ONT Re: Cactus Rules

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

CR.  Note 4

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Given the novelty of the cactus calculus, it is probably
wise to run through a representative sample of the forms
just set down, to note some principles of interpretation,
and to pick up a few clues as to their ordinary language
renderings.  Throughout the rest of this reading it will
be good to recall that "truth", or a boolean valaue of 1,
is represented by a blank string or a blank-labeled node,
while "falsity", or a boolean value of 0, is rendered as
the string "()" or an unlabeled terminal edge, a "spike".

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p , q , r ) ` | ` ` ` ` | `(( p , q , r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_104 ` ` ` | ` ` ` ` | ` ` ` q_151 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

The function q_104 : B^3 -> B is a basic 3-lobe,
interpreted as the "just one false" operator on
three boolean variables, and the function q_151
is its boolean complement or its exact negation.

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` ` ` ` ` |
| ` ` ` p ` ` ` ` ` | ` ` ` ` | ` ` ` | q r ` ` ` |
| ` ` ` o ` ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | q r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p), q , r ) ` | ` ` ` ` | `(((p), q , r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_134 ` ` ` | ` ` ` ` | ` ` ` q_121 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

The operation of q_134 can be understood by asking
what happens if p is true, in effect, if the label
"p" disappears, leaving only its supporting spike.
That spike, the unique false argument on the lobe,
punctures the lobe beneath, if you will, and what
abides is the statement "q r", that is, "q and r".
On the other hand, if p is (), then the branch (p)
appears to be (()), which reduces to true, and so
it disappears instead, leaving just (q, r), which
is tantamount to stating that q is not equal to r.
In sum the cases are:  p q r, (p) q (r), (p)(q) r.
Once again, q_121 is just the complement of q_134.

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o o o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` | | | ` ` ` |
| ` ` ` o o o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | | | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p),(q),(r)) ` | ` ` ` ` | `(((p),(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_22` ` ` ` | ` ` ` ` | ` ` ` q_233 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

The rest of this gang can be dispatched by the same method.
But I want to single out for special mention the form q_22,
the "just one true" operator that is especially handy when
the time comes to specify a partition of the universe into
a number of mutually exclusive and exhaustive territories,
here envisioned to salute the flags p, q, r, respectively.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o