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ONT Re: Cactus Rules




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CR.  Note 9

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In the language of cacti, as in Peirce's existential graphs,
the implication p => q takes the form (p (q)), which can be
parsed in a revealing manner as "not p without q".  Thus it
forms the counterpoint to its counter-exemplary form, p (q),
which may be parsed as "p without q", or just "p and not q".

The parse-graph of (p (q)) is a particular type of tree,
that my school of thought in graph theory nomenclates as
a "painted and rooted tree" (PART).  The symbols from the
alphabet !X! of logical marks, in our case, "p", "q", "r",
are called "paints" as a way of signifying that one can put
as many of them as one likes on a node, or none at all, and
that there is no requirement to use all of the paints of the
given palette !X! on any particular graph.  In my etchings,
the root node is singled out with the amphora sign "@".

The graph of a simple implication can be drawn in any way that
a free rooted tree can be, but it is frequently convenient to
portray it as we see below, partly because of how often we
find ourselves linking implications in stepwise series.

o-------------------------------------------------o
|                                                 |
|                  p           q                  |
|                  o-----------o                  |
|                   \                             |
|                    \                            |
|                     \                           |
|                      \                          |
|                       \                         |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                    ( p ( q ))                   |
o-------------------------------------------------o

Table 5 shows a number of ECAR's that have the form
of simple implications or their logical complements.

Table 5.  Variations on a Theme of Implication
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_207   | q_11001111 | 1 1 0 0 1 1 1 1 |   (p   (q))       |
|         |            |                 |                   |
| q_175   | q_10101111 | 1 0 1 0 1 1 1 1 |   (p        (r))  |
|         |            |                 |                   |
| q_187   | q_10111011 | 1 0 1 1 1 0 1 1 |        (q   (r))  |
|         |            |                 |                   |
| q_243   | q_11110011 | 1 1 1 1 0 0 1 1 |  ((p)   q)        |
|         |            |                 |                   |
| q_245   | q_11110101 | 1 1 1 1 0 1 0 1 |  ((p)        r)   |
|         |            |                 |                   |
| q_221   | q_11011101 | 1 1 0 1 1 1 0 1 |       ((q)   r)   |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_48    | q_00110000 | 0 0 1 1 0 0 0 0 |    p   (q)        |
|         |            |                 |                   |
| q_80    | q_01010000 | 0 1 0 1 0 0 0 0 |    p        (r)   |
|         |            |                 |                   |
| q_68    | q_01000100 | 0 1 0 0 0 1 0 0 |         q   (r)   |
|         |            |                 |                   |
| q_12    | q_00001100 | 0 0 0 0 1 1 0 0 |   (p)   q         |
|         |            |                 |                   |
| q_10    | q_00001010 | 0 0 0 0 1 0 1 0 |   (p)        r    |
|         |            |                 |                   |
| q_34    | q_00100010 | 0 0 1 0 0 0 1 0 |        (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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