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
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| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
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| | 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) |
| | | | |
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Table 4. Singular Propositions and Their Complements
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| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
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| | 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 ) |
| | | | |
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Jon Awbrey
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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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