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__NOTOC__'''Author: [[User:Jon Awbrey|Jon Awbrey]]''' ---- <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> ---- {| align="center" cellpadding="10" | [[File:Diff Log Dyn Sys • Tangent Functor Ferris Wheel 2.0.png|448px]] |} {| style="height:36px; width:100%" | align="left" | ''Stand and unfold yourself.'' | align="right" | Hamlet: Francisco&mdash;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&#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. 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. <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''₈(''u'', ''v'')]]</h4> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of &epsilon;f8|Computation of &epsilon;''f''₈]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Ef8|Computation of E''f''₈]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Df8|Computation of D''f''₈]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of df8 2|Computation of d''f''₈]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of rf8|Computation of r''f''₈]]</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''₉(''u'', ''v'')]]</h4> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of &epsilon;f9|Computation of &epsilon;''f''₉]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Ef9|Computation of E''f''₉]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Df9|Computation of D''f''₉]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of df9 2|Computation of d''f''₉]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of rf9|Computation of r''f''₉]]</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''₁₁(''u'', ''v'')]]</h4> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of &epsilon;f11|Computation of &epsilon;''f''₁₁]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Ef11|Computation of E''f''₁₁]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Df11|Computation of D''f''₁₁]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of df11 2|Computation of d''f''₁₁]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of rf11|Computation of r''f''₁₁]]</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''₁₄(''u'', ''v'')]]</h4> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of &epsilon;f14|Computation of &epsilon;''f''₁₄]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Ef14|Computation of E''f''₁₄]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of Df14|Computation of D''f''₁₄]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of df14 2|Computation of d''f''₁₄]]</h5> <h5 style="margin-left:90px">[[Differential Logic and Dynamic Systems &bull; Appendices#Computation of rf14|Computation of r''f''₁₄]]</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> ---- [[Category:Adaptive systems]] [[Category:Artificial intelligence]] [[Category:Boolean algebra]] [[Category:Boolean functions]] [[Category:Category theory]] [[Category:Combinatorics]] [[Category:Computation theory]] [[Category:Cybernetics]] [[Category:Differential logic]] [[Category:Discrete systems]] [[Category:Dynamical systems]] [[Category:Formal languages]] [[Category:Formal sciences]] [[Category:Formal systems]] [[Category:Functional logic]] [[Category:Graph theory]] [[Category:Group theory]] [[Category:Logic]] [[Category:Logical graphs]] [[Category:Neural networks]] [[Category:Peirce, Charles Sanders]] [[Category:Semiotics]] [[Category:Systems theory]] [[Category:Visualization]]

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