PlanetPhysics/Differential Propositional Calculus Appendix 1

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

Table A1. Propositional Forms on Two Variables

Table A1 lists equivalent expressions for the boolean functions of two variables in a number of different notational systems.

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\multicolumn{7}{Table A1. Propositional Forms on Two Variables } \\ \hline ${\mathcal {L}}_{1}$ & ${\mathcal {L}}_{2}$ && ${\mathcal {L}}_{3}$ & ${\mathcal {L}}_{4}$ & ${\mathcal {L}}_{5}$ & ${\mathcal {L}}_{6}$ \\ \hline & & $x=$ & 1 1 0 0 & & & \\ & & $y=$ & 1 0 1 0 & & & \\ \hline $f_{0}$ & $f_{0000}$ && 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 & $(x\ y)$ & $\operatorname {not~both} \ x\ \operatorname {and} \ y$ & $\lnot x\lor \lnot y$ \\ \hline $f_{8}$ & $f_{1000}$ && 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 & $((~))$ & $\operatorname {true}$ & $1$ \\ \hline

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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.

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\multicolumn{7}{Table A2. Propositional Forms on Two Variables } \\ \hline ${\mathcal {L}}_{1}$ & ${\mathcal {L}}_{2}$ && ${\mathcal {L}}_{3}$ & ${\mathcal {L}}_{4}$ & ${\mathcal {L}}_{5}$ & ${\mathcal {L}}_{6}$ \\ \hline & & $x=$ & 1 1 0 0 & & & \\ & & $y=$ & 1 0 1 0 & & & \\ \hline $f_{0}$ & $f_{0000}$ && 0 0 0 0 & $(~)$ & $\operatorname {false}$ & $0$ \\ \hline $f_{1}$ & $f_{0001}$ && 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 & $x\ y$ & $x\ \operatorname {and} \ y$ & $x\land y$ \\ \hline $f_{3}$ & $f_{0011}$ && 0 0 1 1 & $(x)$ & $\operatorname {not} \ x$ & $\lnot x$ \\ $f_{12}$ & $f_{1100}$ && 1 1 0 0 & $x$ & $x$ & $x$ \\ \hline $f_{6}$ & $f_{0110}$ && 0 1 1 0 & $(x,\ y)$ & $x\ \operatorname {not~equal~to} \ y$ & $x\neq y$ \\ $f_{9}$ & $f_{1001}$ && 1 0 0 1 & $((x,\ y))$ & $x\ \operatorname {equal~to} \ y$ & $x=y$ \\ \hline $f_{5}$ & $f_{0101}$ && 0 1 0 1 & $(y)$ & $\operatorname {not} \ y$ & $\lnot y$ \\ $f_{10}$ & $f_{1010}$ && 1 0 1 0 & $y$ & $y$ & $y$ \\ \hline $f_{7}$ & $f_{0111}$ && 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 & $((x)(y))$ & $x\ \operatorname {or} \ y$ & $x\lor y$ \\ \hline $f_{15}$ & $f_{1111}$ && 1 1 1 1 & $((~))$ & $\operatorname {true}$ & $1$ \\ \hline

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

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

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

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

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

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

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

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\multicolumn{6}{Table A6. $\operatorname {D} f$ Expanded Over Ordinary Features $\{x,y\}$} \\ \hline & $f$ & $\operatorname {D} f|_{x\ y}$ & $\operatorname {D} f|_{x(y)}$ & $\operatorname {D} f|_{(x)y}$ & $\operatorname {D} f|_{(x)(y)}$ \\ \hline $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ \hline $f_{1}$ & $(x)(y)$ & $\operatorname {d} x\ \operatorname {d} y$ & $\operatorname {d} x\ (\operatorname {d} y)$ & $(\operatorname {d} x)\ \operatorname {d} y$ & $((\operatorname {d} x)(\operatorname {d} y))$ \\ $f_{2}$ & $(x)\ y$ & $\operatorname {d} x\ (\operatorname {d} y)$ & $\operatorname {d} x\ \operatorname {d} y$ & $((\operatorname {d} x)(\operatorname {d} y))$ & $(\operatorname {d} x)\ \operatorname {d} y$ \\ $f_{4}$ & $x\ (y)$ & $(\operatorname {d} x)\ \operatorname {d} y$ & $((\operatorname {d} x)(\operatorname {d} y))$ & $\operatorname {d} x\ \operatorname {d} y$ & $\operatorname {d} x\ (\operatorname {d} y)$ \\ $f_{8}$ & $x\ y$ & $((\operatorname {d} x)(\operatorname {d} y))$ & $(\operatorname {d} x)\ \operatorname {d} y$ & $\operatorname {d} x\ (\operatorname {d} y)$ & $\operatorname {d} x\ \operatorname {d} y$ \\ \hline $f_{3}$ & $(x)$ & $\operatorname {d} x$ & $\operatorname {d} x$ & $\operatorname {d} x$ & $\operatorname {d} x$ \\ $f_{12}$ & $x$ & $\operatorname {d} x$ & $\operatorname {d} x$ & $\operatorname {d} x$ & $\operatorname {d} x$ \\ \hline $f_{6}$ & $(x,\ y)$ & $(\operatorname {d} x,\ \operatorname {d} y)$ & $(\operatorname {d} x,\ \operatorname {d} y)$ & $(\operatorname {d} x,\ \operatorname {d} y)$ & $(\operatorname {d} x,\ \operatorname {d} y)$ \\ $f_{9}$ & $((x,\ y))$ & $(\operatorname {d} x,\ \operatorname {d} y)$ & $(\operatorname {d} x,\ \operatorname {d} y)$ & $(\operatorname {d} x,\ \operatorname {d} y)$ & $(\operatorname {d} x,\ \operatorname {d} y)$ \\ \hline $f_{5}$ & $(y)$ & $\operatorname {d} y$ & $\operatorname {d} y$ & $\operatorname {d} y$ & $\operatorname {d} y$ \\ $f_{10}$ & $y$ & $\operatorname {d} y$ & $\operatorname {d} y$ & $\operatorname {d} y$ & $\operatorname {d} y$ \\ \hline $f_{7}$ & $(x\ y)$ & $((\operatorname {d} x)(\operatorname {d} y))$ & $(\operatorname {d} x)\ \operatorname {d} y$ & $\operatorname {d} x\ (\operatorname {d} y)$ & $\operatorname {d} x\ \operatorname {d} y$ \\ $f_{11}$ & $(x\ (y))$ & $(\operatorname {d} x)\ \operatorname {d} y$ & $((\operatorname {d} x)(\operatorname {d} y))$ & $\operatorname {d} x\ \operatorname {d} y$ & $\operatorname {d} x\ (\operatorname {d} y)$ \\ $f_{13}$ & $((x)\ y)$ & $\operatorname {d} x\ (\operatorname {d} y)$ & $\operatorname {d} x\ \operatorname {d} y$ & $((\operatorname {d} x)(\operatorname {d} y))$ & $(\operatorname {d} x)\ \operatorname {d} y$ \\ $f_{14}$ & $((x)(y))$ & $\operatorname {d} x\ \operatorname {d} y$ & $\operatorname {d} x\ (\operatorname {d} y)$ & $(\operatorname {d} x)\ \operatorname {d} y$ & $((\operatorname {d} x)(\operatorname {d} y))$ \\ \hline $f_{15}$ & $((~))$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ \hline

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