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o=============================< QES >=======================o
 
o=============================< QES >=======================o
 
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| [[Image:Equational Inference Bar -- QED.jpg|500px]]
 
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
 
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| [[Image:Praeclarum Theorema CAST 500 x 389 Animation.gif]]
 
| [[Image:Praeclarum Theorema CAST 500 x 389 Animation.gif]]
 
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{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
 
| [[Image:Praeclarum Theorema DNF.jpg|500px]]
 
| [[Image:Praeclarum Theorema DNF.jpg|500px]]
| (3)
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| (32)
 
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o-----------------------------------------------------------o
 
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</pre>
 
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{| align="center" cellpadding="10"
 
{| align="center" cellpadding="10"
| [[Image:Praeclarum Theorema.jpg|500px]] || (1)
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| [[Image:Praeclarum Theorema.jpg|500px]] || (34)
 
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{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
| [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (1)
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| [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (35)
 
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{| align="center" cellpadding="8" style="text-align:center"
 
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Venn Diagram (P (Q)) (P (R)).jpg|500px]] || (2)
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| [[Image:Venn Diagram (P (Q)) (P (R)).jpg|500px]] || (36)
 
|-
 
|-
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}</math>
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}</math>
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{| align="center" cellpadding="8" style="text-align:center"
 
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Venn Diagram (P (Q R)).jpg|500px]] || (3)
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| [[Image:Venn Diagram (P (Q R)).jpg|500px]] || (37)
 
|-
 
|-
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>
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{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)).jpg|500px]] || (4)
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| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)).jpg|500px]] || (38)
 
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{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
 
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg|500px]]
 
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg|500px]]
| (5)
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| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
 
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
 
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{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
 
| [[Image:Logical Graph (P (Q)) (P (R)) DNF.jpg|500px]]
 
| [[Image:Logical Graph (P (Q)) (P (R)) DNF.jpg|500px]]
| (7)
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| (41)
 
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o-----------------------------------------------------------o
 
o-----------------------------------------------------------o
 
</pre>
 
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| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
 
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
 
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{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
| [[Image:Logical Graph (P Q R , (P)).jpg|500px]] || (10)
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| [[Image:Logical Graph (P Q R , (P)).jpg|500px]] || (44)
 
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{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
| [[Image:Logical Graph ((P , P Q R)).jpg|500px]] || (11)
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| [[Image:Logical Graph ((P , P Q R)).jpg|500px]] || (45)
 
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o-----------------------------------------------------------o
 
o-----------------------------------------------------------o
 
</pre>
 
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o-----------------------------------------------------------o
 
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</pre>
 
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o-----------------------------------------------------------o
 
o-----------------------------------------------------------o
 
</pre>
 
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{| align="center" cellpadding="8"
 
{| align="center" cellpadding="8"
 
| [[Image:Logical Graph (( (P (Q)) (P (R)) , (P (Q R)) )).jpg|500px]]
 
| [[Image:Logical Graph (( (P (Q)) (P (R)) , (P (Q R)) )).jpg|500px]]
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| [[Image:Equational Inference Bar -- QED.jpg|500px]]
 
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
 
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{| align="center" cellpadding="6" style="text-align:center"
 
{| align="center" cellpadding="6" style="text-align:center"
| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (2)
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| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (51)
 
|-
 
|-
 
| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))}</math>
 
| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))}</math>
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| &nbsp;
 
| &nbsp;
 
|-
 
|-
| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (3)
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| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (52)
 
|-
 
|-
 
| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{~(} r \texttt{))}</math>
 
| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{~(} r \texttt{))}</math>
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| &nbsp;
 
| &nbsp;
 
|-
 
|-
| [[Image:Venn Diagram (P (R)).jpg|500px]] || (4)
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| [[Image:Venn Diagram (P (R)).jpg|500px]] || (53)
 
|-
 
|-
 
| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{~(} r \texttt{))}</math>
 
| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{~(} r \texttt{))}</math>
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| &nbsp;
 
| &nbsp;
 
|-
 
|-
| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (5)
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| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (54)
 
|-
 
|-
 
| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}</math>
 
| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}</math>
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Figure 12.  Boolean 3-Cube B^3
 
Figure 12.  Boolean 3-Cube B^3
 
</pre>
 
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Figure 14.  Triadic Relation Syll c B^3
 
Figure 14.  Triadic Relation Syll c B^3
 
</pre>
 
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o-----------o o-----------o o-----------o
 
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Figure 17.  Syll c B^3 and its Dyadic Projections
 
Figure 17.  Syll c B^3 and its Dyadic Projections
 
</pre>
 
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o---------------o o---------------o o---------------o
 
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Figure 21.  Tacit Extension te_12_3 (Syll_12)
 
Figure 21.  Tacit Extension te_12_3 (Syll_12)
 
</pre>
 
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Figure 22.  Tacit Extension te_13_2 (Syll_13)
 
Figure 22.  Tacit Extension te_13_2 (Syll_13)
 
</pre>
 
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Figure 23.  Tacit Extension te_23_1 (Syll_23)
 
Figure 23.  Tacit Extension te_23_1 (Syll_23)
 
</pre>
 
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Figure 24.  Syll = te(Syll_12) |^| te(Syll_23)
 
Figure 24.  Syll = te(Syll_12) |^| te(Syll_23)
 
</pre>
 
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<br><sharethis />
 
<br><sharethis />
  −
<!--semantic tags-->
  −
[[Author:=Jon Awbrey|&nbsp;]]
  −
[[Paper Name:=Propositional Equation Reasoning Systems|&nbsp;]]
  −
[[Paper Of::Directory:Jon Awbrey|&nbsp;]]
      
[[Category:Artificial Intelligence]]
 
[[Category:Artificial Intelligence]]
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