ONT Re: Cactus Rules
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 12
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
In the language of cacti, unlike Peirce's alpha graphs,
it is possible to represent the logical functions that
correspond to the difference in truth value and the
equality in truth value of two logical variables
in forms that mention each variable only once.
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p , q)` ` ` | ` ` ` ` | ` ` ((p , q)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_60` ` ` ` | ` ` ` ` | ` ` ` q_195 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
We have already noted the initial variations on the themes
of difference and equality among the forms in Table 2 that
gave the linear propositions and their logical complements.
Table 8 enumerates a few more variations along these lines.
Table 8. More Variations on Difference and Equality
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_96 | q_01100000 | 0 1 1 0 0 0 0 0 | p (q , r) |
| | | | |
| q_72 | q_01001000 | 0 1 0 0 1 0 0 0 | q (p , r) |
| | | | |
| q_40 | q_00101000 | 0 0 1 0 1 0 0 0 | r (p , q) |
| | | | |
| q_144 | q_10010000 | 1 0 0 1 0 0 0 0 | p ((q , r)) |
| | | | |
| q_132 | q_10000100 | 1 0 0 0 0 1 0 0 | q ((p , r)) |
| | | | |
| q_130 | q_10000010 | 1 0 0 0 0 0 1 0 | r ((p , q)) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_6 | q_00000110 | 0 0 0 0 0 1 1 0 | (p) (q , r) |
| | | | |
| q_18 | q_00010010 | 0 0 0 1 0 0 1 0 | (q) (p , r) |
| | | | |
| q_20 | q_00010100 | 0 0 0 1 0 1 0 0 | (r) (p , q) |
| | | | |
| q_9 | q_00001001 | 0 0 0 0 1 0 0 1 | (p) ((q , r)) |
| | | | |
| q_33 | q_00100001 | 0 0 1 0 0 0 0 1 | (q) ((p , r)) |
| | | | |
| q_65 | q_01000001 | 0 1 0 0 0 0 0 1 | (r) ((p , q)) |
| | | | |
o=========o============o=================o===================o
| | | | |
| q_159 | q_10011111 | 1 0 0 1 1 1 1 1 | (p (q , r)) |
| | | | |
| q_183 | q_10110111 | 1 0 1 1 0 1 1 1 | (q (p , r)) |
| | | | |
| q_215 | q_11010111 | 1 1 0 1 0 1 1 1 | (r (p , q)) |
| | | | |
| q_111 | q_01101111 | 0 1 1 0 1 1 1 1 | (p ((q , r))) |
| | | | |
| q_123 | q_01111011 | 0 1 1 1 1 0 1 1 | (q ((p , r))) |
| | | | |
| q_125 | q_01111101 | 0 1 1 1 1 1 0 1 | (r ((p , q))) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_249 | q_11111001 | 1 1 1 1 1 0 0 1 | ((p) (q , r)) |
| | | | |
| q_237 | q_11101101 | 1 1 1 0 1 1 0 1 | ((q) (p , r)) |
| | | | |
| q_235 | q_11101011 | 1 1 1 0 1 0 1 1 | ((r) (p , q)) |
| | | | |
| q_246 | q_11110110 | 1 1 1 1 0 1 1 0 | ((p) ((q , r))) |
| | | | |
| q_222 | q_11011110 | 1 1 0 1 1 1 1 0 | ((q) ((p , r))) |
| | | | |
| q_190 | q_10111110 | 1 0 1 1 1 1 1 0 | ((r) ((p , q))) |
| | | | |
o---------o------------o-----------------o-------------------o
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o