Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a differential logical calculus — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

In accord with the strategy of approaching logical systems in stages, first gaining a foothold in propositional logic and advancing on those grounds, we may set our first stepping stones toward differential logic in differential propositional calculi — propositional calculi extended by sets of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.

What follows is the outline of a sketch on differential propositional calculus intended as an intuitive introduction to the larger subject of differential logic, which amounts in turn to my best effort so far at dealing with the ancient and persistent problems of treating diversity and mutability in logical terms.

Part 1

Casual Introduction

Cactus Calculus

Part 2

Formal_Development

Elementary Notions

Special Classes of Propositions

Linear Propositions

Positive Propositions

Singular Propositions

Differential Extensions

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. 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

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