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Appendices
Logical Translation Rule 1
\(\text{Logical Translation Rule 1}\!\)




\(\text{If}\!\)

\(s ~\text{is a sentence about things in the universe}~ X\)


\(\text{and}\!\)

\(p ~\text{is a proposition} ~:~ X \to \underline\mathbb{B}\)


\(\text{such that:}\!\)



\(\text{L1a.}\!\)

\(\downharpoonleft s \downharpoonright ~=~ p\)


\(\text{then}\!\)

\(\text{the following equations hold:}\!\)



\(\text{L1b}_{00}.\!\)

\(\downharpoonleft \operatorname{false} \downharpoonright\)

\(=\!\)

\((~)\)

\(=\!\)

\(\underline{0} ~:~ X \to \underline\mathbb{B}\)


\(\text{L1b}_{01}.\!\)

\(\downharpoonleft \operatorname{not}~ s \downharpoonright\)

\(=\!\)

\((\downharpoonleft s \downharpoonright)\)

\(=\!\)

\((p) ~:~ X \to \underline\mathbb{B}\)


\(\text{L1b}_{10}.\!\)

\(\downharpoonleft s \downharpoonright\)

\(=\!\)

\(\downharpoonleft s \downharpoonright\)

\(=\!\)

\(p ~:~ X \to \underline\mathbb{B}\)


\(\text{L1b}_{11}.\!\)

\(\downharpoonleft \operatorname{true} \downharpoonright\)

\(=\!\)

\(((~))\)

\(=\!\)

\(\underline{1} ~:~ X \to \underline\mathbb{B}\)


Geometric Translation Rule 1
\(\text{Geometric Translation Rule 1}\!\)




\(\text{If}\!\)

\(Q \subseteq X\)


\(\text{and}\!\)

\(p ~:~ X \to \underline\mathbb{B}\)


\(\text{such that:}\!\)



\(\text{G1a.}\!\)

\(\upharpoonleft Q \upharpoonright ~=~ p\)


\(\text{then}\!\)

\(\text{the following equations hold:}\!\)



\(\text{G1b}_{00}.\!\)

\(\upharpoonleft \varnothing \upharpoonright\)

\(=\!\)

\((~)\)

\(=\!\)

\(\underline{0} ~:~ X \to \underline\mathbb{B}\)


\(\text{G1b}_{01}.\!\)

\(\upharpoonleft {}^{_\sim} Q \upharpoonright\)

\(=\!\)

\((\upharpoonleft Q \upharpoonright)\)

\(=\!\)

\((p) ~:~ X \to \underline\mathbb{B}\)


\(\text{G1b}_{10}.\!\)

\(\upharpoonleft Q \upharpoonright\)

\(=\!\)

\(\upharpoonleft Q \upharpoonright\)

\(=\!\)

\(p ~:~ X \to \underline\mathbb{B}\)


\(\text{G1b}_{11}.\!\)

\(\upharpoonleft X \upharpoonright\)

\(=\!\)

\(((~))\)

\(=\!\)

\(\underline{1} ~:~ X \to \underline\mathbb{B}\)


Logical Translation Rule 2
\(\text{Logical Translation Rule 2}\!\)




\(\text{If}\!\)

\(s, t ~\text{are sentences about things in the universe}~ X\)


\(\text{and}\!\)

\(p, q ~\text{are propositions} ~:~ X \to \underline\mathbb{B}\)


\(\text{such that:}\!\)



\(\text{L2a.}\!\)

\(\downharpoonleft s \downharpoonright ~=~ p \quad \operatorname{and} \quad \downharpoonleft t \downharpoonright ~=~ q\)


\(\text{then}\!\)

\(\text{the following equations hold:}\!\)



\(\text{L2b}_{0}.\!\)

\(\downharpoonleft \operatorname{false} \downharpoonright\)

\(=\!\)

\((~)\)

\(=\!\)

\((~)\)


\(\text{L2b}_{1}.\!\)

\(\downharpoonleft \operatorname{neither}~ s ~\operatorname{nor}~ t \downharpoonright\)

\(=\!\)

\((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright)\)

\(=\!\)

\((p)(q)\!\)


\(\text{L2b}_{2}.\!\)

\(\downharpoonleft \operatorname{not}~ s ~\operatorname{but}~ t \downharpoonright\)

\(=\!\)

\((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright\)

\(=\!\)

\((p) q\!\)


\(\text{L2b}_{3}.\!\)

\(\downharpoonleft \operatorname{not}~ s \downharpoonright\)

\(=\!\)

\((\downharpoonleft s \downharpoonright)\)

\(=\!\)

\((p)\!\)


\(\text{L2b}_{4}.\!\)

\(\downharpoonleft s ~\operatorname{and~not}~ t \downharpoonright\)

\(=\!\)

\(\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright)\)

\(=\!\)

\(p (q)\!\)


\(\text{L2b}_{5}.\!\)

\(\downharpoonleft \operatorname{not}~ t \downharpoonright\)

\(=\!\)

\((\downharpoonleft t \downharpoonright)\)

\(=\!\)

\((q)\!\)


\(\text{L2b}_{6}.\!\)

\(\downharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \downharpoonright\)

\(=\!\)

\((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright)\)

\(=\!\)

\((p, q)\!\)


\(\text{L2b}_{7}.\!\)

\(\downharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \downharpoonright\)

\(=\!\)

\((\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright)\)

\(=\!\)

\((p q)\!\)


\(\text{L2b}_{8}.\!\)

\(\downharpoonleft s ~\operatorname{and}~ t \downharpoonright\)

\(=\!\)

\(\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright\)

\(=\!\)

\(p q\!\)


\(\text{L2b}_{9}.\!\)

\(\downharpoonleft s ~\operatorname{is~equivalent~to}~ t \downharpoonright\)

\(=\!\)

\(((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright))\)

\(=\!\)

\(((p, q))\!\)


\(\text{L2b}_{10}.\!\)

\(\downharpoonleft t \downharpoonright\)

\(=\!\)

\(\downharpoonleft t \downharpoonright\)

\(=\!\)

\(q\!\)


\(\text{L2b}_{11}.\!\)

\(\downharpoonleft s ~\operatorname{implies}~ t \downharpoonright\)

\(=\!\)

\((\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright))\)

\(=\!\)

\((p (q))\!\)


\(\text{L2b}_{12}.\!\)

\(\downharpoonleft s \downharpoonright\)

\(=\!\)

\(\downharpoonleft s \downharpoonright\)

\(=\!\)

\(p\!\)


\(\text{L2b}_{13}.\!\)

\(\downharpoonleft s ~\operatorname{is~implied~by}~ t \downharpoonright\)

\(=\!\)

\(((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright)\)

\(=\!\)

\(((p) q)\!\)


\(\text{L2b}_{14}.\!\)

\(\downharpoonleft s ~\operatorname{or}~ t \downharpoonright\)

\(=\!\)

\(((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright))\)

\(=\!\)

\(((p)(q))\!\)


\(\text{L2b}_{15}.\!\)

\(\downharpoonleft \operatorname{true} \downharpoonright\)

\(=\!\)

\(((~))\)

\(=\!\)

\(((~))\)


Geometric Translation Rule 2
\(\text{Geometric Translation Rule 2}\!\)




\(\text{If}\!\)

\(P, Q \subseteq X\)


\(\text{and}\!\)

\(p, q ~:~ X \to \underline\mathbb{B}\)


\(\text{such that:}\!\)



\(\text{G2a.}\!\)

\(\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q\)


\(\text{then}\!\)

\(\text{the following equations hold:}\!\)



\(\text{G2b}_{0}.\!\)

\(\upharpoonleft \varnothing \upharpoonright\)

\(=\!\)

\((~)\)

\(=\!\)

\((~)\)


\(\text{G2b}_{1}.\!\)

\(\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright\)

\(=\!\)

\((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)\)

\(=\!\)

\((p)(q)\!\)


\(\text{G2b}_{2}.\!\)

\(\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright\)

\(=\!\)

\((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright\)

\(=\!\)

\((p) q\!\)


\(\text{G2b}_{3}.\!\)

\(\upharpoonleft \overline{P} \upharpoonright\)

\(=\!\)

\((\upharpoonleft P \upharpoonright)\)

\(=\!\)

\((p)\!\)


\(\text{G2b}_{4}.\!\)

\(\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright\)

\(=\!\)

\(\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)\)

\(=\!\)

\(p (q)\!\)


\(\text{G2b}_{5}.\!\)

\(\upharpoonleft \overline{Q} \upharpoonright\)

\(=\!\)

\((\upharpoonleft Q \upharpoonright)\)

\(=\!\)

\((q)\!\)


\(\text{G2b}_{6}.\!\)

\(\upharpoonleft P ~+~ Q \upharpoonright\)

\(=\!\)

\((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)\)

\(=\!\)

\((p, q)\!\)


\(\text{G2b}_{7}.\!\)

\(\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright\)

\(=\!\)

\((\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)\)

\(=\!\)

\((p q)\!\)


\(\text{G2b}_{8}.\!\)

\(\upharpoonleft P ~\cap~ Q \upharpoonright\)

\(=\!\)

\(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright\)

\(=\!\)

\(p q\!\)


\(\text{G2b}_{9}.\!\)

\(\upharpoonleft \overline{P ~+~ Q} \upharpoonright\)

\(=\!\)

\(((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))\)

\(=\!\)

\(((p, q))\!\)


\(\text{G2b}_{10}.\!\)

\(\upharpoonleft Q \upharpoonright\)

\(=\!\)

\(\upharpoonleft Q \upharpoonright\)

\(=\!\)

\(q\!\)


\(\text{G2b}_{11}.\!\)

\(\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright\)

\(=\!\)

\((\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright))\)

\(=\!\)

\((p (q))\!\)


\(\text{G2b}_{12}.\!\)

\(\upharpoonleft P \upharpoonright\)

\(=\!\)

\(\upharpoonleft P \upharpoonright\)

\(=\!\)

\(p\!\)


\(\text{G2b}_{13}.\!\)

\(\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright\)

\(=\!\)

\(((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)\)

\(=\!\)

\(((p) q)\!\)


\(\text{G2b}_{14}.\!\)

\(\upharpoonleft P ~\cup~ Q \upharpoonright\)

\(=\!\)

\(((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))\)

\(=\!\)

\(((p)(q))\!\)


\(\text{G2b}_{15}.\!\)

\(\upharpoonleft X \upharpoonright\)

\(=\!\)

\(((~))\)

\(=\!\)

\(((~))\)

