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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 20:46, 28 January 2009

94 bytes removed ,  20:46, 28 January 2009
→‎Geometric Translation Rule 2
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| width="2%"  style="border-top:1px solid black" |  
 
| width="2%"  style="border-top:1px solid black" |  
 
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
 
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
−
| width="84%" style="border-top:1px solid black" |
+
| width="84%" style="border-top:1px solid black" | <math>P, Q \subseteq X</math>
−
<math>s, t ~\text{are sentences about things in the universe}~ X</math>
   
|- style="height:48px"
 
|- style="height:48px"
 
| &nbsp;
 
| &nbsp;
 
| <math>\text{and}\!</math>
 
| <math>\text{and}\!</math>
−
| <math>p, q ~\text{are propositions} ~:~ X \to \underline\mathbb{B}</math>
+
| <math>p, q ~:~ X \to \underline\mathbb{B}</math>
 
|- style="height:48px"
 
|- style="height:48px"
 
| &nbsp;
 
| &nbsp;
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|- style="height:48px"
 
|- style="height:48px"
 
| &nbsp;
 
| &nbsp;
−
| <math>\text{L2a.}\!</math>
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| <math>\text{G2a.}\!</math>
−
| <math>\upharpoonleft s \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft t \upharpoonright ~=~ q</math>
+
| <math>\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q</math>
 
|- style="height:48px"
 
|- style="height:48px"
 
| &nbsp;
 
| &nbsp;
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|- style="height:52px"
 
|- style="height:52px"
 
| width="2%"  style="border-top:1px solid black" | &nbsp;
 
| width="2%"  style="border-top:1px solid black" | &nbsp;
−
| width="14%" style="border-top:1px solid black" align="left" | <math>\text{L2b}_{0}.\!</math>
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| width="14%" style="border-top:1px solid black" align="left" | <math>\text{G2b}_{0}.\!</math>
 
| width="32%" style="border-top:1px solid black" |
 
| width="32%" style="border-top:1px solid black" |
−
<math>\upharpoonleft \operatorname{false} \upharpoonright</math>
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<math>\upharpoonleft \varnothing \upharpoonright</math>
 
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
 
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
 
| width="28%" style="border-top:1px solid black" | <math>(~)</math>
 
| width="28%" style="border-top:1px solid black" | <math>(~)</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{1}.\!</math>
+
| align="left" | <math>\text{G2b}_{1}.\!</math>
 
| <math>\upharpoonleft \operatorname{neither}~ s ~\operatorname{nor}~ t \upharpoonright</math>
 
| <math>\upharpoonleft \operatorname{neither}~ s ~\operatorname{nor}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{2}.\!</math>
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| align="left" | <math>\text{G2b}_{2}.\!</math>
 
| <math>\upharpoonleft \operatorname{not}~ s ~\operatorname{but}~ t \upharpoonright</math>
 
| <math>\upharpoonleft \operatorname{not}~ s ~\operatorname{but}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{3}.\!</math>
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| align="left" | <math>\text{G2b}_{3}.\!</math>
 
| <math>\upharpoonleft \operatorname{not}~ s \upharpoonright</math>
 
| <math>\upharpoonleft \operatorname{not}~ s \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{4}.\!</math>
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| align="left" | <math>\text{G2b}_{4}.\!</math>
 
| <math>\upharpoonleft s ~\operatorname{and~not}~ t \upharpoonright</math>
 
| <math>\upharpoonleft s ~\operatorname{and~not}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{5}.\!</math>
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| align="left" | <math>\text{G2b}_{5}.\!</math>
 
| <math>\upharpoonleft \operatorname{not}~ t \upharpoonright</math>
 
| <math>\upharpoonleft \operatorname{not}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{6}.\!</math>
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| align="left" | <math>\text{G2b}_{6}.\!</math>
 
| <math>\upharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \upharpoonright</math>
 
| <math>\upharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{7}.\!</math>
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| align="left" | <math>\text{G2b}_{7}.\!</math>
 
| <math>\upharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \upharpoonright</math>
 
| <math>\upharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{8}.\!</math>
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| align="left" | <math>\text{G2b}_{8}.\!</math>
 
| <math>\upharpoonleft s ~\operatorname{and}~ t \upharpoonright</math>
 
| <math>\upharpoonleft s ~\operatorname{and}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{9}.\!</math>
+
| align="left" | <math>\text{G2b}_{9}.\!</math>
 
| <math>\upharpoonleft s ~\operatorname{is~equivalent~to}~ t \upharpoonright</math>
 
| <math>\upharpoonleft s ~\operatorname{is~equivalent~to}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{10}.\!</math>
+
| align="left" | <math>\text{G2b}_{10}.\!</math>
 
| <math>\upharpoonleft t \upharpoonright</math>
 
| <math>\upharpoonleft t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{11}.\!</math>
+
| align="left" | <math>\text{G2b}_{11}.\!</math>
 
| <math>\upharpoonleft s ~\operatorname{implies}~ t \upharpoonright</math>
 
| <math>\upharpoonleft s ~\operatorname{implies}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{12}.\!</math>
+
| align="left" | <math>\text{G2b}_{12}.\!</math>
 
| <math>\upharpoonleft s \upharpoonright</math>
 
| <math>\upharpoonleft s \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{13}.\!</math>
+
| align="left" | <math>\text{G2b}_{13}.\!</math>
 
| <math>\upharpoonleft s ~\operatorname{is~implied~by}~ t \upharpoonright</math>
 
| <math>\upharpoonleft s ~\operatorname{is~implied~by}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{14}.\!</math>
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| align="left" | <math>\text{G2b}_{14}.\!</math>
 
| <math>\upharpoonleft s ~\operatorname{or}~ t \upharpoonright</math>
 
| <math>\upharpoonleft s ~\operatorname{or}~ t \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
−
| align="left" | <math>\text{L2b}_{15}.\!</math>
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| align="left" | <math>\text{G2b}_{15}.\!</math>
−
| <math>\upharpoonleft \operatorname{true} \upharpoonright</math>
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| <math>\upharpoonleft X \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>((~))</math>
 
| <math>((~))</math>
Jon Awbrey
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