Inquiry Driven Systems : Appendices

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• Contents • Part 1 • Part 2 • Part 3 • Appendices • References • Document History •

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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

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

\(\text{If}\!\) \(P, Q \subseteq X\)   \(\text{and}\!\) \(p, q ~:~ X \to \underline\mathbb{B}\)   \(\text{such that:}\!\)     \(\text{G2a.}\!\) \(\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q\)   \(\text{then}\!\) \(\text{the following equations hold:}\!\)

\(\text{G2b}_{0}.\!\) \(\upharpoonleft \varnothing \upharpoonright\) \(=\!\) \((~)\) \(=\!\) \((~)\)   \(\text{G2b}_{1}.\!\) \(\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright\) \(=\!\) \((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)\) \(=\!\) \((p)(q)\!\)   \(\text{G2b}_{2}.\!\) \(\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright\) \(=\!\) \((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright\) \(=\!\) \((p) q\!\)   \(\text{G2b}_{3}.\!\) \(\upharpoonleft \overline{P} \upharpoonright\) \(=\!\) \((\upharpoonleft P \upharpoonright)\) \(=\!\) \((p)\!\)   \(\text{G2b}_{4}.\!\) \(\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright\) \(=\!\) \(\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)\) \(=\!\) \(p (q)\!\)   \(\text{G2b}_{5}.\!\) \(\upharpoonleft \overline{Q} \upharpoonright\) \(=\!\) \((\upharpoonleft Q \upharpoonright)\) \(=\!\) \((q)\!\)   \(\text{G2b}_{6}.\!\) \(\upharpoonleft P ~+~ Q \upharpoonright\) \(=\!\) \((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)\) \(=\!\) \((p, q)\!\)   \(\text{G2b}_{7}.\!\) \(\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright\) \(=\!\) \((\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)\) \(=\!\) \((p q)\!\)   \(\text{G2b}_{8}.\!\) \(\upharpoonleft P ~\cap~ Q \upharpoonright\) \(=\!\) \(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright\) \(=\!\) \(p q\!\)   \(\text{G2b}_{9}.\!\) \(\upharpoonleft \overline{P ~+~ Q} \upharpoonright\) \(=\!\) \(((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))\) \(=\!\) \(((p, q))\!\)   \(\text{G2b}_{10}.\!\) \(\upharpoonleft Q \upharpoonright\) \(=\!\) \(\upharpoonleft Q \upharpoonright\) \(=\!\) \(q\!\)   \(\text{G2b}_{11}.\!\) \(\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright\) \(=\!\) \((\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright))\) \(=\!\) \((p (q))\!\)   \(\text{G2b}_{12}.\!\) \(\upharpoonleft P \upharpoonright\) \(=\!\) \(\upharpoonleft P \upharpoonright\) \(=\!\) \(p\!\)   \(\text{G2b}_{13}.\!\) \(\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright\) \(=\!\) \(((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)\) \(=\!\) \(((p) q)\!\)   \(\text{G2b}_{14}.\!\) \(\upharpoonleft P ~\cup~ Q \upharpoonright\) \(=\!\) \(((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))\) \(=\!\) \(((p)(q))\!\)   \(\text{G2b}_{15}.\!\) \(\upharpoonleft X \upharpoonright\) \(=\!\) \(((~))\) \(=\!\) \(((~))\)

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• Contents • Part 1 • Part 2 • Part 3 • Appendices • References • Document History •

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