sub>8</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of df8 2|Computation of d''f''<sub>8</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of rf8|Computation of r''f''<sub>8</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation Summary for Conjunction|Computation Summary for Conjunction]]</h5>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Operator Maps for the Logical Equality f9(u, v)|Operator Maps for the Logical Equality ''f''<sub>9</sub>(''u'', ''v'')]]</h4>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of εf9|Computation of ε''f''<sub>9</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Ef9|Computation of E''f''<sub>9</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Df9|Computation of D''f''<sub>9</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of df9 2|Computation of d''f''<sub>9</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of rf9|Computation of r''f''<sub>9</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation Summary for Equality|Computation Summary for Equality]]</h5>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Operator Maps for the Logical Implication f11(u, v)|Operator Maps for the Logical Implication ''f''<sub>11</sub>(''u'', ''v'')]]</h4>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of εf11|Computation of ε''f''<sub>11</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Ef11|Computation of E''f''<sub>11</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Df11|Computation of D''f''<sub>11</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of df11 2|Computation of d''f''<sub>11</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of rf11|Computation of r''f''<sub>11</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation Summary for Implication|Computation Summary for Implication]]</h5>
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems • Appendices#Operator Maps for the Logical Disjunction f14(u, v)|Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(''u'', ''v'')]]</h4>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of εf14|Computation of ε''f''<sub>14</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Ef14|Computation of E''f''<sub>14</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of Df14|Computation of D''f''<sub>14</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of df14 2|Computation of d''f''<sub>14</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation of rf14|Computation of r''f''<sub>14</sub>]]</h5>
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems • Appendices#Computation Summary for Disjunction|Computation Summary for Disjunction]]</h5>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 4. Source Materials|Appendix 4. Source Materials]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • Appendices#Appendix 5. Various Definitions of the Tangent Vector|Appendix 5. Various Definitions of the Tangent Vector]]</h3>
<h2>[[Differential Logic and Dynamic Systems • References#References|References]]</h2>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • References#Works Cited|Works Cited]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • References#Works Consulted|Works Consulted]]</h3>
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems • References#Incidental Works|Incidental Works]]</h3>
<h2>[[Differential Logic and Dynamic Systems • Document History#Document History|Document History]]</h2>
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<div align="center">
• [[Differential Logic and Dynamic Systems • Overview|Overview]]
• [[Differential Logic and Dynamic Systems • Part 1|Part 1]]
• [[Differential Logic and Dynamic Systems • Part 2|Part 2]]
• [[Differential Logic and Dynamic Systems • Part 3|Part 3]]
• [[Differential Logic and Dynamic Systems • Part 4|Part 4]]
• [[Differential Logic and Dynamic Systems • Part 5|Part 5]]
• [[Differential Logic and Dynamic Systems • Appendices|Appendices]]
• [[Differential Logic and Dynamic Systems • References|References]]
• [[Differential Logic and Dynamic Systems • Document History|Document History]]
•
</div>
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