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ONT Re: Cactus Rules




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CR.  Note 8

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In any k-dimensional universe of discourse X% = [x_1, ..., x_k]
there are two other (2^k)-clans of propositions that ordinarily
merit special attention.  These are the "positive" propositions
and the "singular" propositions, tabulated for the present case
k = 3 in Tables 3 and 4, respectively, as usual throwing in the
logical complements just for good measure.

Table 3.  Positive Propositions and Their Complements
o---------o------------o-----------------o-------------------o
| L_1     | L_2        | L_3             | L_4               |
|         |            |                 |                   |
| Decimal | Binary     | Vector          | Cactus            |
o---------o------------o-----------------o-------------------o
|         |          p : 1 1 1 1 0 0 0 0 |                   |
|         |          q : 1 1 0 0 1 1 0 0 |                   |
|         |          r : 1 0 1 0 1 0 1 0 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_255   | q_11111111 | 1 1 1 1 1 1 1 1 |       (( ))       |
|         |            |                 |                   |
| q_240   | q_11110000 | 1 1 1 1 0 0 0 0 |    p              |
|         |            |                 |                   |
| q_204   | q_11001100 | 1 1 0 0 1 1 0 0 |         q         |
|         |            |                 |                   |
| q_170   | q_10101010 | 1 0 1 0 1 0 1 0 |              r    |
|         |            |                 |                   |
| q_192   | q_11000000 | 1 1 0 0 0 0 0 0 |    p    q         |
|         |            |                 |                   |
| q_160   | q_10100000 | 1 0 1 0 0 0 0 0 |    p         r    |
|         |            |                 |                   |
| q_136   | q_10001000 | 1 0 0 0 1 0 0 0 |         q    r    |
|         |            |                 |                   |
| q_128   | q_10000000 | 1 0 0 0 0 0 0 0 |    p    q    r    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_1     | q_00000000 | 0 0 0 0 0 0 0 0 |        ( )        |
|         |            |                 |                   |
| q_15    | q_00001111 | 0 0 0 0 1 1 1 1 |   (p)             |
|         |            |                 |                   |
| q_51    | q_00110011 | 0 0 1 1 0 0 1 1 |        (q)        |
|         |            |                 |                   |
| q_85    | q_01010101 | 0 1 0 1 0 1 0 1 |             (r)   |
|         |            |                 |                   |
| q_63    | q_00111111 | 0 0 1 1 1 1 1 1 |   (p    q)        |
|         |            |                 |                   |
| q_95    | q_01011111 | 0 1 0 1 1 1 1 1 |   (p         r)   |
|         |            |                 |                   |
| q_119   | q_01110111 | 0 1 1 1 0 1 1 1 |        (q    r)   |
|         |            |                 |                   |
| q_127   | q_01111111 | 0 1 1 1 1 1 1 1 |   (p    q    r)   |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Table 4.  Singular Propositions and Their Complements
o---------o------------o-----------------o-------------------o
| L_1     | L_2        | L_3             | L_4               |
|         |            |                 |                   |
| Decimal | Binary     | Vector          | Cactus            |
o---------o------------o-----------------o-------------------o
|         |          p : 1 1 1 1 0 0 0 0 |                   |
|         |          q : 1 1 0 0 1 1 0 0 |                   |
|         |          r : 1 0 1 0 1 0 1 0 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_1     | q_00000001 | 0 0 0 0 0 0 0 1 |   (p)  (q)  (r)   |
|         |            |                 |                   |
| q_2     | q_00000010 | 0 0 0 0 0 0 1 0 |   (p)  (q)   r    |
|         |            |                 |                   |
| q_4     | q_00000100 | 0 0 0 0 0 1 0 0 |   (p)   q   (r)   |
|         |            |                 |                   |
| q_8     | q_00001000 | 0 0 0 0 1 0 0 0 |   (p)   q    r    |
|         |            |                 |                   |
| q_16    | q_00010000 | 0 0 0 1 0 0 0 0 |    p   (q)  (r)   |
|         |            |                 |                   |
| q_32    | q_00100000 | 0 0 1 0 0 0 0 0 |    p   (q)   r    |
|         |            |                 |                   |
| q_64    | q_01000000 | 0 1 0 0 0 0 0 0 |    p    q   (r)   |
|         |            |                 |                   |
| q_128   | q_10000000 | 1 0 0 0 0 0 0 0 |    p    q    r    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_254   | q_11111110 | 1 1 1 1 1 1 1 0 |  ((p)  (q)   r))  |
|         |            |                 |                   |
| q_253   | q_11111101 | 1 1 1 1 1 1 0 1 |  ((p)  (q)   r )  |
|         |            |                 |                   |
| q_251   | q_11111011 | 1 1 1 1 1 0 1 1 |  ((p)   q   (r))  |
|         |            |                 |                   |
| q_247   | q_11110111 | 1 1 1 1 0 1 1 1 |  ((p)   q    r )  |
|         |            |                 |                   |
| q_239   | q_11101111 | 1 1 1 0 1 1 1 1 |  ( p   (q)  (r))  |
|         |            |                 |                   |
| q_223   | q_11011111 | 1 1 0 1 1 1 1 1 |  ( p   (q)   r )  |
|         |            |                 |                   |
| q_191   | q_10111111 | 1 0 1 1 1 1 1 1 |  ( p    q   (r))  |
|         |            |                 |                   |
| q_127   | q_01111111 | 0 1 1 1 1 1 1 1 |  ( p    q    r )  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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