Inquiry Driven Systems • Appendix A1

Author: Jon Awbrey

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• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Part 6 • Part 7 • Part 8 • Part 9 • Part 10 • Part 11 • Part 12 • Part 13 • Part 14 • Part 15 • Part 16 • Appendix A1 • Appendix A2 • References • Document History •

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Translation Rules

Logical Translation Rule 1

${\text{Logical Translation Rule 1}}$

${\text{If}}$ $s~{\text{is a sentence about things in the universe}}~X$   ${\text{and}}$ $p~{\text{is a proposition}}~:~X\to {\underline {\mathbb {B} }}$   ${\text{such that:}}$     ${\text{L1a.}}$ $\downharpoonleft s\downharpoonright ~=~p$   ${\text{then}}$ ${\text{the following equations hold:}}$

${\text{L1b}}_{00}.$ $\downharpoonleft \mathrm {false} \downharpoonright$ $=$ $(~)$ $=$ ${\underline {0}}~:~X\to {\underline {\mathbb {B} }}$   ${\text{L1b}}_{01}.$ $\downharpoonleft \mathrm {not} ~s\downharpoonright$ $=$ $(\downharpoonleft s\downharpoonright )$ $=$ $(p)~:~X\to {\underline {\mathbb {B} }}$   ${\text{L1b}}_{10}.$ $\downharpoonleft s\downharpoonright$ $=$ $\downharpoonleft s\downharpoonright$ $=$ $p~:~X\to {\underline {\mathbb {B} }}$   ${\text{L1b}}_{11}.$ $\downharpoonleft \mathrm {true} \downharpoonright$ $=$ $((~))$ $=$ ${\underline {1}}~:~X\to {\underline {\mathbb {B} }}$

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

${\text{Logical Translation Rule 2}}$

${\text{If}}$ $s,t~{\text{are sentences about things in the universe}}~X$   ${\text{and}}$ $p,q~{\text{are propositions}}~:~X\to {\underline {\mathbb {B} }}$   ${\text{such that:}}$     ${\text{L2a.}}$ $\downharpoonleft s\downharpoonright ~=~p\quad \mathrm {and} \quad \downharpoonleft t\downharpoonright ~=~q$   ${\text{then}}$ ${\text{the following equations hold:}}$

${\text{L2b}}_{0}.$ $\downharpoonleft \mathrm {false} \downharpoonright$ $=$ $(~)$ $=$ $(~)$   ${\text{L2b}}_{1}.$ $\downharpoonleft \mathrm {neither} ~s~\mathrm {nor} ~t\downharpoonright$ $=$ $(\downharpoonleft s\downharpoonright )(\downharpoonleft t\downharpoonright )$ $=$ $(p)(q)$   ${\text{L2b}}_{2}.$ $\downharpoonleft \mathrm {not} ~s~\mathrm {but} ~t\downharpoonright$ $=$ $(\downharpoonleft s\downharpoonright )\downharpoonleft t\downharpoonright$ $=$ $(p)q$   ${\text{L2b}}_{3}.$ $\downharpoonleft \mathrm {not} ~s\downharpoonright$ $=$ $(\downharpoonleft s\downharpoonright )$ $=$ $(p)$   ${\text{L2b}}_{4}.$ $\downharpoonleft s~\mathrm {and~not} ~t\downharpoonright$ $=$ $\downharpoonleft s\downharpoonright (\downharpoonleft t\downharpoonright )$ $=$ $p(q)$   ${\text{L2b}}_{5}.$ $\downharpoonleft \mathrm {not} ~t\downharpoonright$ $=$ $(\downharpoonleft t\downharpoonright )$ $=$ $(q)$   ${\text{L2b}}_{6}.$ $\downharpoonleft s~\mathrm {or} ~t,~\mathrm {not~both} \downharpoonright$ $=$ $(\downharpoonleft s\downharpoonright ~,~\downharpoonleft t\downharpoonright )$ $=$ $(p,q)$   ${\text{L2b}}_{7}.$ $\downharpoonleft \mathrm {not~both} ~s~\mathrm {and} ~t\downharpoonright$ $=$ $(\downharpoonleft s\downharpoonright ~\downharpoonleft t\downharpoonright )$ $=$ $(pq)$   ${\text{L2b}}_{8}.$ $\downharpoonleft s~\mathrm {and} ~t\downharpoonright$ $=$ $\downharpoonleft s\downharpoonright ~\downharpoonleft t\downharpoonright$ $=$ $pq$   ${\text{L2b}}_{9}.$ $\downharpoonleft s~\mathrm {is~equivalent~to} ~t\downharpoonright$ $=$ $((\downharpoonleft s\downharpoonright ~,~\downharpoonleft t\downharpoonright ))$ $=$ $((p,q))$   ${\text{L2b}}_{10}.$ $\downharpoonleft t\downharpoonright$ $=$ $\downharpoonleft t\downharpoonright$ $=$ $q$   ${\text{L2b}}_{11}.$ $\downharpoonleft s~\mathrm {implies} ~t\downharpoonright$ $=$ $(\downharpoonleft s\downharpoonright (\downharpoonleft t\downharpoonright ))$ $=$ $(p(q))$   ${\text{L2b}}_{12}.$ $\downharpoonleft s\downharpoonright$ $=$ $\downharpoonleft s\downharpoonright$ $=$ $p$   ${\text{L2b}}_{13}.$ $\downharpoonleft s~\mathrm {is~implied~by} ~t\downharpoonright$ $=$ $((\downharpoonleft s\downharpoonright )\downharpoonleft t\downharpoonright )$ $=$ $((p)q)$   ${\text{L2b}}_{14}.$ $\downharpoonleft s~\mathrm {or} ~t\downharpoonright$ $=$ $((\downharpoonleft s\downharpoonright )(\downharpoonleft t\downharpoonright ))$ $=$ $((p)(q))$   ${\text{L2b}}_{15}.$ $\downharpoonleft \mathrm {true} \downharpoonright$ $=$ $((~))$ $=$ $((~))$

Geometric Translation Rule 2

${\text{Geometric Translation Rule 2}}$

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

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

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• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Part 6 • Part 7 • Part 8 • Part 9 • Part 10 • Part 11 • Part 12 • Part 13 • Part 14 • Part 15 • Part 16 • Appendix A1 • Appendix A2 • References • Document History •

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