Inquiry Driven Systems : Appendices

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

 $$\text{Geometric Translation Rule 2}\!$$
 $$\text{If}\!$$ $$P, Q \subseteq X$$ $$\text{and}\!$$ $$p, q ~:~ X \to \underline\mathbb{B}$$ $$\text{such that:}\!$$ $$\text{G2a.}\!$$ $$\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q$$ $$\text{then}\!$$ $$\text{the following equations hold:}\!$$
 $$\text{G2b}_{0}.\!$$ $$\upharpoonleft \varnothing \upharpoonright$$ $$=\!$$ $$(~)$$ $$=\!$$ $$(~)$$ $$\text{G2b}_{1}.\!$$ $$\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright$$ $$=\!$$ $$(\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)$$ $$=\!$$ $$(p)(q)\!$$ $$\text{G2b}_{2}.\!$$ $$\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright$$ $$=\!$$ $$(\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright$$ $$=\!$$ $$(p) q\!$$ $$\text{G2b}_{3}.\!$$ $$\upharpoonleft \overline{P} \upharpoonright$$ $$=\!$$ $$(\upharpoonleft P \upharpoonright)$$ $$=\!$$ $$(p)\!$$ $$\text{G2b}_{4}.\!$$ $$\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright$$ $$=\!$$ $$\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)$$ $$=\!$$ $$p (q)\!$$ $$\text{G2b}_{5}.\!$$ $$\upharpoonleft \overline{Q} \upharpoonright$$ $$=\!$$ $$(\upharpoonleft Q \upharpoonright)$$ $$=\!$$ $$(q)\!$$ $$\text{G2b}_{6}.\!$$ $$\upharpoonleft P ~+~ Q \upharpoonright$$ $$=\!$$ $$(\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)$$ $$=\!$$ $$(p, q)\!$$ $$\text{G2b}_{7}.\!$$ $$\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright$$ $$=\!$$ $$(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)$$ $$=\!$$ $$(p q)\!$$ $$\text{G2b}_{8}.\!$$ $$\upharpoonleft P ~\cap~ Q \upharpoonright$$ $$=\!$$ $$\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright$$ $$=\!$$ $$p q\!$$ $$\text{G2b}_{9}.\!$$ $$\upharpoonleft \overline{P ~+~ Q} \upharpoonright$$ $$=\!$$ $$((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))$$ $$=\!$$ $$((p, q))\!$$ $$\text{G2b}_{10}.\!$$ $$\upharpoonleft Q \upharpoonright$$ $$=\!$$ $$\upharpoonleft Q \upharpoonright$$ $$=\!$$ $$q\!$$ $$\text{G2b}_{11}.\!$$ $$\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright$$ $$=\!$$ $$(\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright))$$ $$=\!$$ $$(p (q))\!$$ $$\text{G2b}_{12}.\!$$ $$\upharpoonleft P \upharpoonright$$ $$=\!$$ $$\upharpoonleft P \upharpoonright$$ $$=\!$$ $$p\!$$ $$\text{G2b}_{13}.\!$$ $$\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright$$ $$=\!$$ $$((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)$$ $$=\!$$ $$((p) q)\!$$ $$\text{G2b}_{14}.\!$$ $$\upharpoonleft P ~\cup~ Q \upharpoonright$$ $$=\!$$ $$((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))$$ $$=\!$$ $$((p)(q))\!$$ $$\text{G2b}_{15}.\!$$ $$\upharpoonleft X \upharpoonright$$ $$=\!$$ $$((~))$$ $$=\!$$ $$((~))$$