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

Revision as of 20:22, 12 August 2009

172 bytes added ,  20:22, 12 August 2009
→‎Variations on a theme of transitivity: number figures
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<pre>
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Table 21.  Composite and Compiled Order Relations
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Table 43.  Composite and Compiled Order Relations
 
o---------o------------o-----------------o----------------o-------------o
 
o---------o------------o-----------------o----------------o-------------o
 
| L_1    | L_2        | L_3            | L_4            | L_5        |
 
| L_1    | L_2        | L_3            | L_4            | L_5        |
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o---------o------------o-----------------o----------------o-------------o
 
o---------o------------o-----------------o----------------o-------------o
 
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q_207.  (p (q))
 
q_207.  (p (q))
 
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q_187.  (q (r))
 
q_187.  (q (r))
 
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q_175.  (p (r))
 
q_175.  (p (r))
 
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q_139.  (p (q))(q (r))
 
q_139.  (p (q))(q (r))
 
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o-----------------------------o
 
o-----------------------------o
 
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o-----------------------------o
 
o-----------------------------o
 
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o-----------------------------o
 
o-----------------------------o
 
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o-----------------------------o
 
o-----------------------------o
 
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o-----------------------------o
 
o-----------------------------o
 
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o-------------o-------------o-------------o-------------o
 
o-------------o-------------o-------------o-------------o
 
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The analogies between derivation, differentiation, implication, projection, and others sorts of information reducing operation will undergo extensive development in the remainder and sequel of the present discussion.
 
The analogies between derivation, differentiation, implication, projection, and others sorts of information reducing operation will undergo extensive development in the remainder and sequel of the present discussion.
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We were in the middle of discussing the relationships between information preserving rules of inference and information destroying rules of inference — folks of a 3-basket philosophical bent will no doubt be asking, "And what of information creating rules of inference?", but there I must wait for some signs of enlightenment, desiring not to tread on the rules of that succession.
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We were in the middle of discussing the relationships between information preserving rules of inference and information destroying rules of inference &mdash; folks of a 3-basket philosophical bent will no doubt be asking, "And what of information creating rules of inference?", but there I must wait for some signs of enlightenment, desiring not to tread on the rules of that succession.
    
The contrast between the information destroying and the information preserving versions of the transitive rule of inference led us to examine the relationships among several boolean functions, namely, those that qualify locally as the elementary cellular automata rules ''q''<sub>139</sub>, ''q''<sub>175</sub>, ''q''<sub>187</sub>, ''q''<sub>207</sub>.
 
The contrast between the information destroying and the information preserving versions of the transitive rule of inference led us to examine the relationships among several boolean functions, namely, those that qualify locally as the elementary cellular automata rules ''q''<sub>139</sub>, ''q''<sub>175</sub>, ''q''<sub>187</sub>, ''q''<sub>207</sub>.
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Figure 28-a.  Boolean 3-Cube B^3
 
Figure 28-a.  Boolean 3-Cube B^3
 
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o-----------------------o
 
o-----------------------o
 
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Figure 28-c.  Triadic Relation Syll c B^3
 
Figure 28-c.  Triadic Relation Syll c B^3
 
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o-----------------------o
 
o-----------------------o
 
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o-----------o o-----------o o-----------o
 
o-----------o o-----------o o-----------o
 
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Figure 29-c.  Syll c B^3 and its Dyadic Projections
 
Figure 29-c.  Syll c B^3 and its Dyadic Projections
 
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o-----------------------o
 
o-----------------------o
 
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o-----------o o-----------o o-----------o
 
o-----------o o-----------o o-----------o
 
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o---------------o o---------------o o---------------o
 
o---------------o o---------------o o---------------o
 
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Figure 30-d.  Tacit Extension TE_12_3 (Syll_12)
 
Figure 30-d.  Tacit Extension TE_12_3 (Syll_12)
 
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Figure 30-e.  Tacit Extension TE_13_2 (Syll_13)
 
Figure 30-e.  Tacit Extension TE_13_2 (Syll_13)
 
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Figure 30-f.  Tacit Extension TE_23_1 (Syll_23)
 
Figure 30-f.  Tacit Extension TE_23_1 (Syll_23)
 
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Figure 31.  Syll = TE(Syll_12) |^| TE(Syll_23)
 
Figure 31.  Syll = TE(Syll_12) |^| TE(Syll_23)
 
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Jon Awbrey
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