Inquiry Driven Systems : Appendices

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Author: Jon Awbrey


ContentsPart 1Part 2Part 3Part 4Part 5Part 6Part 7Part 8Part 9Part 10AppendicesReferencesDocument History


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 =\! ((~)) =\! ((~))



ContentsPart 1Part 2Part 3Part 4Part 5Part 6Part 7Part 8Part 9Part 10AppendicesReferencesDocument History