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Directory:Jon Awbrey/Papers/Dynamics And Logic (view source)

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3,164 bytes removed ,  14:14, 22 June 2009
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A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
 
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
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<math>\begin{array}{lccl}
 
<math>\begin{array}{lccl}
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For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B}</math> that is defined by the following equation:
 
For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B}</math> that is defined by the following equation:
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<math>\begin{array}{l}
 
<math>\begin{array}{l}
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In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:
 
In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:
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<math>\begin{array}{l}
 
<math>\begin{array}{l}
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Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to "multiply things out" in the usual manner to arrive at the following result:
 
Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to "multiply things out" in the usual manner to arrive at the following result:
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<math>\begin{matrix}
 
<math>\begin{matrix}
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To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math>  Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:
 
To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math>  Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:
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| align="center" |
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| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
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<pre>
  −
o-------------------------------------------------o
  −
|                                                |
  −
|                  p  dp q  dq                  |
  −
|                  o---o o---o                  |
  −
|                    \  | |  /                    |
  −
|                    \ | | /                    |
  −
|                      \| |/                      |
  −
|                      @=@                      |
  −
|                                                |
  −
o-------------------------------------------------o
  −
| Ef =           (p, dp) (q, dq)                 |
  −
o-------------------------------------------------o
  −
</pre>
   
|-
 
|-
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| align="center" |
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| [[Image:Cactus Graph Ef@PQ = (dP)(dQ).jpg|500px]]
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<pre>
  −
o-------------------------------------------------o
  −
|                                                |
  −
|                      dp    dq                  |
  −
|                  o---o o---o                  |
  −
|                    \  | |  /                    |
  −
|                    \ | | /                    |
  −
|                      \| |/                      |
  −
|                      @=@                      |
  −
|                                                |
  −
o-------------------------------------------------o
  −
| Ef|pq =            (dp) (dq)                   |
  −
o-------------------------------------------------o
  −
</pre>
   
|-
 
|-
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| align="center" |
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| [[Image:Cactus Graph Ef@P(Q) = (dP)dQ.jpg|500px]]
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<pre>
  −
o-------------------------------------------------o
  −
|                                                |
  −
|                        o                      |
  −
|                      dp |  dq                  |
  −
|                  o---o o---o                  |
  −
|                    \  | |  /                    |
  −
|                    \ | | /                    |
  −
|                      \| |/                      |
  −
|                      @=@                       |
  −
|                                                |
  −
o-------------------------------------------------o
  −
| Ef|p(q) =         (dp) dq                    |
  −
o-------------------------------------------------o
  −
</pre>
   
|-
 
|-
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| align="center" |
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| [[Image:Cactus Graph Ef@(P)Q = dP(dQ).jpg|500px]]
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<pre>
  −
o-------------------------------------------------o
  −
|                                                |
  −
|                  o                            |
  −
|                  |  dp    dq                  |
  −
|                  o---o o---o                  |
  −
|                    \  | |  /                    |
  −
|                    \ | | /                    |
  −
|                      \| |/                      |
  −
|                      @=@                       |
  −
|                                                |
  −
o-------------------------------------------------o
  −
| Ef|(p)q =           dp  (dq)                   |
  −
o-------------------------------------------------o
  −
</pre>
   
|-
 
|-
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| align="center" |
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| [[Image:Cactus Graph Ef@(P)(Q) = dP dQ.jpg|500px]]
−
<pre>
  −
o-------------------------------------------------o
  −
|                                                |
  −
|                  o    o                      |
  −
|                  |  dp |  dq                  |
  −
|                  o---o o---o                  |
  −
|                    \  | |  /                    |
  −
|                    \ | | /                    |
  −
|                      \| |/                      |
  −
|                      @=@                      |
  −
|                                                |
  −
o-------------------------------------------------o
  −
| Ef|(p)(q) =         dp  dq                    |
  −
o-------------------------------------------------o
  −
</pre>
   
|}
 
|}
    
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\operatorname{E}f.</math>
 
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\operatorname{E}f.</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
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Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>
 
Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>
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{| align="center" cellspacing="20" style="text-align:center"
 
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
 
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
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<math>\begin{array}{rcccccc}
 
<math>\begin{array}{rcccccc}
Jon Awbrey
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