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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)

Revision as of 20:25, 14 August 2009

301 bytes added ,  20:25, 14 August 2009
→‎Variations on a theme of transitivity: format table
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{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
|+ style="height:30px" | <math>\text{Table 48.} ~~ [| f_{207} |] ~=~ [| p \le q |]</math>
 
|+ style="height:30px" | <math>\text{Table 48.} ~~ [| f_{207} |] ~=~ [| p \le q |]</math>
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|- style="height:40px"
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|- style="height:40px; background:#f0f0ff"
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
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{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
|+ style="height:30px" | <math>\text{Table 49.} ~~ [| f_{187} |] ~=~ [| q \le r |]</math>
 
|+ style="height:30px" | <math>\text{Table 49.} ~~ [| f_{187} |] ~=~ [| q \le r |]</math>
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|- style="height:40px"
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|- style="height:40px; background:#f0f0ff"
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
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{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
|+ style="height:30px" | <math>\text{Table 50.} ~~ [| f_{175} |] ~=~ [| p \le r |]</math>
 
|+ style="height:30px" | <math>\text{Table 50.} ~~ [| f_{175} |] ~=~ [| p \le r |]</math>
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|- style="height:40px"
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|- style="height:40px; background:#f0f0ff"
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
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{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
|+ style="height:30px" | <math>\text{Table 51.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]</math>
 
|+ style="height:30px" | <math>\text{Table 51.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]</math>
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|- style="height:40px"
+
|- style="height:40px; background:#f0f0ff"
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
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{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 
|+ style="height:30px" | <math>\text{Table 52.} ~~ \text{Syllogism Relation}</math>
 
|+ style="height:30px" | <math>\text{Table 52.} ~~ \text{Syllogism Relation}</math>
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|- style="height:40px"
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|- style="height:40px; background:#f0f0ff"
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>p\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
 
| style="border-bottom:1px solid black" | <math>q\!</math>
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One of the first questions that we might ask about a 3-adic relation, in this case <math>\operatorname{Syll},</math> is whether it is ''determined by'' its 2-adic projections.  I will illustrate what this means in the present case.
 
One of the first questions that we might ask about a 3-adic relation, in this case <math>\operatorname{Syll},</math> is whether it is ''determined by'' its 2-adic projections.  I will illustrate what this means in the present case.
   −
Table&nbsp;25-b repeats the relation <math>\operatorname{Syll}</math> in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of <math>\operatorname{Syll}</math> in the next three columns.  For instance, <math>\operatorname{Syll}_{pq}</math> is the 2-adic projection of <math>\operatorname{Syll}</math> on the <math>pq\!</math> plane that is arrived at by deleting the <math>r\!</math> column and counting each 2-tuple that results just one time.  Likewise, <math>\operatorname{Syll}_{pr}</math> is obtained by deleting the <math>q\!</math> column and <math>\operatorname{Syll}_{qr}</math> is derived by deleting the <math>p\!</math> column, ignoring whatever duplicate pairs may result.  The final row of the right three columns gives the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}</math> that indicate the 2-adic relations that result from these projections.
+
Table&nbsp;53 repeats the relation <math>\operatorname{Syll}</math> in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of <math>\operatorname{Syll}</math> in the next three columns.  For instance, <math>\operatorname{Syll}_{pq}</math> is the 2-adic projection of <math>\operatorname{Syll}</math> on the <math>pq\!</math> plane that is arrived at by deleting the <math>r\!</math> column and counting each 2-tuple that results just one time.  Likewise, <math>\operatorname{Syll}_{pr}</math> is obtained by deleting the <math>q\!</math> column and <math>\operatorname{Syll}_{qr}</math> is derived by deleting the <math>p\!</math> column, ignoring whatever duplicate pairs may result.  The final row of the right three columns gives the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}</math> that indicate the 2-adic relations that result from these projections.
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{| align="center" cellpadding="10" style="text-align:center; width:90%"
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<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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|+ style="height:30px" | <math>\text{Table 53.} ~~ \text{Dyadic Projections of the Syllogism Relation}</math>
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|- style="height:40px; background:#f0f0ff"
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| <math>\operatorname{Syll}</math>
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| <math>\operatorname{Syll}_{pq}</math>
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| <math>\operatorname{Syll}_{pr}</math>
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| <math>\operatorname{Syll}_{qr}</math>
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|-
 +
|
 +
<math>\begin{matrix}
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0~0~0 \\ 0~0~1 \\ 0~1~1 \\ 1~1~1
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\end{matrix}</math>
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|
 +
<math>\begin{matrix}
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0~0 \\ 0~0 \\ 0~1 \\ 1~1
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\end{matrix}</math>
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|
 +
<math>\begin{matrix}
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0~0 \\ 0~1 \\ 0~1 \\ 1~1
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\end{matrix}</math>
 
|
 
|
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<pre>
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<math>\begin{matrix}
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Table 25-b.  Dyadic Projections of the Syllogism Relation
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0~0 \\ 0~1 \\ 1~1 \\ 1~1
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o-------------o-------------o-------------o-------------o
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\end{matrix}</math>
−
|   Syll    |  Syll_pq  |  Syll_pr  |  Syll_qr  |
+
|- style="height:40px; background:#f0f0ff"
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o-------------o-------------o-------------o-------------o
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| <math>p \le q \le r</math>
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|    000    |    00      |    00      |    00      |
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| <math>\texttt{(} p \texttt{~(} q \texttt{))}</math>
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|    001    |    00      |    01      |    01      |
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| <math>\texttt{(} p \texttt{~(} r \texttt{))}</math>
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|    011    |    01      |    01      |    11      |
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| <math>\texttt{(} q \texttt{~(} r \texttt{))}</math>
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|    111    |    11      |    11      |    11      |
  −
o-------------o-------------o-------------o-------------o
  −
| p =< q =< r |   (p (q))   |   (p (r))   |   (q (r))   |
  −
o-------------o-------------o-------------o-------------o
  −
</pre>
  −
| (53)
   
|}
 
|}
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 +
<br>
    
Let us make the simple observation that taking a projection, in our framework, deleting a column from a relational table, is like taking a derivative in differential calculus.  What it means is that our attempt to return to the integral from whence the derivative was derived will in general encounter an indefinite variation on account of the circumstance that real information may have been destroyed by the derivation.
 
Let us make the simple observation that taking a projection, in our framework, deleting a column from a relational table, is like taking a derivative in differential calculus.  What it means is that our attempt to return to the integral from whence the derivative was derived will in general encounter an indefinite variation on account of the circumstance that real information may have been destroyed by the derivation.
Jon Awbrey
12,080

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