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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (1)  | + | | (23)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 544: | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (2)  | + | | (24)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (1)  | + | | (25)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (2)  | + | | (26)  | 
|   | |-  |   | |-  | 
|   | |  |   | |  | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (3)  | + | | (27)  | 
|   | |-  |   | |-  | 
|   | |  |   | |  | 
| Line 666: | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (4)  | + | | (28)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
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|   | o=============================< QES >=======================o  |   | o=============================< QES >=======================o  | 
|   | </pre>  |   | </pre>  | 
| − | | (5)  | + | | (29)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
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|   | | [[Image:Equational Inference Bar -- QED.jpg|500px]]  |   | | [[Image:Equational Inference Bar -- QED.jpg|500px]]  | 
|   | |}  |   | |}  | 
| − | | (1)  | + | | (30)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
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|   | | [[Image:Praeclarum Theorema CAST 500 x 389 Animation.gif]]  |   | | [[Image:Praeclarum Theorema CAST 500 x 389 Animation.gif]]  | 
|   | |}  |   | |}  | 
| − | | (2)  | + | | (31)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 824: | 
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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)  | + | | (32)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 845: | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (4)  | + | | (33)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 863: | 
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|   |  |   |  | 
|   | {| align="center" cellpadding="10"  |   | {| align="center" cellpadding="10"  | 
| − | | [[Image:Praeclarum Theorema.jpg|500px]] || (1)  | + | | [[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)  | + | | [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (35)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 907: | 
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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)  | + | | [[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>  | 
| Line 919: | 
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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)  | + | | [[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>  | 
| Line 953: | 
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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)  | + | | [[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)  | + | | (39)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,023: | 
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|   | | [[Image:Equational Inference Bar -- DNF.jpg|500px]]  |   | | [[Image:Equational Inference Bar -- DNF.jpg|500px]]  | 
|   | |}  |   | |}  | 
| − | | (6)  | + | | (40)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,030: | 
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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)  | + | | (41)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,049: | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (8)  | + | | (42)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
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|   | | [[Image:Equational Inference Bar -- DNF.jpg|500px]]  |   | | [[Image:Equational Inference Bar -- DNF.jpg|500px]]  | 
|   | |}  |   | |}  | 
| − | | (9)  | + | | (43)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
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|   |  |   |  | 
|   | {| align="center" cellpadding="8"  |   | {| align="center" cellpadding="8"  | 
| − | | [[Image:Logical Graph (P Q R , (P)).jpg|500px]] || (10)  | + | | [[Image:Logical Graph (P Q R , (P)).jpg|500px]] || (44)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,108: | 
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|   |  |   |  | 
|   | {| align="center" cellpadding="8"  |   | {| align="center" cellpadding="8"  | 
| − | | [[Image:Logical Graph ((P , P Q R)).jpg|500px]] || (11)  | + | | [[Image:Logical Graph ((P , P Q R)).jpg|500px]] || (45)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,128: | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (12)  | + | | (46)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,147: | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (13)  | + | | (47)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,178: | 
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|   | o-----------------------------------------------------------o  |   | o-----------------------------------------------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (14)  | + | | (48)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,213: | 
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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]]  | 
| − | | (15)  | + | | (49)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,292: | 
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|   | | [[Image:Equational Inference Bar -- QED.jpg|500px]]  |   | | [[Image:Equational Inference Bar -- QED.jpg|500px]]  | 
|   | |}  |   | |}  | 
| − | | (16)  | + | | (50)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,609: | 
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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)  | + | | [[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>  | 
| Line 1,615: | 
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|   | |    |   | |    | 
|   | |-  |   | |-  | 
| − | | [[Image:Venn Diagram (Q (R)).jpg|500px]] || (3)  | + | | [[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>  | 
| Line 1,621: | 
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|   | |    |   | |    | 
|   | |-  |   | |-  | 
| − | | [[Image:Venn Diagram (P (R)).jpg|500px]] || (4)  | + | | [[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>  | 
| Line 1,627: | 
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|   | |    |   | |    | 
|   | |-  |   | |-  | 
| − | | [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (5)  | + | | [[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>  | 
| Line 1,908: | 
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|   | Figure 12.  Boolean 3-Cube B^3  |   | Figure 12.  Boolean 3-Cube B^3  | 
|   | </pre>  |   | </pre>  | 
| − | | (12)  | + | | (55)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,926: | 
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|   | o-----------------------o  |   | o-----------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (13)  | + | | (56)  | 
|   | |-  |   | |-  | 
|   | |  |   | |  | 
| Line 1,961: | 
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|   | Figure 14.  Triadic Relation Syll c B^3  |   | Figure 14.  Triadic Relation Syll c B^3  | 
|   | </pre>  |   | </pre>  | 
| − | | (14)  | + | | (57)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 1,979: | 
Line 1,979: | 
|   | o-----------------------o  |   | o-----------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (15)  | + | | (58)  | 
|   | |-  |   | |-  | 
|   | |  |   | |  | 
| Line 1,996: | 
Line 1,996: | 
|   | o-----------o o-----------o o-----------o  |   | o-----------o o-----------o o-----------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (16)  | + | | (59)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 2,059: | 
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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>  |   | </pre>  | 
| − | | (17)  | + | | (60)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 2,077: | 
Line 2,077: | 
|   | o-----------------------o  |   | o-----------------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (18)  | + | | (61)  | 
|   | |-  |   | |-  | 
|   | |  |   | |  | 
| Line 2,094: | 
Line 2,094: | 
|   | o-----------o o-----------o o-----------o  |   | o-----------o o-----------o o-----------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (19)  | + | | (62)  | 
|   | |-  |   | |-  | 
|   | |  |   | |  | 
| Line 2,116: | 
Line 2,116: | 
|   | o---------------o o---------------o o---------------o  |   | o---------------o o---------------o o---------------o  | 
|   | </pre>  |   | </pre>  | 
| − | | (20)  | + | | (63)  | 
|   | |-  |   | |-  | 
|   | |  |   | |  | 
| Line 2,161: | 
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|   | Figure 21.  Tacit Extension te_12_3 (Syll_12)  |   | Figure 21.  Tacit Extension te_12_3 (Syll_12)  | 
|   | </pre>  |   | </pre>  | 
| − | | (21)  | + | | (64)  | 
|   | |-  |   | |-  | 
|   | |  |   | |  | 
| Line 2,215: | 
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|   | Figure 22.  Tacit Extension te_13_2 (Syll_13)  |   | Figure 22.  Tacit Extension te_13_2 (Syll_13)  | 
|   | </pre>  |   | </pre>  | 
| − | | (22)  | + | | (65)  | 
|   | |-  |   | |-  | 
|   | |  |   | |  | 
| Line 2,260: | 
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|   | Figure 23.  Tacit Extension te_23_1 (Syll_23)  |   | Figure 23.  Tacit Extension te_23_1 (Syll_23)  | 
|   | </pre>  |   | </pre>  | 
| − | | (23)  | + | | (66)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 2,338: | 
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|   | Figure 24.  Syll = te(Syll_12) |^| te(Syll_23)  |   | Figure 24.  Syll = te(Syll_12) |^| te(Syll_23)  | 
|   | </pre>  |   | </pre>  | 
| − | | (24)  | + | | (67)  | 
|   | |}  |   | |}  | 
|   |  |   |  | 
| Line 2,846: | 
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|   |  |   |  | 
|   | <br><sharethis />  |   | <br><sharethis />  | 
| − | 
  |   | 
| − | <!--semantic tags-->
  |   | 
| − | [[Author:=Jon Awbrey| ]]
  |   | 
| − | [[Paper Name:=Propositional Equation Reasoning Systems| ]]
  |   | 
| − | [[Paper Of::Directory:Jon Awbrey| ]]
  |   | 
|   |  |   |  | 
|   | [[Category:Artificial Intelligence]]  |   | [[Category:Artificial Intelligence]]  |