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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 19:12, 28 January 2009

539 bytes removed ,  19:12, 28 January 2009
→‎Logical Translation Rule 2
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===Logical Translation Rule 2===
 
===Logical Translation Rule 2===
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<pre>
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Logical Translation Rule 2
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If S, T are sentences
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about things in the universe U
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and P, Q are propositions: U -> B, such that:
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L2a. [S] = P  and  [T] = Q,
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  −
then the following equations hold:
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L2b00. [False] = () = 0 : U->B.
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L2b01. [Neither S nor T] = ([S])([T]) = (P)(Q).
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L2b02. [Not S, but T] = ([S])[T] = (P) Q.
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L2b03. [Not S] = ([S]) = (P).
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L2b04. [S and not T] = [S]([T]) = P (Q).
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L2b05. [Not T] = ([T]) = (Q).
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L2b06. [S or T, not both] = ([S], [T]) = (P, Q).
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L2b07. [Not both S and T] = ([S].[T]) = (P Q).
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L2b08. [S and T] = [S].[T] = P.Q.
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L2b09. [S <=> T] = (([S], [T])) = ((P, Q)).
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L2b10. [T] = [T] = Q.
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L2b11. [S => T] = ([S]([T])) = (P (Q)).
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L2b12. [S] = [S] = P.
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L2b13. [S <= T] = (([S]) [T]) = ((P) Q).
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L2b14. [S or T] = (([S])([T])) = ((P)(Q)).
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L2b15. [True] = (()) = 1 : U->B.
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</pre>
      
<br>
 
<br>
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| 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>
 
| width="14%" style="border-top:1px solid black" align="left" | <math>\text{L2b}_{0}.\!</math>
−
| width="28%" style="border-top:1px solid black" |
+
| width="32%" style="border-top:1px solid black" |
 
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
 
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
 
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
 
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
−
| width="24%" 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="4%"  style="border-top:1px solid black" | <math>=\!</math>
−
| width="24%" style="border-top:1px solid black" |
+
| width="16%" style="border-top:1px solid black" | <math>(~)</math>
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<math>\underline{0} ~:~ X \to \underline\mathbb{B}</math>
   
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
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| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{3}.\!</math>
 
| align="left" | <math>\text{L2b}_{3}.\!</math>
−
| <math>\downharpoonleft \operatorname{not}~ s\downharpoonright</math>
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| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>(\downharpoonleft s \downharpoonright)</math>
 
| <math>(\downharpoonleft s \downharpoonright)</math>
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| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{4}.\!</math>
 
| align="left" | <math>\text{L2b}_{4}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
+
| <math>\downharpoonleft s ~\operatorname{and~not}~ t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
+
| <math>\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
+
| <math>p (q)\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{5}.\!</math>
 
| align="left" | <math>\text{L2b}_{5}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft \operatorname{not}~ t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>(\downharpoonleft t \downharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
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| <math>(q)\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{6}.\!</math>
 
| align="left" | <math>\text{L2b}_{6}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
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| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>(\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
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| <math>(p, q)\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{7}.\!</math>
 
| align="left" | <math>\text{L2b}_{7}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>(\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
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| <math>(p q)\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{8}.\!</math>
 
| align="left" | <math>\text{L2b}_{8}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft s ~\operatorname{and}~ t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
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| <math>p q\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{9}.\!</math>
 
| align="left" | <math>\text{L2b}_{9}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft s ~\operatorname{is~equivalent~to}~ t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright))</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
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| <math>((p, q))\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{10}.\!</math>
 
| align="left" | <math>\text{L2b}_{10}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>\downharpoonleft t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
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| <math>q\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{11}.\!</math>
 
| align="left" | <math>\text{L2b}_{11}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft s ~\operatorname{implies}~ t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>(\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright))</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
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| <math>(p (q))\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{12}.\!</math>
 
| align="left" | <math>\text{L2b}_{12}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft s \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>\downharpoonleft s \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
+
| <math>p\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{13}.\!</math>
 
| align="left" | <math>\text{L2b}_{13}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft s ~\operatorname{is~implied~by}~ t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
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| <math>((p) q)\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L2b}_{14}.\!</math>
 
| align="left" | <math>\text{L2b}_{14}.\!</math>
−
| <math>\downharpoonleft \ldots \downharpoonright</math>
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| <math>\downharpoonleft s ~\operatorname{or}~ t \downharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots \downharpoonleft \ldots \downharpoonright \ldots</math>
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| <math>((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright))</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\ldots</math>
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| <math>((p)(q))\!</math>
 
|- style="height:52px"
 
|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
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| <math>((~))</math>
 
| <math>((~))</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\underline{1} ~:~ X \to \underline\mathbb{B}</math>
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| <math>((~))</math>
 
|}
 
|}
 
|}
 
|}
Jon Awbrey
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