# differential propositional calculus : appendix 1

Note. The following Tables are best viewed in the Page Image mode.

## 0.1 Table A1. Propositional Forms on Two Variables

 Table A1. Propositional Forms on Two Variables $\mathcal{L}_{1}$ $\mathcal{L}_{2}$ $\mathcal{L}_{3}$ $\mathcal{L}_{4}$ $\mathcal{L}_{5}$ $\mathcal{L}_{6}$ $x=$ 1 1 0 0 $y=$ 1 0 1 0 $f_{0}$ $f_{0000}$ 0 0 0 0 $(~{})$ $\operatorname{false}$ $0$ $f_{1}$ $f_{0001}$ 0 0 0 1 $(x)(y)$ $\operatorname{neither}\ x\ \operatorname{nor}\ y$ $\lnot x\land\lnot y$ $f_{2}$ $f_{0010}$ 0 0 1 0 $(x)\ y$ $y\ \operatorname{without}\ x$ $\lnot x\land y$ $f_{3}$ $f_{0011}$ 0 0 1 1 $(x)$ $\operatorname{not}\ x$ $\lnot x$ $f_{4}$ $f_{0100}$ 0 1 0 0 $x\ (y)$ $x\ \operatorname{without}\ y$ $x\land\lnot y$ $f_{5}$ $f_{0101}$ 0 1 0 1 $(y)$ $\operatorname{not}\ y$ $\lnot y$ $f_{6}$ $f_{0110}$ 0 1 1 0 $(x,\ y)$ $x\ \operatorname{not~{}equal~{}to}\ y$ $x\neq y$ $f_{7}$ $f_{0111}$ 0 1 1 1 $(x\ y)$ $\operatorname{not~{}both}\ x\ \operatorname{and}\ y$ $\lnot x\lor\lnot y$ $f_{8}$ $f_{1000}$ 1 0 0 0 $x\ y$ $x\ \operatorname{and}\ y$ $x\land y$ $f_{9}$ $f_{1001}$ 1 0 0 1 $((x,\ y))$ $x\ \operatorname{equal~{}to}\ y$ $x=y$ $f_{10}$ $f_{1010}$ 1 0 1 0 $y$ $y$ $y$ $f_{11}$ $f_{1011}$ 1 0 1 1 $(x\ (y))$ $\operatorname{not}\ x\ \operatorname{without}\ y$ $x\Rightarrow y$ $f_{12}$ $f_{1100}$ 1 1 0 0 $x$ $x$ $x$ $f_{13}$ $f_{1101}$ 1 1 0 1 $((x)\ y)$ $\operatorname{not}\ y\ \operatorname{without}\ x$ $x\Leftarrow y$ $f_{14}$ $f_{1110}$ 1 1 1 0 $((x)(y))$ $x\ \operatorname{or}\ y$ $x\lor y$ $f_{15}$ $f_{1111}$ 1 1 1 1 $((~{}))$ $\operatorname{true}$ $1$

## 0.2 Table A2. Propositional Forms on Two Variables

Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.

 Table A2. Propositional Forms on Two Variables $\mathcal{L}_{1}$ $\mathcal{L}_{2}$ $\mathcal{L}_{3}$ $\mathcal{L}_{4}$ $\mathcal{L}_{5}$ $\mathcal{L}_{6}$ $x=$ 1 1 0 0 $y=$ 1 0 1 0 $f_{0}$ $f_{0000}$ 0 0 0 0 $(~{})$ $\operatorname{false}$ $0$ $f_{1}$ $f_{0001}$ 0 0 0 1 $(x)(y)$ $\operatorname{neither}\ x\ \operatorname{nor}\ y$ $\lnot x\land\lnot y$ $f_{2}$ $f_{0010}$ 0 0 1 0 $(x)\ y$ $y\ \operatorname{without}\ x$ $\lnot x\land y$ $f_{4}$ $f_{0100}$ 0 1 0 0 $x\ (y)$ $x\ \operatorname{without}\ y$ $x\land\lnot y$ $f_{8}$ $f_{1000}$ 1 0 0 0 $x\ y$ $x\ \operatorname{and}\ y$ $x\land y$ $f_{3}$ $f_{0011}$ 0 0 1 1 $(x)$ $\operatorname{not}\ x$ $\lnot x$ $f_{12}$ $f_{1100}$ 1 1 0 0 $x$ $x$ $x$ $f_{6}$ $f_{0110}$ 0 1 1 0 $(x,\ y)$ $x\ \operatorname{not~{}equal~{}to}\ y$ $x\neq y$ $f_{9}$ $f_{1001}$ 1 0 0 1 $((x,\ y))$ $x\ \operatorname{equal~{}to}\ y$ $x=y$ $f_{5}$ $f_{0101}$ 0 1 0 1 $(y)$ $\operatorname{not}\ y$ $\lnot y$ $f_{10}$ $f_{1010}$ 1 0 1 0 $y$ $y$ $y$ $f_{7}$ $f_{0111}$ 0 1 1 1 $(x\ y)$ $\operatorname{not~{}both}\ x\ \operatorname{and}\ y$ $\lnot x\lor\lnot y$ $f_{11}$ $f_{1011}$ 1 0 1 1 $(x\ (y))$ $\operatorname{not}\ x\ \operatorname{without}\ y$ $x\Rightarrow y$ $f_{13}$ $f_{1101}$ 1 1 0 1 $((x)\ y)$ $\operatorname{not}\ y\ \operatorname{without}\ x$ $x\Leftarrow y$ $f_{14}$ $f_{1110}$ 1 1 1 0 $((x)(y))$ $x\ \operatorname{or}\ y$ $x\lor y$ $f_{15}$ $f_{1111}$ 1 1 1 1 $((~{}))$ $\operatorname{true}$ $1$

## 0.3 Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{\operatorname{d}x,\operatorname{d}y\}$

Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{\operatorname{d}x,\operatorname{d}y\}$
$\operatorname{T}_{11}$ $\operatorname{T}_{10}$ $\operatorname{T}_{01}$ $\operatorname{T}_{00}$
$f$ $\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$ $\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}$ $\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}$ $\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$
$f_{0}$ $(~{})$ $(~{})$ $(~{})$ $(~{})$ $(~{})$
$f_{1}$ $(x)(y)$ $x\ y$ $x\ (y)$ $(x)\ y$ $(x)(y)$
$f_{2}$ $(x)\ y$ $x\ (y)$ $x\ y$ $(x)(y)$ $(x)\ y$
$f_{4}$ $x\ (y)$ $(x)\ y$ $(x)(y)$ $x\ y$ $x\ (y)$
$f_{8}$ $x\ y$ $(x)(y)$ $(x)\ y$ $x\ (y)$ $x\ y$
$f_{3}$ $(x)$ $x$ $x$ $(x)$ $(x)$
$f_{12}$ $x$ $(x)$ $(x)$ $x$ $x$
$f_{6}$ $(x,\ y)$ $(x,\ y)$ $((x,\ y))$ $((x,\ y))$ $(x,\ y)$
$f_{9}$ $((x,\ y))$ $((x,\ y))$ $(x,\ y)$ $(x,\ y)$ $((x,\ y))$
$f_{5}$ $(y)$ $y$ $(y)$ $y$ $(y)$
$f_{10}$ $y$ $(y)$ $y$ $(y)$ $y$
$f_{7}$ $(x\ y)$ $((x)(y))$ $((x)\ y)$ $(x\ (y))$ $(x\ y)$
$f_{11}$ $(x\ (y))$ $((x)\ y)$ $((x)(y))$ $(x\ y)$ $(x\ (y))$
$f_{13}$ $((x)\ y)$ $(x\ (y))$ $(x\ y)$ $((x)(y))$ $((x)\ y)$
$f_{14}$ $((x)(y))$ $(x\ y)$ $(x\ (y))$ $((x)\ y)$ $((x)(y))$
$f_{15}$ $((~{}))$ $((~{}))$ $((~{}))$ $((~{}))$ $((~{}))$
Fixed Point  (http://planetmath.org/FixedPoint) Total: 4 4 4 16

## 0.4 Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{\operatorname{d}x,\operatorname{d}y\}$

Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{\operatorname{d}x,\operatorname{d}y\}$
$f$ $\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ $\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}$ $\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}$ $\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$
$f_{0}$ $(~{})$ $(~{})$ $(~{})$ $(~{})$ $(~{})$
$f_{1}$ $(x)(y)$ $((x,\ y))$ $(y)$ $(x)$ $(~{})$
$f_{2}$ $(x)\ y$ $(x,\ y)$ $y$ $(x)$ $(~{})$
$f_{4}$ $x\ (y)$ $(x,\ y)$ $(y)$ $x$ $(~{})$
$f_{8}$ $x\ y$ $((x,\ y))$ $y$ $x$ $(~{})$
$f_{3}$ $(x)$ $((~{}))$ $((~{}))$ $(~{})$ $(~{})$
$f_{12}$ $x$ $((~{}))$ $((~{}))$ $(~{})$ $(~{})$
$f_{6}$ $(x,\ y)$ $(~{})$ $((~{}))$ $((~{}))$ $(~{})$
$f_{9}$ $((x,\ y))$ $(~{})$ $((~{}))$ $((~{}))$ $(~{})$
$f_{5}$ $(y)$ $((~{}))$ $(~{})$ $((~{}))$ $(~{})$
$f_{10}$ $y$ $((~{}))$ $(~{})$ $((~{}))$ $(~{})$
$f_{7}$ $(x\ y)$ $((x,\ y))$ $y$ $x$ $(~{})$
$f_{11}$ $(x\ (y))$ $(x,\ y)$ $(y)$ $x$ $(~{})$
$f_{13}$ $((x)\ y)$ $(x,\ y)$ $y$ $(x)$ $(~{})$
$f_{14}$ $((x)(y))$ $((x,\ y))$ $(y)$ $(x)$ $(~{})$
$f_{15}$ $((~{}))$ $(~{})$ $(~{})$ $(~{})$ $(~{})$

## 0.5 Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{x,y\}$

Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{x,y\}$
$f$ $\operatorname{E}f|_{x\ y}$ $\operatorname{E}f|_{x(y)}$ $\operatorname{E}f|_{(x)y}$ $\operatorname{E}f|_{(x)(y)}$
$f_{0}$ $(~{})$ $(~{})$ $(~{})$ $(~{})$ $(~{})$
$f_{1}$ $(x)(y)$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $(\operatorname{d}x)(\operatorname{d}y)$
$f_{2}$ $(x)\ y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $(\operatorname{d}x)(\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$
$f_{4}$ $x\ (y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $(\operatorname{d}x)(\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$
$f_{8}$ $x\ y$ $(\operatorname{d}x)(\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$
$f_{3}$ $(x)$ $\operatorname{d}x$ $\operatorname{d}x$ $(\operatorname{d}x)$ $(\operatorname{d}x)$
$f_{12}$ $x$ $(\operatorname{d}x)$ $(\operatorname{d}x)$ $\operatorname{d}x$ $\operatorname{d}x$
$f_{6}$ $(x,\ y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $((\operatorname{d}x,\ \operatorname{d}y))$ $((\operatorname{d}x,\ \operatorname{d}y))$ $(\operatorname{d}x,\ \operatorname{d}y)$
$f_{9}$ $((x,\ y))$ $((\operatorname{d}x,\ \operatorname{d}y))$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $((\operatorname{d}x,\ \operatorname{d}y))$
$f_{5}$ $(y)$ $\operatorname{d}y$ $(\operatorname{d}y)$ $\operatorname{d}y$ $(\operatorname{d}y)$
$f_{10}$ $y$ $(\operatorname{d}y)$ $\operatorname{d}y$ $(\operatorname{d}y)$ $\operatorname{d}y$
$f_{7}$ $(x\ y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $((\operatorname{d}x)\ \operatorname{d}y)$ $(\operatorname{d}x\ (\operatorname{d}y))$ $(\operatorname{d}x\ \operatorname{d}y)$
$f_{11}$ $(x\ (y))$ $((\operatorname{d}x)\ \operatorname{d}y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x\ \operatorname{d}y)$ $(\operatorname{d}x\ (\operatorname{d}y))$
$f_{13}$ $((x)\ y)$ $(\operatorname{d}x\ (\operatorname{d}y))$ $(\operatorname{d}x\ \operatorname{d}y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $((\operatorname{d}x)\ \operatorname{d}y)$
$f_{14}$ $((x)(y))$ $(\operatorname{d}x\ \operatorname{d}y)$ $(\operatorname{d}x\ (\operatorname{d}y))$ $((\operatorname{d}x)\ \operatorname{d}y)$ $((\operatorname{d}x)(\operatorname{d}y))$
$f_{15}$ $((~{}))$ $((~{}))$ $((~{}))$ $((~{}))$ $((~{}))$

## 0.6 Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{x,y\}$

Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{x,y\}$
$f$ $\operatorname{D}f|_{x\ y}$ $\operatorname{D}f|_{x(y)}$ $\operatorname{D}f|_{(x)y}$ $\operatorname{D}f|_{(x)(y)}$
$f_{0}$ $(~{})$ $(~{})$ $(~{})$ $(~{})$ $(~{})$
$f_{1}$ $(x)(y)$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$
$f_{2}$ $(x)\ y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$
$f_{4}$ $x\ (y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$
$f_{8}$ $x\ y$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$
$f_{3}$ $(x)$ $\operatorname{d}x$ $\operatorname{d}x$ $\operatorname{d}x$ $\operatorname{d}x$
$f_{12}$ $x$ $\operatorname{d}x$ $\operatorname{d}x$ $\operatorname{d}x$ $\operatorname{d}x$
$f_{6}$ $(x,\ y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$
$f_{9}$ $((x,\ y))$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$
$f_{5}$ $(y)$ $\operatorname{d}y$ $\operatorname{d}y$ $\operatorname{d}y$ $\operatorname{d}y$
$f_{10}$ $y$ $\operatorname{d}y$ $\operatorname{d}y$ $\operatorname{d}y$ $\operatorname{d}y$
$f_{7}$ $(x\ y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$
$f_{11}$ $(x\ (y))$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$
$f_{13}$ $((x)\ y)$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$
$f_{14}$ $((x)(y))$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$
$f_{15}$ $((~{}))$ $(~{})$ $(~{})$ $(~{})$ $(~{})$
 Title differential propositional calculus : appendix 1 Canonical name DifferentialPropositionalCalculusAppendix1 Date of creation 2013-11-16 13:40:11 Last modified on 2013-11-16 13:40:11 Owner Jon Awbrey (15246) Last modified by Jon Awbrey (15246) Numerical id 20 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