Re: Cactus Rules
Jon Awbrey <[email protected]> Sat, 20 Mar 2004 12:16:23 -0500
| Newsgroups | gmane.comp.inquiry,gmane.comp.misc.ontology.general |
|---|---|
| Message-ID | <[email protected]> |
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