Registered: Feb 2004
Posts: 551

CR. Note 1

With an eye toward the aims of the NKS Forum, I've begun to work out
a translation of the "elementary cellular automaton rules" (ECAR's),
in effect, just the boolean functions of abstract type q : B^3 -> B,
into cactus language, and I'll post a selection of my working notes
here. By way of the briefest possible reminder, this cactus syntax,
in its existential interpretation and its traverse-string redaction,
uses just two series of k-adic connectives, first, the concatenation
of k expressions is read as their k-adic logical conjunction, second,
a bracket of the form (e_1, ..., e_k) is read to say that exactly one
of the k expressions e_1, ..., e_k is false. I may sometimes refer to
this bracket as a k-adic "boundary operator" or a k-place "cactus lobe".

Reference Material:

http://atlas.wolfram.com/
http://atlas.wolfram.com/01/01/
http://atlas.wolfram.com/01/01/views/3/TableView.html
http://atlas.wolfram.com/01/01/views/87/TableView.html
http://atlas.wolfram.com/01/01/views/172/TableView.html

Incidental Musement:

http://www.pinball.com/games/cactus/

Jon Awbrey

Latest Syntactic Description of the 256 Rules Attached Below:

Attachment: cactus rules -- final.txt
This has been downloaded 1608 time(s).

Last edited by Jon Awbrey on 04-06-2004 at 03:00 AM

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 2

One of the first things I note is that several whole families
of otherwise enigmatic and obscurely expressed rules take on
remarkably simple and transparently related expressions in
the cactus syntax.

For example, Table 1 exhibits the cactus syntax for
an especially interesting family of ECAR's, that is,
boolean maps of the concrete shape [p, q, r] -> [q],
or the abstract type q_j : B^3 -> B.

Table 1. A Family of Propositional Forms On Three Variables
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_22 ` `| q_00010110 | 0 0 0 1 0 1 1 0 | `((p), (q), (r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_41 ` `| q_00101001 | 0 0 1 0 1 0 0 1 | `((p), (q), `r ) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_73 ` `| q_01001001 | 0 1 0 0 1 0 0 1 | `((p), `q , (r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_134 ` | q_10000110 | 1 0 0 0 0 1 1 0 | `((p), `q , `r ) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_97 ` `| q_01100001 | 0 1 1 0 0 0 0 1 | `( p , (q), (r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_146 ` | q_10010010 | 1 0 0 1 0 0 1 0 | `( p , (q), `r ) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_148 ` | q_10010100 | 1 0 0 1 0 1 0 0 | `( p , `q , (r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_104 ` | q_01101000 | 0 1 1 0 1 0 0 0 | `( p , `q , `r ) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_233 ` | q_11101001 | 1 1 1 0 1 0 0 1 | (((p), (q), (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_214 ` | q_11010110 | 1 1 0 1 0 1 1 0 | (((p), (q), `r )) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_182 ` | q_10110110 | 1 0 1 1 0 1 1 0 | (((p), `q , (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_121 ` | q_01111001 | 0 1 1 1 1 0 0 1 | (((p), `q , `r )) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_158 ` | q_10011110 | 1 0 0 1 1 1 1 0 | (( p , (q), (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_109 ` | q_01101101 | 0 1 1 0 1 1 0 1 | (( p , (q), `r )) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_107 ` | q_01101011 | 0 1 1 0 1 0 1 1 | (( p , `q , (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_151 ` | q_10010111 | 1 0 0 1 0 1 1 1 | (( p , `q , `r )) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o

I invite the Reader to compare these expressions with their
corresponding numbers, the same boolean functions expressed
in terms of operators from the set {And, Or, Xor, Not}, for
example, as shown in the "Wolfram Atlas of Simple Programs":

http://atlas.wolfram.com/01/01/views/172/TableView.html

Jon Awbrey

Last edited by Jon Awbrey on 03-17-2004 at 04:16 PM

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 3

Here are the parse-graph portraits of the family of cacti
that we examined last time, listed in complementary pairs.

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

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

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` p | r ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p | r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p ,(q), r ) ` | ` ` ` ` | `(( p ,(q), r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_146 ` ` ` | ` ` ` ` | ` ` ` q_109 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` p q | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p q | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p , q ,(r)) ` | ` ` ` ` | `(( p , q ,(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_148 ` ` ` | ` ` ` ` | ` ` ` q_107 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o o ` ` ` ` |
| ` ` ` p q ` ` ` ` | ` ` ` ` | ` ` ` | | r ` ` ` |
| ` ` ` o o ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | | r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p),(q), r ) ` | ` ` ` ` | `(((p),(q), r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_41` ` ` ` | ` ` ` ` | ` ` ` q_214 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` | q | ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | q | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p), q ,(r)) ` | ` ` ` ` | `(((p), q ,(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_73` ` ` ` | ` ` ` ` | ` ` ` q_182 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o o ` ` ` |
| ` ` ` ` q r ` ` ` | ` ` ` ` | ` ` ` p | | ` ` ` |
| ` ` ` ` o o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p | | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p ,(q),(r)) ` | ` ` ` ` | `(( p ,(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_97` ` ` ` | ` ` ` ` | ` ` ` q_158 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

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

As I work through the 256 ECAR's or functions q_j : B^3 -> B,
I will keep an updated copy of my worksheet as an attachment
to the first posting on this thread at the NKS Forum website:

Re: http://forum.wolframscience.com/sho...tid=810#post810
In: http://forum.wolframscience.com/sho...s=&threadid=256

The interested reader is invited to help check this work,
as errors are almost inevitable in this type of exercise.
Plus, I can't always get expressions that are as elegant
as I might like, and it may be that other eyes would see
forms more economical than the ones that strike me first.

Jon Awbrey

Last edited by Jon Awbrey on 03-16-2004 at 08:16 PM

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 4

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

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 5

So long as we're seeing the sights at Cactus Junction,
we might as well take a gander at a computational way
to assay the import of any ole cactus expression that
comes down the pike. Way out here, and elsewhere, too,
the computational clarification of a formal expression
is claimed to yield its canonical or its "normal" form.
Finer distinctions can be weighed, of course, and there
is always the problem of just how, exactly, and, indeed,
even whether such forms will be forthcoming from a given
cut of syntax for a given objective domain, or any other
wide open space. But the notion of a "normal form" is
cast in the right direction, and so it'll do for now.

By way of example, let's examine the subtype of cactoid expression
that is typified by q_97 and its complement q_158, and that hardly
got its just deserts in the way of attention the last time around.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` s_1 ` s_k ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `g` ` `|` ` `|` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `o-----o-...-o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

We can reason out the meaning of all such expressions
by using the case analysis tactic that we used before.
If g is true, then it's just like "g" wasn't there at
all, and the expression comes down to the case below:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` s_1 ` ` s_k ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o` ` ` `o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `|` ` ` `|` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o--...--o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

But this expresses the "just one true" condition that partitions
the remaining space, that is to say, the space where g is true,
into k sectors where each of the s_j in its own turn is true.

On the other hand, in the case that g is false, we are left
with a (k+1)-lobe that is known to bear this one bare spike:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` s_1 ` s_k ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `o` ` `o` ` `o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` ` `|` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `o-----o-...-o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

If that expression as a whole is going to turn out to be true,
then there can be only one expression that evaluates to false
on its argument list, and since we already have it in custody,
we know that the remaining arguments, (s_1), ..., (s_k), will
all have to be true. In effect, the spike collapses the lobe
to a node, leaving a conjunction of the negations of the s_j.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `s_1` ` `s_k` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o `...` o ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

In summation, we have the following interpretation:
If g is true, then exactly one of the s_j is true;
if g is false, then all of the s_j are false, too.

That is not yet a method that would be amenable to
computational routine, but it does get us part way.

Jon Awbrey

Last edited by Jon Awbrey on 03-19-2004 at 05:16 PM

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 6

Within each space of boolean functions {f : B^k -> B},
altogether ranking a cardinality of 2^(2^k) functions,
there are several standard subsets of cardinality 2^k
that rate special mention and study. One such subset
is the space of linear functions, known algebraically
as the set of "homomorphisms" {hom : B^k -> B} or the
"dual space" X*, because it is dual to the coordinate
space X of "points" or "vectors" in B^k.

In the present setting, where k = 3, we may expect to find
2^3 = 8 linear functions of the abstract type h : B^3 -> B.

Table 2 shows the q_j that are linear functions, together
with their boolean complements or their logical negations.

Table 2. Linear Propositions and Their Complements
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_0 ` ` | q_00000000 | 0 0 0 0 0 0 0 0 | ` ` ` `( )` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_240 ` | q_11110000 | 1 1 1 1 0 0 0 0 | ` `p` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_204 ` | q_11001100 | 1 1 0 0 1 1 0 0 | ` ` ` ` q ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_170 ` | q_10101010 | 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_60 ` `| q_00111100 | 0 0 1 1 1 1 0 0 | ` (p , `q)` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_90 ` `| q_01011010 | 0 1 0 1 1 0 1 0 | ` (p , ` ` ` r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_102 ` | q_01100110 | 0 1 1 0 0 1 1 0 | ` ` ` `(q , `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_150 ` | q_10010110 | 1 0 0 1 0 1 1 0 | ` (p , (q , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_255 ` | q_11111111 | 1 1 1 1 1 1 1 1 | ` ` ` (( )) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_15 ` `| q_00001111 | 0 0 0 0 1 1 1 1 | ` (p) ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_51 ` `| q_00110011 | 0 0 1 1 0 0 1 1 | ` ` ` `(q)` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_85 ` `| q_01010101 | 0 1 0 1 0 1 0 1 | ` ` ` ` ` ` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_195 ` | q_11000011 | 1 1 0 0 0 0 1 1 | `((p , `q)) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_165 ` | q_10100101 | 1 0 1 0 0 1 0 1 | `((p , ` ` ` r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_153 ` | q_10011001 | 1 0 0 1 1 0 0 1 | ` ` ` ((q , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_105 ` | q_01101001 | 0 1 1 0 1 0 0 1 | `((p , (q , `r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o

The Figures that follow give a representative selection
of the corresponding cacti in all their greenest glory.

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `( )` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_0` ` ` ` | ` ` ` ` | ` ` ` q_255 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` `(p)` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_240 ` ` ` | ` ` ` ` | ` ` ` q_15` ` ` ` |
o-------------------o ` ` ` ` o-------------------o

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

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` r ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o---o ` ` |
| ` ` ` ` q ` r ` ` | ` ` ` ` | ` ` ` p `\ /` ` ` |
| ` ` ` ` o---o ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p `\ /` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p , (q , r)) ` | ` ` ` ` | `((p , (q , r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_150 ` ` ` | ` ` ` ` | ` ` ` q_105 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

BeannachtaÃ_ na Féile Pádraig oraibh go leir!

Jon Awbrey

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 7

Had I been thinking ahead, I might have mentioned this first,
but now that aspects of algebra and geometry have intruded on
our logical paradise, in the guise of the dual space X*, let's
give belated notice to one family of propositions that have been
basic to our enterprise all along, whether we noticed them or not.

In a k-dimensional universe of discourse X% = [x_1, ..., x_k] the
position space X = <|x_1, ..., x_k|> is isomorphic to B^k and the
proposition space X^ = (X -> B) = {f : X -> B} bears the abstract
type B^k -> B. In algebra and geometry, as a rule, one tends to
take position spaces and function spaces together in pairs, and
so we assign the universe X% a "stereotype" of <B^k, B^k -> B>,
or B^k +-> B, for short. I like to think of these spaces as
the "paint layer" X and "draw layer" X^ of the universe X%.

What I need to make a point of at this point is that the k-set
of logical features !X! = {x_1, ..., x_k} that we invoke as the
basis of the universe of discourse also constitutes an important
family of propositions x_j : B^k -> B, for j = 1 to k. These are
called by any one of several different names: "basic propositions",
"coordinate projections", or "simple propositions".

Table 0 accords this family of simple propositions their
formal recognition, for the present case of 3 dimensions.

Table 0. Simple Propositions
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_240 ` | q_11110000 | 1 1 1 1 0 0 0 0 | ` `p` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_204 ` | q_11001100 | 1 1 0 0 1 1 0 0 | ` ` ` ` q ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_170 ` | q_10101010 | 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o

Of course, we've already seen this 3-set of basic propositions
numbered among the (2^3)-set of linear propositions in Table 2.

Additional discussion of these underpinnings can be found here:

| Jon Awbrey, "Differential Logic and Dynamic Systems"
| http://stderr.org/pipermail/inquiry...thread.html#478
| http://stderr.org/pipermail/inquiry...thread.html#553

Especially:

DLOG D2. http://stderr.org/pipermail/inquiry...May/000480.html
DLOG D5. http://stderr.org/pipermail/inquiry...May/000483.html

With that out of the way, I'll try to
get back to the main event next time.

Jon Awbrey

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 8

In any k-dimensional universe of discourse X% = [x_1, ..., x_k]
there are two other (2^k)-clans of propositions that ordinarily
merit special attention. These are the "positive" propositions
and the "singular" propositions, tabulated for the present case
k = 3 in Tables 3 and 4, respectively, as usual throwing in the
logical complements just for good measure.

Table 3. Positive Propositions and Their Complements
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_255 ` | q_11111111 | 1 1 1 1 1 1 1 1 | ` ` ` (( )) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_240 ` | q_11110000 | 1 1 1 1 0 0 0 0 | ` `p` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_204 ` | q_11001100 | 1 1 0 0 1 1 0 0 | ` ` ` ` q ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_170 ` | q_10101010 | 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_192 ` | q_11000000 | 1 1 0 0 0 0 0 0 | ` `p` ` q ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_160 ` | q_10100000 | 1 0 1 0 0 0 0 0 | ` `p` ` ` ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_136 ` | q_10001000 | 1 0 0 0 1 0 0 0 | ` ` ` ` q ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_128 ` | q_10000000 | 1 0 0 0 0 0 0 0 | ` `p` ` q ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_0 ` ` | q_00000000 | 0 0 0 0 0 0 0 0 | ` ` ` `( )` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_15` ` | q_00001111 | 0 0 0 0 1 1 1 1 | ` (p) ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_51` ` | q_00110011 | 0 0 1 1 0 0 1 1 | ` ` ` `(q)` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_85` ` | q_01010101 | 0 1 0 1 0 1 0 1 | ` ` ` ` ` ` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_63` ` | q_00111111 | 0 0 1 1 1 1 1 1 | ` (p` ` q)` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_95` ` | q_01011111 | 0 1 0 1 1 1 1 1 | ` (p` ` ` ` `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_119 ` | q_01110111 | 0 1 1 1 0 1 1 1 | ` ` ` `(q ` `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_127 ` | q_01111111 | 0 1 1 1 1 1 1 1 | ` (p` ` q ` `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o

Table 4. Singular Propositions and Their Complements
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_1 ` ` | q_00000001 | 0 0 0 0 0 0 0 1 | ` (p) `(q)` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_2 ` ` | q_00000010 | 0 0 0 0 0 0 1 0 | ` (p) `(q)` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_4 ` ` | q_00000100 | 0 0 0 0 0 1 0 0 | ` (p) ` q ` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_8 ` ` | q_00001000 | 0 0 0 0 1 0 0 0 | ` (p) ` q ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_16` ` | q_00010000 | 0 0 0 1 0 0 0 0 | ` `p` `(q)` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_32` ` | q_00100000 | 0 0 1 0 0 0 0 0 | ` `p` `(q)` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_64` ` | q_01000000 | 0 1 0 0 0 0 0 0 | ` `p` ` q ` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_128 ` | q_10000000 | 1 0 0 0 0 0 0 0 | ` `p` ` q ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_254 ` | q_11111110 | 1 1 1 1 1 1 1 0 | `((p) `(q)` `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_253 ` | q_11111101 | 1 1 1 1 1 1 0 1 | `((p) `(q)` `r )` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_251 ` | q_11111011 | 1 1 1 1 1 0 1 1 | `((p) ` q ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_247 ` | q_11110111 | 1 1 1 1 0 1 1 1 | `((p) ` q ` `r )` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_239 ` | q_11101111 | 1 1 1 0 1 1 1 1 | `( p` `(q)` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_223 ` | q_11011111 | 1 1 0 1 1 1 1 1 | `( p` `(q)` `r )` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_191 ` | q_10111111 | 1 0 1 1 1 1 1 1 | `( p` ` q ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_127 ` | q_01111111 | 0 1 1 1 1 1 1 1 | `( p` ` q ` `r )` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o

Jon Awbrey

Last edited by Jon Awbrey on 03-19-2004 at 03:34 AM

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 9

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

Last edited by Jon Awbrey on 03-19-2004 at 02:40 PM

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 10

Table 6. More 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_176 ` | q_10110000 | 1 0 1 1 0 0 0 0 | ` `p` `(q ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_208 ` | q_11010000 | 1 1 0 1 0 0 0 0 | ` `p` `(r ` (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_11` ` | q_00001011 | 0 0 0 0 1 0 1 1 | ` (p) `(q ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_13` ` | q_00001101 | 0 0 0 0 1 1 0 1 | ` (p) `(r ` (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_140 ` | q_10001100 | 1 0 0 0 1 1 0 0 | ` `q` `(p ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_196 ` | q_11000100 | 1 1 0 0 0 1 0 0 | ` `q` `(r ` (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_35` ` | q_00100011 | 0 0 1 0 0 0 1 1 | ` (q) `(p ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_49` ` | q_00110001 | 0 0 1 1 0 0 0 1 | ` (q) `(r ` (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_138 ` | q_10001010 | 1 0 0 0 1 0 1 0 | ` `r` `(p ` (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_162 ` | q_10100010 | 1 0 1 0 0 0 1 0 | ` `r` `(q ` (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_69` ` | q_01000101 | 0 1 0 0 0 1 0 1 | ` (r) `(p ` (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_81` ` | q_01010001 | 0 1 0 1 0 0 0 1 | ` (r) `(q ` (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_79` ` | q_01001111 | 0 1 0 0 1 1 1 1 | `( p` `(q ` (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_47` ` | q_00101111 | 0 0 1 0 1 1 1 1 | `( p` `(r ` (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_244 ` | q_11110100 | 1 1 1 1 0 1 0 0 | `((p) `(q ` (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_242 ` | q_11110010 | 1 1 1 1 0 0 1 0 | `((p) `(r ` (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_115 ` | q_01110011 | 0 1 1 1 0 0 1 1 | `( q` `(p ` (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_59` ` | q_00111011 | 0 0 1 1 1 0 1 1 | `( q` `(r ` (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_220 ` | q_11011100 | 1 1 0 1 1 1 0 0 | `((q) `(p ` (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_206 ` | q_11001110 | 1 1 0 0 1 1 1 0 | `((q) `(r ` (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_117 ` | q_01110101 | 0 1 1 1 0 1 0 1 | `( r` `(p ` (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_93` ` | q_01011101 | 0 1 0 1 1 1 0 1 | `( r` `(q ` (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_186 ` | q_10111010 | 1 0 1 1 1 0 1 0 | `((r) `(p ` (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_174 ` | q_10101110 | 1 0 1 0 1 1 1 0 | `((r) `(q ` (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o

Jon Awbrey

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 11

Table 7. Conjunctive Implications and Their Complements
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_139 ` | q_10001011 | 1 0 0 0 1 0 1 1 | ` (p (q))(q (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_141 ` | q_10001101 | 1 0 0 0 1 1 0 1 | ` (p (r))(r (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_177 ` | q_10110001 | 1 0 1 1 0 0 0 1 | ` (q (r))(r (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_163 ` | q_10100011 | 1 0 1 0 0 0 1 1 | ` (q (p))(p (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_197 ` | q_11000101 | 1 1 0 0 0 1 0 1 | ` (r (p))(p (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_209 ` | q_11010001 | 1 1 0 1 0 0 0 1 | ` (r (q))(q (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_116 ` | q_01110100 | 0 1 1 1 0 1 0 0 | `((p (q))(q (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_114 ` | q_01110010 | 0 1 1 1 0 0 1 0 | `((p (r))(r (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_78` ` | q_01001110 | 0 1 0 0 1 1 1 0 | `((q (r))(r (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_92` ` | q_01011100 | 0 1 0 1 1 1 0 0 | `((q (p))(p (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_58` ` | q_00111010 | 0 0 1 1 1 0 1 0 | `((r (p))(p (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_46` ` | q_00101110 | 0 0 1 0 1 1 1 0 | `((r (q))(q (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o

Jon Awbrey

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 12

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

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 13

Table 9. Conjunctive Differences and Equalities
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_24` ` | q_00011000 | 0 0 0 1 1 0 0 0 | ` (p, q)` (p, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_36` ` | q_00100100 | 0 0 1 0 0 1 0 0 | ` (p, q)` (q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_66` ` | q_01000010 | 0 1 0 0 0 0 1 0 | ` (p, r)` (q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_129 ` | q_10000001 | 1 0 0 0 0 0 0 1 | `((p, q))((q, r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
o---------o------------o-----------------o--------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_231 ` | q_11100111 | 1 1 1 0 0 1 1 1 | ( (p, q)` (p, r) ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_219 ` | q_11011011 | 1 1 0 1 1 0 1 1 | ( (p, q)` (q, r) ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_189 ` | q_10111101 | 1 0 1 1 1 1 0 1 | ( (p, r)` (q, r) ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_126 ` | q_01111110 | 0 1 1 1 1 1 1 0 | (((p, q))((q, r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
o---------o------------o-----------------o--------------------o

Jon Awbrey

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 14

I will explain my concept of "thematization"
or "thematic extension" after I copy out the
series of Tables that is formed on its basis.
In the meantime, here is a general exposition:

| Jon Awbrey, "Differential Logic and Dynamic Systems"
| DLOG D28. http://suo.ieee.org/ontology/msg04826.html
| DLOG D29. http://suo.ieee.org/ontology/msg04827.html
| DLOG D30. http://suo.ieee.org/ontology/msg04828.html
| DLOG D31. http://suo.ieee.org/ontology/msg04829.html
| DLOG D32. http://suo.ieee.org/ontology/msg04830.html
| DLOG D33. http://suo.ieee.org/ontology/msg04832.html

In order to make the pattern of their construction
more evident, I have left the expressions of the
thematic extensions in their unreduced forms.

Table 10. Thematic Extensions: [q, r] -> [p, q, r]
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_15` ` | q_00001111 | 0 0 0 0 1 1 1 1 | ((p , ` `( )` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_30` ` | q_00011110 | 0 0 0 1 1 1 1 0 | ((p , `(q) (r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_45` ` | q_00101101 | 0 0 1 0 1 1 0 1 | ((p , `(q)` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_60` ` | q_00111100 | 0 0 1 1 1 1 0 0 | ((p , `(q)` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_75` ` | q_01001011 | 0 1 0 0 1 0 1 1 | ((p , ` q `(r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_90` ` | q_01011010 | 0 1 0 1 1 0 1 0 | ((p , ` ` `(r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_105 ` | q_01101001 | 0 1 1 0 1 0 0 1 | ((p , `(q , r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_120 ` | q_01111000 | 0 1 1 1 1 0 0 0 | ((p , `(q ` r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_135 ` | q_10000111 | 1 0 0 0 0 1 1 1 | ((p , ` q ` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_150 ` | q_10010110 | 1 0 0 1 0 1 1 0 | ((p , ((q , r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_165 ` | q_10100101 | 1 0 1 0 0 1 0 1 | ((p , ` ` ` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_180 ` | q_10110100 | 1 0 1 1 0 1 0 0 | ((p , `(q `(r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_195 ` | q_11000011 | 1 1 0 0 0 0 1 1 | ((p , ` q ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_210 ` | q_11010010 | 1 1 0 1 0 0 1 0 | ((p , ((q)` r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_225 ` | q_11100001 | 1 1 1 0 0 0 0 1 | ((p , ((q) (r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_240 ` | q_11110000 | 1 1 1 1 0 0 0 0 | ((p , ` ` ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o

Jon Awbrey

Report this post to a moderator | IP: Logged

Registered: Feb 2004
Posts: 551

CR. Note 15

Table 11. Thematic Extensions: [p, r] -> [p, q, r]
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_51` ` | q_00110011 | 0 0 1 1 0 0 1 1 | ((q , ` `( )` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_54` ` | q_00110110 | 0 0 1 1 0 1 1 0 | ((q , `(p) (r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_57` ` | q_00111001 | 0 0 1 1 1 0 0 1 | ((q , `(p)` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_60` ` | q_00111100 | 0 0 1 1 1 1 0 0 | ((q , `(p)` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_99` ` | q_01100011 | 0 1 1 0 0 0 1 1 | ((q , ` p `(r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_102 ` | q_01100110 | 0 1 1 0 0 1 1 0 | ((q , ` ` `(r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_105 ` | q_01101001 | 0 1 1 0 1 0 0 1 | ((q , `(p , r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_108 ` | q_01101100 | 0 1 1 0 1 1 0 0 | ((q , `(p ` r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_147 ` | q_10010011 | 1 0 0 1 0 0 1 1 | ((q , ` p ` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_150 ` | q_10010110 | 1 0 0 1 0 1 1 0 | ((q , ((p , r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_153 ` | q_10011001 | 1 0 0 1 1 0 0 1 | ((q , ` ` ` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_156 ` | q_10011100 | 1 0 0 1 1 1 0 0 | ((q , `(p `(r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_195 ` | q_11000011 | 1 1 0 0 0 0 1 1 | ((q , ` p ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_198 ` | q_11000110 | 1 1 0 0 0 1 1 0 | ((q , ((p)` r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_201 ` | q_11001001 | 1 1 0 0 1 0 0 1 | ((q , ((p) (r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_204 ` | q_11001100 | 1 1 0 0 1 1 0 0 | ((q , ` ` ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o

Jon Awbrey

Report this post to a moderator | IP: Logged