[Inquiry] Re: Cactus Rules

[Jon Awbrey]

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