# differential propositional calculus : appendix 2

The actions of the difference operator (http://planetmath.org/FiniteDifference) $\operatorname{D}$ and the tangent operator (http://planetmath.org/TangentMap) $\operatorname{d}$ on the 16 propositional forms in two variables are shown in the Tables below.

Table A7 expands the resulting differential forms over a logical basis:

$\{(\operatorname{d}x)(\operatorname{d}y),\ \operatorname{d}x\,(\operatorname{d% }y),\ (\operatorname{d}x)\,\operatorname{d}y,\ \operatorname{d}x\,% \operatorname{d}y\}.$

This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive cells of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:

$\partial x=\operatorname{d}x\,(\operatorname{d}y)$ and $\partial y=(\operatorname{d}x)\,\operatorname{d}y.$

Table A8 expands the resulting differential forms over an algebraic basis:

$\{1,\ \operatorname{d}x,\ \operatorname{d}y,\ \operatorname{d}x\,\operatorname% {d}y\}.$

This set consists of the positive propositions in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the positive differential basis.

## 0.1 Table A7. Differential Forms Expanded on a Logical Basis

Table A7. Differential Forms Expanded on a Logical Basis
$f$ $\operatorname{D}f$ $\operatorname{d}f$
$f_{0}$ $(~{})$ $0$ $0$
$\begin{smallmatrix}f_{1}\\ f_{2}\\ f_{4}\\ f_{8}\\ \end{smallmatrix}$ $\begin{smallmatrix}(x)&(y)\\ (x)&y\\ x&(y)\\ x&y\\ \end{smallmatrix}$ $\begin{smallmatrix}(y)&\operatorname{d}x\ (\operatorname{d}y)&+&(x)&(% \operatorname{d}x)\ \operatorname{d}y&+&((x,y))&\operatorname{d}x\ % \operatorname{d}y\\ y&\operatorname{d}x\ (\operatorname{d}y)&+&(x)&(\operatorname{d}x)\ % \operatorname{d}y&+&(x,y)&\operatorname{d}x\ \operatorname{d}y\\ (y)&\operatorname{d}x\ (\operatorname{d}y)&+&x&(\operatorname{d}x)\ % \operatorname{d}y&+&(x,y)&\operatorname{d}x\ \operatorname{d}y\\ y&\operatorname{d}x\ (\operatorname{d}y)&+&x&(\operatorname{d}x)\ % \operatorname{d}y&+&((x,y))&\operatorname{d}x\ \operatorname{d}y\\ \end{smallmatrix}$ $\begin{smallmatrix}(y)&\partial x&+&(x)&\partial y\\ y&\partial x&+&(x)&\partial y\\ (y)&\partial x&+&x&\partial y\\ y&\partial x&+&x&\partial y\\ \end{smallmatrix}$
$\begin{smallmatrix}f_{3}\\ f_{12}\\ \end{smallmatrix}$ $\begin{smallmatrix}(x)\\ x\\ \end{smallmatrix}$ $\begin{smallmatrix}\operatorname{d}x\ (\operatorname{d}y)&+&\operatorname{d}x% \ \operatorname{d}y\\ \operatorname{d}x\ (\operatorname{d}y)&+&\operatorname{d}x\ \operatorname{d}y% \\ \end{smallmatrix}$ $\begin{smallmatrix}\partial x\\ \partial x\\ \end{smallmatrix}$
$\begin{smallmatrix}f_{6}\\ f_{9}\\ \end{smallmatrix}$ $\begin{smallmatrix}(x,&y)\\ ((x,&y))\\ \end{smallmatrix}$ $\begin{smallmatrix}\operatorname{d}x\ (\operatorname{d}y)&+&(\operatorname{d}x% )\ \operatorname{d}y\\ \operatorname{d}x\ (\operatorname{d}y)&+&(\operatorname{d}x)\ \operatorname{d}% y\\ \end{smallmatrix}$ $\begin{smallmatrix}\partial x&+&\partial y\\ \partial x&+&\partial y\\ \end{smallmatrix}$
$\begin{smallmatrix}f_{5}\\ f_{10}\\ \end{smallmatrix}$ $\begin{smallmatrix}(y)\\ y\\ \end{smallmatrix}$ $\begin{smallmatrix}(\operatorname{d}x)\ \operatorname{d}y&+&\operatorname{d}x% \ \operatorname{d}y\\ (\operatorname{d}x)\ \operatorname{d}y&+&\operatorname{d}x\ \operatorname{d}y% \\ \end{smallmatrix}$ $\begin{smallmatrix}\partial y\\ \partial y\\ \end{smallmatrix}$
$\begin{smallmatrix}f_{7}\\ f_{11}\\ f_{13}\\ f_{14}\\ \end{smallmatrix}$ $\begin{smallmatrix}(x&y)\\ (x&(y))\\ ((x)&y)\\ ((x)&(y))\\ \end{smallmatrix}$ $\begin{smallmatrix}y&\operatorname{d}x\ (\operatorname{d}y)&+&x&(\operatorname% {d}x)\ \operatorname{d}y&+&((x,y))&\operatorname{d}x\ \operatorname{d}y\\ (y)&\operatorname{d}x\ (\operatorname{d}y)&+&x&(\operatorname{d}x)\ % \operatorname{d}y&+&(x,y)&\operatorname{d}x\ \operatorname{d}y\\ y&\operatorname{d}x\ (\operatorname{d}y)&+&(x)&(\operatorname{d}x)\ % \operatorname{d}y&+&(x,y)&\operatorname{d}x\ \operatorname{d}y\\ (y)&\operatorname{d}x\ (\operatorname{d}y)&+&(x)&(\operatorname{d}x)\ % \operatorname{d}y&+&((x,y))&\operatorname{d}x\ \operatorname{d}y\\ \end{smallmatrix}$ $\begin{smallmatrix}y&\partial x&+&x&\partial y\\ (y)&\partial x&+&x&\partial y\\ y&\partial x&+&(x)&\partial y\\ (y)&\partial x&+&(x)&\partial y\\ \end{smallmatrix}$
$f_{15}$ $((~{}))$ $0$ $0$

## 0.2 Table A8. Differential Forms Expanded on an Algebraic Basis

Table A8. Differential Forms Expanded on an Algebraic Basis
$f$ $\operatorname{D}f$ $\operatorname{d}f$
$f_{0}$ $(~{})$ $0$ $0$
$\begin{smallmatrix}f_{1}\\ f_{2}\\ f_{4}\\ f_{8}\\ \end{smallmatrix}$ $\begin{smallmatrix}(x)&(y)\\ (x)&y\\ x&(y)\\ x&y\\ \end{smallmatrix}$ $\begin{smallmatrix}(y)&\operatorname{d}x&+&(x)&\operatorname{d}y&+&% \operatorname{d}x\ \operatorname{d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y&+&\operatorname{d}x\ \operatorname% {d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y&+&\operatorname{d}x\ \operatorname% {d}y\\ y&\operatorname{d}x&+&x&\operatorname{d}y&+&\operatorname{d}x\ \operatorname{d% }y\\ \end{smallmatrix}$ $\begin{smallmatrix}(y)&\operatorname{d}x&+&(x)&\operatorname{d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y\\ y&\operatorname{d}x&+&x&\operatorname{d}y\\ \end{smallmatrix}$
$\begin{smallmatrix}f_{3}\\ f_{12}\\ \end{smallmatrix}$ $\begin{smallmatrix}(x)\\ x\\ \end{smallmatrix}$ $\begin{smallmatrix}\operatorname{d}x\\ \operatorname{d}x\\ \end{smallmatrix}$ $\begin{smallmatrix}\operatorname{d}x\\ \operatorname{d}x\\ \end{smallmatrix}$
$\begin{smallmatrix}f_{6}\\ f_{9}\\ \end{smallmatrix}$ $\begin{smallmatrix}(x,&y)\\ ((x,&y))\\ \end{smallmatrix}$ $\begin{smallmatrix}\operatorname{d}x&+&\operatorname{d}y\\ \operatorname{d}x&+&\operatorname{d}y\\ \end{smallmatrix}$ $\begin{smallmatrix}\operatorname{d}x&+&\operatorname{d}y\\ \operatorname{d}x&+&\operatorname{d}y\\ \end{smallmatrix}$
$\begin{smallmatrix}f_{5}\\ f_{10}\\ \end{smallmatrix}$ $\begin{smallmatrix}(y)\\ y\\ \end{smallmatrix}$ $\begin{smallmatrix}\operatorname{d}y\\ \operatorname{d}y\\ \end{smallmatrix}$ $\begin{smallmatrix}\operatorname{d}y\\ \operatorname{d}y\\ \end{smallmatrix}$
$\begin{smallmatrix}f_{7}\\ f_{11}\\ f_{13}\\ f_{14}\\ \end{smallmatrix}$ $\begin{smallmatrix}(x&y)\\ (x&(y))\\ ((x)&y)\\ ((x)&(y))\\ \end{smallmatrix}$ $\begin{smallmatrix}y&\operatorname{d}x&+&x&\operatorname{d}y&+&\operatorname{d% }x\ \operatorname{d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y&+&\operatorname{d}x\ \operatorname% {d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y&+&\operatorname{d}x\ \operatorname% {d}y\\ (y)&\operatorname{d}x&+&(x)&\operatorname{d}y&+&\operatorname{d}x\ % \operatorname{d}y\\ \end{smallmatrix}$ $\begin{smallmatrix}y&\operatorname{d}x&+&x&\operatorname{d}y\\ (y)&\operatorname{d}x&+&x&\operatorname{d}y\\ y&\operatorname{d}x&+&(x)&\operatorname{d}y\\ (y)&\operatorname{d}x&+&(x)&\operatorname{d}y\\ \end{smallmatrix}$
$f_{15}$ $((~{}))$ $0$ $0$
 Title differential propositional calculus : appendix 2 Canonical name DifferentialPropositionalCalculusAppendix2 Date of creation 2013-03-22 18:09:13 Last modified on 2013-03-22 18:09:13 Owner Jon Awbrey (15246) Last modified by Jon Awbrey (15246) Numerical id 15 Author Jon Awbrey (15246) Entry type Application Classification msc 53A40 Classification msc 39A12 Classification msc 34G99 Classification msc 03B44 Classification msc 03B42 Classification msc 03B05 Related topic DifferentialLogic Related topic MinimalNegationOperator Related topic PropositionalCalculus Related topic ZerothOrderLogic