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Directory:Jon Awbrey/Papers/Syntactic Transformations (view source)

Revision as of 15:08, 13 February 2009

2,089 bytes added ,  15:08, 13 February 2009
→‎Syntactic Transformation Rules: format rule box
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<br>
 
<br>
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<pre>
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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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Evaluation Rule 1
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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If f, g : U -> B
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|- style="height:50px; text-align:right"
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| width="98%" | <math>\operatorname{Evaluation~Rule~1}</math>
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and u C U,
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| width="2%"  | &nbsp;
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|}
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then the following are equivalent:
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|-
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|
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E1a. f(u) = g(u). :V1a
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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::
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|- style="height:50px"
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E1b. f(u) <=> g(u). :V1b
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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::
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| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
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E1c. (( f(u) , g(u) )). :V1c
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| width="84%" style="border-top:1px solid black" | <math>f, g ~:~ X \to \underline\mathbb{B}</math>
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:$1a
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|- style="height:50px"
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::
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| &nbsp;
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E1d. (( f , g ))$(u). :$1b
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| <math>\text{and}\!</math>
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</pre>
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| <math>x ~\in~ X</math>
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|- style="height:50px"
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<br>
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| &nbsp;
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| <math>\text{then}\!</math>
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<pre>
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| <math>\text{the following are equivalent:}\!</math>
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Evaluation Rule 1
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|}
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|-
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If S, T are sentences
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|
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about things in the universe U,
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:10px"
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f, g are propositions: U -> B,
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="14%" style="border-top:1px solid black" | &nbsp;
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and u C U,
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| width="64%" style="border-top:1px solid black" | &nbsp;
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| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
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then the following are equivalent:
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|- style="height:40px"
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| &nbsp;
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E1a. f(u) = g(u). :V1a
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| <math>\operatorname{E1a.}</math>
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::
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| <math>f(x) ~=~ g(x)</math>
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E1b. f(u) <=> g(u). :V1b
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{E1a~:~V1a}</math>
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::
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|- style="height:20px"
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E1c. (( f(u) , g(u) )). :V1c
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| colspan="3" | &nbsp;
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:$1a
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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::
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|- style="height:40px"
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E1d. (( f , g ))$(u). :$1b
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| &nbsp;
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</pre>
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| <math>\operatorname{E1b.}</math>
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| <math>f(x) ~\Leftrightarrow~ g(x)</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{E1b~:~V1b}</math>
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|- style="height:20px"
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| colspan="3" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{E1c.}</math>
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| <math>\underline{((}~ f(x) ~,~ g(x) ~\underline{))}</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{E1c~:~V1c}</math></p>
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<p><math>\operatorname{E1c~:~$1a}</math></p>
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|- style="height:20px"
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| colspan="3" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:40px"
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| &nbsp;
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| <math>\operatorname{E1d.}</math>
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| <math>\underline{((}~ f ~,~ g ~\underline{))}^\$ (x)</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{E1d~:~$1b}</math>
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|- style="height:10px"
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| colspan="3" | &nbsp;
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| style="border-left:1px solid black" | &nbsp;
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|}
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|}
    
<br>
 
<br>
Jon Awbrey
12,080

edits

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