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| o-----------------------------------------------------------o | | o-----------------------------------------------------------o |
| </pre> | | </pre> |
− | | (1) | + | | (23) |
| |} | | |} |
| | | |
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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) |
| |- | | |- |
| | | | | |
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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) |
| |} | | |} |
| | | |
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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) |
| |} | | |} |
| | | |
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| o-----------------------------------------------------------o | | o-----------------------------------------------------------o |
| </pre> | | </pre> |
− | | (4) | + | | (33) |
| |} | | |} |
| | | |
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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) |
| |} | | |} |
| | | |
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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> |
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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> |
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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) |
| |} | | |} |
| | | |
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| | [[Image:Equational Inference Bar -- DNF.jpg|500px]] | | | [[Image:Equational Inference Bar -- DNF.jpg|500px]] |
| |} | | |} |
− | | (6) | + | | (40) |
| |} | | |} |
| | | |
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Line 1,030: |
| {| 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) |
| |} | | |} |
| | | |
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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) |
| |} | | |} |
| | | |
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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) |
| |} | | |} |
| | | |
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| o-----------------------------------------------------------o | | o-----------------------------------------------------------o |
| </pre> | | </pre> |
− | | (12) | + | | (46) |
| |} | | |} |
| | | |
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| o-----------------------------------------------------------o | | o-----------------------------------------------------------o |
| </pre> | | </pre> |
− | | (13) | + | | (47) |
| |} | | |} |
| | | |
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| o-----------------------------------------------------------o | | o-----------------------------------------------------------o |
| </pre> | | </pre> |
− | | (14) | + | | (48) |
| |} | | |} |
| | | |
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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) |
| |} | | |} |
| | | |
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| | [[Image:Equational Inference Bar -- QED.jpg|500px]] | | | [[Image:Equational Inference Bar -- QED.jpg|500px]] |
| |} | | |} |
− | | (16) | + | | (50) |
| |} | | |} |
| | | |
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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: |
Line 1,627: |
| | | | | |
| |- | | |- |
− | | [[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: |
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| 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: |
Line 2,161: |
| 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: |
Line 2,215: |
| 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: |
Line 2,260: |
| 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: |
Line 2,338: |
| 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]] |