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User:Jon Awbrey/SCRATCHPAD (view source)
Revision as of 19:38, 28 January 2009
, 19:38, 28 January 2009→Geometric Translation Rule 2
G2b15. {U} = (()) = 1 : U->B.
G2b15. {U} = (()) = 1 : U->B.
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{| 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%"
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{| 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%" |
|}
|-
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px"
| width="2%" style="border-top:1px solid black" |
| 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"
|
| <math>\text{and}\!</math>
| <math>p, q ~\text{are propositions} ~:~ X \to \underline\mathbb{B}</math>
|- style="height:48px"
|
| <math>\text{such that:}\!</math>
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|- style="height:48px"
|
| <math>\text{L2a.}\!</math>
| <math>\downharpoonleft s \downharpoonright ~=~ p \quad \operatorname{and} \quad \downharpoonleft t \downharpoonright ~=~ q</math>
|- style="height:48px"
|
| <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" |
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| 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"
|
| align="left" | <math>\text{L2b}_{15}.\!</math>
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
| <math>=\!</math>
| <math>((~))</math>
| <math>=\!</math>
| <math>((~))</math>
|}
|}
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