MyWikiBiz, Author Your Legacy — Wednesday April 09, 2025
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Appendices
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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