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Directory:Jon Awbrey/Papers/Differential Logic (view source)

Revision as of 19:53, 9 June 2009

909 bytes added ,  19:53, 9 June 2009
→‎Note 11: markup
Line 2,926: Line 2,926:     
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
−
| <math>\mathrm{G} ~=~ \mathrm{e} + \mathrm{f} + \mathrm{g} + \mathrm{h}</math>
+
| <math>\mathrm{G} ~=~ \mathrm{e} ~+~ \mathrm{f} ~+~ \mathrm{g} ~+~ \mathrm{h}</math>
 
|}
 
|}
   Line 2,939: Line 2,939:  
& + & \mathrm{g}:\mathrm{g}
 
& + & \mathrm{g}:\mathrm{g}
 
& + & \mathrm{h}:\mathrm{h}
 
& + & \mathrm{h}:\mathrm{h}
−
\\
+
\\[4pt]
 
& + & \mathrm{e}:\mathrm{f}
 
& + & \mathrm{e}:\mathrm{f}
 
& + & \mathrm{f}:\mathrm{e}
 
& + & \mathrm{f}:\mathrm{e}
 
& + & \mathrm{g}:\mathrm{h}
 
& + & \mathrm{g}:\mathrm{h}
 
& + & \mathrm{h}:\mathrm{g}
 
& + & \mathrm{h}:\mathrm{g}
−
\\
+
\\[4pt]
 
& + & \mathrm{e}:\mathrm{g}
 
& + & \mathrm{e}:\mathrm{g}
 
& + & \mathrm{f}:\mathrm{h}
 
& + & \mathrm{f}:\mathrm{h}
 
& + & \mathrm{g}:\mathrm{e}
 
& + & \mathrm{g}:\mathrm{e}
 
& + & \mathrm{h}:\mathrm{f}
 
& + & \mathrm{h}:\mathrm{f}
−
\\
+
\\[4pt]
 
& + & \mathrm{e}:\mathrm{h}
 
& + & \mathrm{e}:\mathrm{h}
 
& + & \mathrm{f}:\mathrm{g}
 
& + & \mathrm{f}:\mathrm{g}
Line 2,963: Line 2,963:  
Working through the construction for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular post-representations:
 
Working through the construction for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular post-representations:
   −
<pre>
+
{| align="center" cellpadding="6" width="90%"
−
  e = e:e + f:f + g:g + h:h
+
|
−
 
+
<math>\begin{matrix}
−
  f = e:f + f:e + g:h + h:g
+
\mathrm{e}
−
 
+
& = & \mathrm{e}:\mathrm{e}
−
  g = e:g + f:h + g:e + h:f
+
& + & \mathrm{f}:\mathrm{f}
−
 
+
& + & \mathrm{g}:\mathrm{g}
−
  h = e:h + f:g + g:f + h:e
+
& + & \mathrm{h}:\mathrm{h}
−
</pre>
+
\\[4pt]
 +
\mathrm{f}
 +
& = & \mathrm{e}:\mathrm{f}
 +
& + & \mathrm{f}:\mathrm{e}
 +
& + & \mathrm{g}:\mathrm{h}
 +
& + & \mathrm{h}:\mathrm{g}
 +
\\[4pt]
 +
\mathrm{g}
 +
& = & \mathrm{e}:\mathrm{g}
 +
& + & \mathrm{f}:\mathrm{h}
 +
& + & \mathrm{g}:\mathrm{e}
 +
& + & \mathrm{h}:\mathrm{f}
 +
\\[4pt]
 +
\mathrm{h}
 +
& = & \mathrm{e}:\mathrm{h}
 +
& + & \mathrm{f}:\mathrm{g}
 +
& + & \mathrm{g}:\mathrm{f}
 +
& + & \mathrm{h}:\mathrm{e}
 +
\end{matrix}</math>
 +
|}
    
So if somebody asks you, say, "What is <math>\operatorname{g}</math>?", you can say, "I don't know for certain, but in practice its effects go a bit like this:
 
So if somebody asks you, say, "What is <math>\operatorname{g}</math>?", you can say, "I don't know for certain, but in practice its effects go a bit like this:
Line 2,983: Line 3,002:  
Working through this alternative for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular ante-representations:
 
Working through this alternative for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular ante-representations:
   −
<pre>
+
{| align="center" cellpadding="6" width="90%"
−
  e = e:e + f:f + g:g + h:h
+
|
−
 
+
<math>\begin{matrix}
−
  f = f:e + e:f + h:g + g:h
+
\mathrm{e}
−
 
+
& = & \mathrm{e}:\mathrm{e}
−
  g = g:e + h:f + e:g + f:h
+
& + & \mathrm{f}:\mathrm{f}
−
 
+
& + & \mathrm{g}:\mathrm{g}
−
  h = h:e + g:f + f:g + e:h
+
& + & \mathrm{h}:\mathrm{h}
−
</pre>
+
\\[4pt]
 +
\mathrm{f}
 +
& = & \mathrm{f}:\mathrm{e}
 +
& + & \mathrm{e}:\mathrm{f}
 +
& + & \mathrm{h}:\mathrm{g}
 +
& + & \mathrm{g}:\mathrm{h}
 +
\\[4pt]
 +
\mathrm{g}
 +
& = & \mathrm{g}:\mathrm{e}
 +
& + & \mathrm{h}:\mathrm{f}
 +
& + & \mathrm{e}:\mathrm{g}
 +
& + & \mathrm{f}:\mathrm{h}
 +
\\[4pt]
 +
\mathrm{h}
 +
& = & \mathrm{h}:\mathrm{e}
 +
& + & \mathrm{g}:\mathrm{f}
 +
& + & \mathrm{f}:\mathrm{g}
 +
& + & \mathrm{e}:\mathrm{h}
 +
\end{matrix}</math>
 +
|}
    
Your paraphrastic interpretation of what this all means would come out precisely the same as before.
 
Your paraphrastic interpretation of what this all means would come out precisely the same as before.
Jon Awbrey
12,080

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