Difference between revisions of "Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices"

MyWikiBiz, Author Your Legacy — Friday November 29, 2024
Jump to navigationJump to search
(add [appendices] to navigation bars)
(→‎Appendices: + Tables)
Line 15: Line 15:
 
==Appendices==
 
==Appendices==
  
 +
===Logical Translation Rule 1===
 +
 +
<br>
 +
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 +
|- style="height:48px; text-align:right"
 +
| width="98%" | <math>\text{Logical Translation Rule 1}\!</math>
 +
| width="2%"  | &nbsp;
 +
|}
 +
|-
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 +
|- style="height:48px"
 +
| width="2%"  style="border-top:1px solid black" | &nbsp;
 +
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
 +
| width="80%" style="border-top:1px solid black" |
 +
<math>s ~\text{is a sentence about things in the universe X}</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{and}\!</math>
 +
| <math>p ~\text{is a proposition} ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{such that:}\!</math>
 +
| &nbsp;
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{L1a.}\!</math>
 +
| <math>\downharpoonleft s \downharpoonright ~=~ p</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{then}\!</math>
 +
| <math>\text{the following equations hold:}\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
 +
|- style="height:52px"
 +
| width="2%"  style="border-top:1px solid black" | &nbsp;
 +
| width="18%" style="border-top:1px solid black" align="left" | <math>\text{L1b}_{00}.\!</math>
 +
| width="20%" style="border-top:1px solid black" |
 +
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
 +
| width="5%"  style="border-top:1px solid black" | <math>=\!</math>
 +
| width="20%" style="border-top:1px solid black" | <math>(~)</math>
 +
| width="5%"  style="border-top:1px solid black" | <math>=\!</math>
 +
| width="30%" style="border-top:1px solid black" |
 +
<math>\underline{0} ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L1b}_{01}.\!</math>
 +
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p) ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L1b}_{10}.\!</math>
 +
| <math>\downharpoonleft s \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\downharpoonleft s \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>p ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L1b}_{11}.\!</math>
 +
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((~))</math>
 +
| <math>=\!</math>
 +
| <math>\underline{1} ~:~ X \to \underline\mathbb{B}</math>
 +
|}
 +
|}
 +
 +
<br>
 +
 +
===Geometric Translation Rule 1===
 +
 +
<br>
 +
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 +
|- style="height:48px; text-align:right"
 +
| width="98%" | <math>\text{Geometric Translation Rule 1}\!</math>
 +
| width="2%"  | &nbsp;
 +
|}
 +
|-
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 +
|- style="height:48px"
 +
| width="2%"  style="border-top:1px solid black" | &nbsp;
 +
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
 +
| width="80%" style="border-top:1px solid black" | <math>Q \subseteq X</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{and}\!</math>
 +
| <math>p ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{such that:}\!</math>
 +
| &nbsp;
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{G1a.}\!</math>
 +
| <math>\upharpoonleft Q \upharpoonright ~=~ p</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{then}\!</math>
 +
| <math>\text{the following equations hold:}\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
 +
|- style="height:52px"
 +
| width="2%"  style="border-top:1px solid black" | &nbsp;
 +
| width="18%" style="border-top:1px solid black" align="left" | <math>\text{G1b}_{00}.\!</math>
 +
| width="20%" style="border-top:1px solid black" |
 +
<math>\upharpoonleft \varnothing \upharpoonright</math>
 +
| width="5%"  style="border-top:1px solid black" | <math>=\!</math>
 +
| width="20%" style="border-top:1px solid black" | <math>(~)</math>
 +
| width="5%"  style="border-top:1px solid black" | <math>=\!</math>
 +
| width="30%" style="border-top:1px solid black" |
 +
<math>\underline{0} ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G1b}_{01}.\!</math>
 +
| <math>\upharpoonleft {}^{_\sim} Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\upharpoonleft Q \upharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p) ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G1b}_{10}.\!</math>
 +
| <math>\upharpoonleft Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\upharpoonleft Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>p ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G1b}_{11}.\!</math>
 +
| <math>\upharpoonleft X \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((~))</math>
 +
| <math>=\!</math>
 +
| <math>\underline{1} ~:~ X \to \underline\mathbb{B}</math>
 +
|}
 +
|}
 +
 +
<br>
 +
 +
===Logical Translation Rule 2===
 +
 +
<br>
 +
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 +
|- style="height:48px; text-align:right"
 +
| width="98%" | <math>\text{Logical Translation Rule 2}\!</math>
 +
| width="2%"  | &nbsp;
 +
|}
 +
|-
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 +
|- style="height:48px"
 +
| width="2%"  style="border-top:1px solid black" | &nbsp;
 +
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
 +
| width="84%" style="border-top:1px solid black" |
 +
<math>s, t ~\text{are sentences about things in the universe}~ X</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{and}\!</math>
 +
| <math>p, q ~\text{are propositions} ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{such that:}\!</math>
 +
| &nbsp;
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{L2a.}\!</math>
 +
| <math>\downharpoonleft s \downharpoonright ~=~ p \quad \operatorname{and} \quad \downharpoonleft t \downharpoonright ~=~ q</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{then}\!</math>
 +
| <math>\text{the following equations hold:}\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
 +
|- style="height:52px"
 +
| width="2%"  style="border-top:1px solid black" | &nbsp;
 +
| width="14%" style="border-top:1px solid black" align="left" | <math>\text{L2b}_{0}.\!</math>
 +
| width="32%" style="border-top:1px solid black" |
 +
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
 +
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
 +
| width="28%" style="border-top:1px solid black" | <math>(~)</math>
 +
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
 +
| width="16%" style="border-top:1px solid black" | <math>(~)</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{1}.\!</math>
 +
| <math>\downharpoonleft \operatorname{neither}~ s ~\operatorname{nor}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p)(q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{2}.\!</math>
 +
| <math>\downharpoonleft \operatorname{not}~ s ~\operatorname{but}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(p) q\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{3}.\!</math>
 +
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{4}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{and~not}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>p (q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{5}.\!</math>
 +
| <math>\downharpoonleft \operatorname{not}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{6}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p, q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{7}.\!</math>
 +
| <math>\downharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{8}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{and}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>p q\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{9}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{is~equivalent~to}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright))</math>
 +
| <math>=\!</math>
 +
| <math>((p, q))\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{10}.\!</math>
 +
| <math>\downharpoonleft t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\downharpoonleft t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>q\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{11}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{implies}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright))</math>
 +
| <math>=\!</math>
 +
| <math>(p (q))\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{12}.\!</math>
 +
| <math>\downharpoonleft s \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\downharpoonleft s \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>p\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{13}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{is~implied~by}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>((p) q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{14}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{or}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright))</math>
 +
| <math>=\!</math>
 +
| <math>((p)(q))\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{15}.\!</math>
 +
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((~))</math>
 +
| <math>=\!</math>
 +
| <math>((~))</math>
 +
|}
 +
|}
 +
 +
<br>
 +
 +
===Geometric Translation Rule 2===
 +
 +
<br>
 +
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 +
|- style="height:48px; text-align:right"
 +
| width="98%" | <math>\text{Geometric Translation Rule 2}\!</math>
 +
| width="2%"  | &nbsp;
 +
|}
 +
|-
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 +
|- style="height:48px"
 +
| width="2%"  style="border-top:1px solid black" | &nbsp;
 +
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
 +
| width="84%" style="border-top:1px solid black" | <math>P, Q \subseteq X</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{and}\!</math>
 +
| <math>p, q ~:~ X \to \underline\mathbb{B}</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{such that:}\!</math>
 +
| &nbsp;
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{G2a.}\!</math>
 +
| <math>\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q</math>
 +
|- style="height:48px"
 +
| &nbsp;
 +
| <math>\text{then}\!</math>
 +
| <math>\text{the following equations hold:}\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
 +
|- style="height:52px"
 +
| width="2%"  style="border-top:1px solid black" | &nbsp;
 +
| width="14%" style="border-top:1px solid black" align="left" | <math>\text{G2b}_{0}.\!</math>
 +
| width="32%" style="border-top:1px solid black" |
 +
<math>\upharpoonleft \varnothing \upharpoonright</math>
 +
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
 +
| width="28%" style="border-top:1px solid black" | <math>(~)</math>
 +
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
 +
| width="16%" style="border-top:1px solid black" | <math>(~)</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{1}.\!</math>
 +
| <math>\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p)(q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{2}.\!</math>
 +
| <math>\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(p) q\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{3}.\!</math>
 +
| <math>\upharpoonleft \overline{P} \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\upharpoonleft P \upharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{4}.\!</math>
 +
| <math>\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>p (q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{5}.\!</math>
 +
| <math>\upharpoonleft \overline{Q} \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\upharpoonleft Q \upharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{6}.\!</math>
 +
| <math>\upharpoonleft P ~+~ Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p, q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{7}.\!</math>
 +
| <math>\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{8}.\!</math>
 +
| <math>\upharpoonleft P ~\cap~ Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>p q\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{9}.\!</math>
 +
| <math>\upharpoonleft \overline{P ~+~ Q} \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))</math>
 +
| <math>=\!</math>
 +
| <math>((p, q))\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{10}.\!</math>
 +
| <math>\upharpoonleft Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\upharpoonleft Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>q\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{11}.\!</math>
 +
| <math>\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright))</math>
 +
| <math>=\!</math>
 +
| <math>(p (q))\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{12}.\!</math>
 +
| <math>\upharpoonleft P \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\upharpoonleft P \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>p\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{13}.\!</math>
 +
| <math>\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>((p) q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{14}.\!</math>
 +
| <math>\upharpoonleft P ~\cup~ Q \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))</math>
 +
| <math>=\!</math>
 +
| <math>((p)(q))\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{G2b}_{15}.\!</math>
 +
| <math>\upharpoonleft X \upharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((~))</math>
 +
| <math>=\!</math>
 +
| <math>((~))</math>
 +
|}
 +
|}
 +
 +
<br>
  
 
----
 
----

Revision as of 20:30, 10 September 2010

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


\(\text{Geometric Translation Rule 2}\!\)  
  \(\text{If}\!\) \(P, Q \subseteq X\)
  \(\text{and}\!\) \(p, q ~:~ X \to \underline\mathbb{B}\)
  \(\text{such that:}\!\)  
  \(\text{G2a.}\!\) \(\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q\)
  \(\text{then}\!\) \(\text{the following equations hold:}\!\)
  \(\text{G2b}_{0}.\!\)

\(\upharpoonleft \varnothing \upharpoonright\)

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



<sharethis />

Retrieved from "https://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems_:_Appendices&oldid=124025"