MyWikiBiz, Author Your Legacy — Thursday October 03, 2024
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Appendices
Logical Translation Rule 1
\(\text{Logical Translation Rule 1}\!\)
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\(\text{If}\!\)
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\(s ~\text{is a sentence about things in the universe}~ X\)
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\(\text{and}\!\)
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\(p ~\text{is a proposition} ~:~ X \to \underline\mathbb{B}\)
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\(\text{such that:}\!\)
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\(\text{L1a.}\!\)
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\(\downharpoonleft s \downharpoonright ~=~ p\)
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\(\text{then}\!\)
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\(\text{the following equations hold:}\!\)
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\(\text{L1b}_{00}.\!\)
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\(\downharpoonleft \operatorname{false} \downharpoonright\)
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\(=\!\)
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\((~)\)
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\(=\!\)
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\(\underline{0} ~:~ X \to \underline\mathbb{B}\)
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\(\text{L1b}_{01}.\!\)
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\(\downharpoonleft \operatorname{not}~ s \downharpoonright\)
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\(=\!\)
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\((\downharpoonleft s \downharpoonright)\)
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\(=\!\)
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\((p) ~:~ X \to \underline\mathbb{B}\)
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\(\text{L1b}_{10}.\!\)
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\(\downharpoonleft s \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft s \downharpoonright\)
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\(=\!\)
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\(p ~:~ X \to \underline\mathbb{B}\)
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\(\text{L1b}_{11}.\!\)
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\(\downharpoonleft \operatorname{true} \downharpoonright\)
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\(=\!\)
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\(((~))\)
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\(=\!\)
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\(\underline{1} ~:~ X \to \underline\mathbb{B}\)
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Geometric Translation Rule 1
\(\text{Geometric Translation Rule 1}\!\)
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\(\text{If}\!\)
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\(Q \subseteq X\)
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\(\text{and}\!\)
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\(p ~:~ X \to \underline\mathbb{B}\)
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\(\text{such that:}\!\)
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\(\text{G1a.}\!\)
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\(\upharpoonleft Q \upharpoonright ~=~ p\)
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\(\text{then}\!\)
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\(\text{the following equations hold:}\!\)
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\(\text{G1b}_{00}.\!\)
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\(\upharpoonleft \varnothing \upharpoonright\)
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\(=\!\)
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\((~)\)
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\(=\!\)
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\(\underline{0} ~:~ X \to \underline\mathbb{B}\)
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\(\text{G1b}_{01}.\!\)
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\(\upharpoonleft {}^{_\sim} Q \upharpoonright\)
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\(=\!\)
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\((\upharpoonleft Q \upharpoonright)\)
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\(=\!\)
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\((p) ~:~ X \to \underline\mathbb{B}\)
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\(\text{G1b}_{10}.\!\)
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\(\upharpoonleft Q \upharpoonright\)
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\(=\!\)
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\(\upharpoonleft Q \upharpoonright\)
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\(=\!\)
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\(p ~:~ X \to \underline\mathbb{B}\)
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\(\text{G1b}_{11}.\!\)
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\(\upharpoonleft X \upharpoonright\)
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\(=\!\)
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\(((~))\)
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\(=\!\)
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\(\underline{1} ~:~ X \to \underline\mathbb{B}\)
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Logical Translation Rule 2
\(\text{Logical Translation Rule 2}\!\)
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\(\text{If}\!\)
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\(s, t ~\text{are sentences about things in the universe}~ X\)
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\(\text{and}\!\)
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\(p, q ~\text{are propositions} ~:~ X \to \underline\mathbb{B}\)
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\(\text{such that:}\!\)
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\(\text{L2a.}\!\)
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\(\downharpoonleft s \downharpoonright ~=~ p \quad \operatorname{and} \quad \downharpoonleft t \downharpoonright ~=~ q\)
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\(\text{then}\!\)
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\(\text{the following equations hold:}\!\)
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\(\text{L2b}_{0}.\!\)
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\(\downharpoonleft \operatorname{false} \downharpoonright\)
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\(=\!\)
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\((~)\)
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\(=\!\)
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\((~)\)
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\(\text{L2b}_{1}.\!\)
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\(\downharpoonleft \operatorname{neither}~ s ~\operatorname{nor}~ t \downharpoonright\)
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\(=\!\)
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\((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright)\)
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\(=\!\)
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\((p)(q)\!\)
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\(\text{L2b}_{2}.\!\)
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\(\downharpoonleft \operatorname{not}~ s ~\operatorname{but}~ t \downharpoonright\)
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\(=\!\)
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\((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright\)
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\(=\!\)
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\((p) q\!\)
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\(\text{L2b}_{3}.\!\)
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\(\downharpoonleft \operatorname{not}~ s \downharpoonright\)
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\(=\!\)
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\((\downharpoonleft s \downharpoonright)\)
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\(=\!\)
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\((p)\!\)
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\(\text{L2b}_{4}.\!\)
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\(\downharpoonleft s ~\operatorname{and~not}~ t \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright)\)
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\(=\!\)
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\(p (q)\!\)
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\(\text{L2b}_{5}.\!\)
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\(\downharpoonleft \operatorname{not}~ t \downharpoonright\)
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\(=\!\)
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\((\downharpoonleft t \downharpoonright)\)
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\(=\!\)
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\((q)\!\)
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\(\text{L2b}_{6}.\!\)
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\(\downharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \downharpoonright\)
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\(=\!\)
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\((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright)\)
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\(=\!\)
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\((p, q)\!\)
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\(\text{L2b}_{7}.\!\)
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\(\downharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \downharpoonright\)
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\(=\!\)
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\((\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright)\)
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\(=\!\)
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\((p q)\!\)
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\(\text{L2b}_{8}.\!\)
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\(\downharpoonleft s ~\operatorname{and}~ t \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright\)
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\(=\!\)
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\(p q\!\)
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\(\text{L2b}_{9}.\!\)
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\(\downharpoonleft s ~\operatorname{is~equivalent~to}~ t \downharpoonright\)
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\(=\!\)
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\(((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright))\)
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\(=\!\)
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\(((p, q))\!\)
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\(\text{L2b}_{10}.\!\)
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\(\downharpoonleft t \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft t \downharpoonright\)
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\(=\!\)
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\(q\!\)
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\(\text{L2b}_{11}.\!\)
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\(\downharpoonleft s ~\operatorname{implies}~ t \downharpoonright\)
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\(=\!\)
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\((\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright))\)
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\(=\!\)
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\((p (q))\!\)
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\(\text{L2b}_{12}.\!\)
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\(\downharpoonleft s \downharpoonright\)
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\(=\!\)
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\(\downharpoonleft s \downharpoonright\)
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\(=\!\)
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\(p\!\)
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\(\text{L2b}_{13}.\!\)
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\(\downharpoonleft s ~\operatorname{is~implied~by}~ t \downharpoonright\)
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\(=\!\)
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\(((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright)\)
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\(=\!\)
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\(((p) q)\!\)
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\(\text{L2b}_{14}.\!\)
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\(\downharpoonleft s ~\operatorname{or}~ t \downharpoonright\)
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\(=\!\)
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\(((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright))\)
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\(=\!\)
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\(((p)(q))\!\)
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\(\text{L2b}_{15}.\!\)
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\(\downharpoonleft \operatorname{true} \downharpoonright\)
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\(=\!\)
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\(((~))\)
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\(=\!\)
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\(((~))\)
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Geometric Translation Rule 2
\(\text{Geometric Translation Rule 2}\!\)
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\(\text{If}\!\)
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\(P, Q \subseteq X\)
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\(\text{and}\!\)
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\(p, q ~:~ X \to \underline\mathbb{B}\)
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\(\text{such that:}\!\)
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\(\text{G2a.}\!\)
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\(\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q\)
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\(\text{then}\!\)
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\(\text{the following equations hold:}\!\)
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\(\text{G2b}_{0}.\!\)
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\(\upharpoonleft \varnothing \upharpoonright\)
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\(=\!\)
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\((~)\)
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\(=\!\)
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\((~)\)
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\(\text{G2b}_{1}.\!\)
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\(\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright\)
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\(=\!\)
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\((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)\)
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\(=\!\)
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\((p)(q)\!\)
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\(\text{G2b}_{2}.\!\)
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\(\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright\)
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\(=\!\)
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\((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright\)
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\(=\!\)
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\((p) q\!\)
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\(\text{G2b}_{3}.\!\)
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\(\upharpoonleft \overline{P} \upharpoonright\)
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\(=\!\)
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\((\upharpoonleft P \upharpoonright)\)
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\(=\!\)
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\((p)\!\)
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\(\text{G2b}_{4}.\!\)
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\(\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright\)
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\(=\!\)
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\(\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)\)
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\(=\!\)
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\(p (q)\!\)
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\(\text{G2b}_{5}.\!\)
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\(\upharpoonleft \overline{Q} \upharpoonright\)
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\(=\!\)
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\((\upharpoonleft Q \upharpoonright)\)
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\(=\!\)
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\((q)\!\)
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\(\text{G2b}_{6}.\!\)
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\(\upharpoonleft P ~+~ Q \upharpoonright\)
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\(=\!\)
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\((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)\)
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\(=\!\)
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\((p, q)\!\)
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\(\text{G2b}_{7}.\!\)
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\(\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright\)
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\(=\!\)
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\((\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)\)
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\(=\!\)
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\((p q)\!\)
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\(\text{G2b}_{8}.\!\)
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\(\upharpoonleft P ~\cap~ Q \upharpoonright\)
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\(=\!\)
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\(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright\)
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\(=\!\)
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\(p q\!\)
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\(\text{G2b}_{9}.\!\)
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\(\upharpoonleft \overline{P ~+~ Q} \upharpoonright\)
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\(=\!\)
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\(((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))\)
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\(=\!\)
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\(((p, q))\!\)
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\(\text{G2b}_{10}.\!\)
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\(\upharpoonleft Q \upharpoonright\)
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\(=\!\)
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\(\upharpoonleft Q \upharpoonright\)
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\(=\!\)
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\(q\!\)
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\(\text{G2b}_{11}.\!\)
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\(\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright\)
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\(=\!\)
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\((\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright))\)
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\(=\!\)
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\((p (q))\!\)
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\(\text{G2b}_{12}.\!\)
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\(\upharpoonleft P \upharpoonright\)
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\(=\!\)
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\(\upharpoonleft P \upharpoonright\)
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\(=\!\)
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\(p\!\)
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\(\text{G2b}_{13}.\!\)
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\(\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright\)
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\(=\!\)
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\(((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)\)
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\(=\!\)
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\(((p) q)\!\)
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\(\text{G2b}_{14}.\!\)
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\(\upharpoonleft P ~\cup~ Q \upharpoonright\)
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\(=\!\)
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\(((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))\)
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\(=\!\)
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\(((p)(q))\!\)
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\(\text{G2b}_{15}.\!\)
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\(\upharpoonleft X \upharpoonright\)
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\(=\!\)
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\(((~))\)
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\(=\!\)
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\(((~))\)
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