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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 22:52, 27 January 2009

2,398 bytes added ,  22:52, 27 January 2009
→‎Logical Translation Rule 1
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===Logical Translation Rule 1===
 
===Logical Translation Rule 1===
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<pre>
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Logical Translation Rule 1
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If S is a sentence
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about things in the universe U
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and P is a proposition : U -> B, such that:
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L1a. [S]  =  P,
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then the following equations hold:
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L1b00. [False] = () = 0 : U->B.
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L1b01. [Not S] = ([S]) = (P) : U->B.
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L1b10. [S] = [S] = P : U->B.
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L1b11. [True] = (()) = 1 : U->B.
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</pre>
      
<br>
 
<br>
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| <math>p ~:~ X \to \underline\mathbb{B}</math>
 
| <math>p ~:~ X \to \underline\mathbb{B}</math>
 
|- style="height:56px"
 
|- style="height:56px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L1b}_{11}.\!</math>
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| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
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| <math>=\!</math>
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| <math>((~))</math>
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| <math>=\!</math>
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| <math>\underline{1} ~:~ X \to \underline\mathbb{B}</math>
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|}
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|}
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 +
<br>
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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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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:48px; text-align:right"
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| width="98%" | <math>\text{Logical Translation Rule 1}\!</math>
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| width=2%"  | &nbsp;
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|}
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|-
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:48px"
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
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| width="80%" style="border-top:1px solid black" |
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<math>s ~\text{is a sentence about things in the universe X}</math>
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|- style="height:48px"
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| &nbsp;
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| <math>\text{and}\!</math>
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| <math>p ~\text{is a proposition} ~:~ X \to \underline\mathbb{B}</math>
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|- style="height:48px"
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| &nbsp;
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| <math>\text{such that:}\!</math>
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| &nbsp;
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|- style="height:48px"
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| &nbsp;
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| <math>\text{L1a.}\!</math>
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| <math>\downharpoonleft s \downharpoonright ~=~ p</math>
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|- style="height:48px"
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| &nbsp;
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| <math>\text{then}\!</math>
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| <math>\text{the following equations hold:}\!</math>
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|}
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|-
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|
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{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
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|- style="height:52px"
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="18%" style="border-top:1px solid black" align="left" | <math>\text{L1b}_{00}.\!</math>
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| width="20%" style="border-top:1px solid black" |
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<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
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| width="5%" style="border-top:1px solid black" | <math>=\!</math>
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| width="20%" style="border-top:1px solid black" | <math>(~)</math>
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| width="5%" style="border-top:1px solid black" | <math>=\!</math>
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| width="30%" style="border-top:1px solid black" |
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<math>\underline{0} ~:~ X \to \underline\mathbb{B}</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{L1b}_{01}.\!</math>
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| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
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| <math>=\!</math>
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| <math>(\downharpoonleft s \downharpoonright)</math>
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| <math>=\!</math>
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| <math>(p) ~:~ X \to \underline\mathbb{B}</math>
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|- style="height:52px"
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| &nbsp;
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| align="left" | <math>\text{L1b}_{10}.\!</math>
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| <math>\downharpoonleft s \downharpoonright</math>
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| <math>=\!</math>
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| <math>\downharpoonleft s \downharpoonright</math>
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| <math>=\!</math>
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| <math>p ~:~ X \to \underline\mathbb{B}</math>
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|- style="height:52px"
 
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{L1b}_{11}.\!</math>
 
| align="left" | <math>\text{L1b}_{11}.\!</math>
Jon Awbrey
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