Difference between revisions of "Differential Logic and Dynamic Systems • Overview"

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__NOTOC__'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
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----
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<div align="center">
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&bull; [[Differential Logic and Dynamic Systems &bull; Overview|Overview]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 1|Part&nbsp;1]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 2|Part&nbsp;2]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 3|Part&nbsp;3]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 4|Part&nbsp;4]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 5|Part&nbsp;5]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Appendices|Appendices]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; References|References]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Document History|Document&nbsp;History]]
 +
&bull;
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</div>
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----
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{| align="center" cellpadding="10"
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| [[File:Diff Log Dyn Sys • Tangent Functor Ferris Wheel 2.0.png|448px]]
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|}
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{| style="height:36px; width:100%"
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| align="left"  | ''Stand and unfold yourself.''
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| align="right" | Hamlet: Francisco&mdash;1.1.2
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|}
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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&#8209;off between dynamic paradigms and symbolic paradigms.&nbsp; Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system's state through time.&nbsp; 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.&nbsp; Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus.&nbsp; The work laid out in this report is intended to address that lack.
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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.&nbsp; 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.
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 +
<h2>[[Differential Logic and Dynamic Systems &bull; Part 1#Review and Transition|Review and Transition]]</h2>
 +
 
 +
<h2>[[Differential Logic and Dynamic Systems &bull; 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 &bull; Part 1#Qualitative Logic and Quantitative Analogy|Qualitative Logic and Quantitative Analogy]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; Part 1#Special Classes of Propositions|Special Classes of Propositions]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 1#Basis Relativity and Type Ambiguity|Basis Relativity and Type Ambiguity]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; 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 &bull; 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 &bull; 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 &bull; Part 1#Tables of Propositional Forms|Tables of Propositional Forms]]</h3>
 +
 
 +
<h2>[[Differential Logic and Dynamic Systems &bull; 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 &bull; 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 &bull; Part 1#An Interlude on the Path|An Interlude on the Path]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 1#The Extended Universe of Discourse|The Extended Universe of Discourse]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 1#Intentional Propositions|Intentional Propositions]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 1#Life on Easy Street|Life on Easy Street]]</h3>
 +
 
 +
<h2>[[Differential Logic and Dynamic Systems &bull; 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 &bull; Part 2#A One-Dimensional Universe|A One-Dimensional Universe]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 2#Example 1. A Square Rigging|Example 1. A Square Rigging]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 2#Back to the Feature|Back to the Feature]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 2#Tacit Extensions|Tacit Extensions]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 2#Example 2. Drives and Their Vicissitudes|Example 2. Drives and Their Vicissitudes]]</h3>
 +
 
 +
<h2>[[Differential Logic and Dynamic Systems &bull; Part 3#Transformations of Discourse|Transformations of Discourse]]</h2>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; 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 &bull; 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 &bull; 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 &bull; Part 3#Thematization : Venn Diagrams|Thematization : Venn Diagrams]]</h4>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 3#Thematization : Truth Tables|Thematization : Truth Tables]]</h4>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 3#Propositional Transformations|Propositional Transformations]]</h3>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 3#Alias and Alibi Transformations|Alias and Alibi Transformations]]</h4>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 3#Transformations of General Type|Transformations of General Type]]</h4>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 3#Analytic Expansions : Operators and Functors|Analytic Expansions : Operators and Functors]]</h3>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 3#Operators on Propositions and Transformations|Operators on Propositions and Transformations]]</h4>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; Part 3#The Secant Operator : E|The Secant Operator : E]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 3#The Radius Operator : e|The Radius Operator : e]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 3#The Phantom of the Operators : &eta;|The Phantom of the Operators : &eta;]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 3#The Chord Operator : D|The Chord Operator : D]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 3#The Tangent Operator : T|The Tangent Operator : T]]</h5>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Part 4#Transformations of Type B² &rarr; B¹|Transformations of Type B² &rarr; B¹]]</h3>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 4#Analytic Expansion of Conjunction|Analytic Expansion of Conjunction]]</h4>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 4#Tacit Extension of Conjunction|Tacit Extension of Conjunction]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 4#Enlargement Map of Conjunction|Enlargement Map of Conjunction]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; Part 4#Difference Map of Conjunction|Difference Map of Conjunction]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 4#Differential of Conjunction|Differential of Conjunction]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 4#Remainder of Conjunction|Remainder of Conjunction]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 4#Summary of Conjunction|Summary of Conjunction]]</h5>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 4#Analytic Series : Coordinate Method|Analytic Series : Coordinate Method]]</h4>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 4#Analytic Series : Recap|Analytic Series : Recap]]</h4>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 4#Terminological Interlude|Terminological Interlude]]</h4>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; Part 4#Operator Maps : Areal Views|Operator Maps : Areal Views]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Part 4#Operator Maps : Box Views|Operator Maps : Box Views]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; 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 &bull; Part 5#Transformations of Type B² &rarr; B²|Transformations of Type B² &rarr; B²]]</h3>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 5#Logical Transformations|Logical Transformations]]</h4>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 5#Local Transformations|Local Transformations]]</h4>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; Part 5#Difference Operators and Tangent Functors|Difference Operators and Tangent Functors]]</h4>
 +
 
 +
<h2>[[Differential Logic and Dynamic Systems &bull; Part 5#Epilogue, Enchoiry, Exodus|Epilogue, Enchoiry, Exodus]]</h2>
 +
 
 +
<h2>[[Differential Logic and Dynamic Systems &bull; Appendices#Appendices|Appendices]]</h2>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; 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 &bull; 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 &bull; 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 &bull; 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 &bull; 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 &bull; 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 &bull; Appendices#Appendix 2. Differential Forms|Appendix 2. Differential Forms]]</h3>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; 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 &bull; 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 &bull; 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 &bull; 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 &bull; 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 &bull; Appendices#Appendix 3. Computational Details|Appendix 3. Computational Details]]</h3>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; Appendices#Computation of &epsilon;f8|Computation of &epsilon;''f''<sub>8</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Ef8|Computation of E''f''<sub>8</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Df8|Computation of D''f''<sub>8</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of df8 2|Computation of d''f''<sub>8</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of rf8|Computation of r''f''<sub>8</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation Summary for Conjunction|Computation Summary for Conjunction]]</h5>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; Appendices#Computation of &epsilon;f9|Computation of &epsilon;''f''<sub>9</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Ef9|Computation of E''f''<sub>9</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Df9|Computation of D''f''<sub>9</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of df9 2|Computation of d''f''<sub>9</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of rf9|Computation of r''f''<sub>9</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation Summary for Equality|Computation Summary for Equality]]</h5>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; Appendices#Computation of &epsilon;f11|Computation of &epsilon;''f''<sub>11</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Ef11|Computation of E''f''<sub>11</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Df11|Computation of D''f''<sub>11</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of df11 2|Computation of d''f''<sub>11</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of rf11|Computation of r''f''<sub>11</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation Summary for Implication|Computation Summary for Implication]]</h5>
 +
 
 +
<h4 style="margin-left:60px">[[Differential Logic and Dynamic Systems &bull; 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 &bull; Appendices#Computation of &epsilon;f14|Computation of &epsilon;''f''<sub>14</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Ef14|Computation of E''f''<sub>14</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Df14|Computation of D''f''<sub>14</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of df14 2|Computation of d''f''<sub>14</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of rf14|Computation of r''f''<sub>14</sub>]]</h5>
 +
 
 +
<h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation Summary for Disjunction|Computation Summary for Disjunction]]</h5>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Appendices#Appendix 4. Source Materials|Appendix 4. Source Materials]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; Appendices#Appendix 5. Various Definitions of the Tangent Vector|Appendix 5. Various Definitions of the Tangent Vector]]</h3>
 +
 
 +
<h2>[[Differential Logic and Dynamic Systems &bull; References#References|References]]</h2>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; References#Works Cited|Works Cited]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; References#Works Consulted|Works Consulted]]</h3>
 +
 
 +
<h3 style="margin-left:30px">[[Differential Logic and Dynamic Systems &bull; References#Incidental Works|Incidental Works]]</h3>
 +
 
 +
<h2>[[Differential Logic and Dynamic Systems &bull; Document History#Document History|Document History]]</h2>
 +
 
 +
----
 +
<div align="center">
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Overview|Overview]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 1|Part&nbsp;1]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 2|Part&nbsp;2]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 3|Part&nbsp;3]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 4|Part&nbsp;4]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Part 5|Part&nbsp;5]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Appendices|Appendices]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; References|References]]
 +
&bull; [[Differential Logic and Dynamic Systems &bull; Document History|Document&nbsp;History]]
 +
&bull;
 +
</div>
 +
----
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Latest revision as of 16:40, 23 January 2026

Author: Jon Awbrey

Diff Log Dyn Sys • Tangent Functor Ferris Wheel 2.0.png
Stand and unfold yourself. 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.

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

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

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

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

Taking Aim at Higher Dimensional Targets

Transformations of Type B² → B²

Logical Transformations

Local Transformations

Difference Operators and Tangent Functors

Epilogue, Enchoiry, Exodus

Appendices

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

References

Works Cited

Works Consulted

Incidental Works

Document History

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