[Jets And Sharks] - A New Kind of Science: The NKS Forum

A New Kind of Science: The NKS Forum

Pages:1


Jets And Sharks

(Click here to view the original thread with full colors/images)


Posted by: Jon Awbrey

JAS. Note 1

A couple of threads in the NKS Forum that have invoked the themes of
"cellular automata" (CA's) and "neural architectures" (NA's) all in
the same breath have brought to mind this old example application,
in which the "boundary operator" of the form "(_,_,_,...)" that
is the chief construct of the cactus language is used to emulate
the logical effects of neural pools in McClelland and Rumelhart's
"Jets and Sharks" example. An introduction to the cactus syntax
can be found at the following sites:

DLOG. http://stderr.org/pipermail/inquiry...thread.html#478
PERS. http://forum.wolframscience.com/sho...hp?threadid=297

Excerpt from "Theme One: A Program of Inquiry",
By Jon Awbrey and Susan Awbrey, August 9, 1989.

Jets and Sharks

The propositional calculus that is based on the boundary operator
can be interpreted in a way that resembles the logic of activation
states and competition constraints in certain neural network models.
One way to do this is by interpreting the blank or unmarked state as
the resting state of a neural pool, the bound or marked state as its
activated state, and by representing a mutually inhibitory pool of
neurons A, B, C in the expression "(A, B, C)". To illustrate this
possibility, we transcribe a well-known example from the parallel
distributed processing literature (McClelland & Rumelhart, 1988)
and work through two of the associated exercises as portrayed
in Existential Graph format.

o---------------------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( art ` `),( al ` ),( sam `),( clyde ),( mike `),` ` ` ` ` ` ` ` |
| ` `( jim ` `),( greg ),( john ),( doug `),( lance ),` ` ` ` ` ` ` ` |
| ` `( george ),( pete ),( fred ),( gene `),( ralph ),` ` ` ` ` ` ` ` |
| ` `( phil ` ),( ike `),( nick ),( don ` ),( ned ` ),( karl ), ` ` ` |
| ` `( ken ` `),( earl ),( rick ),( ol ` `),( neal `),( dave )) ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( jets , sharks ) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( jets ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( art ` `),( al ` ),( sam `),( clyde ),( mike `),` ` ` ` ` ` ` ` |
| ` `( jim ` `),( greg ),( john ),( doug `),( lance ),` ` ` ` ` ` ` ` |
| ` `( george ),( pete ),( fred ),( gene `),( ralph ))` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( sharks ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( phil ),( ike `),( nick ),( don ),( ned `),( karl ),` ` ` ` ` ` |
| ` `( ken `),( earl ),( rick ),( ol `),( neal ),( dave ))` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( 20's ),( 30's ),( 40's ))` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( 20's ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( sam ` `),( jim `),( greg ),( john ),( lance ), ` ` ` ` ` ` ` ` |
| ` `( george ),( pete ),( fred ),( gene ),( ken ` )) ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( 30's ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( al ` ),( mike ),( doug ),( ralph ),( phil ),( ike `),` ` ` ` ` |
| ` `( nick ),( don `),( ned `),( rick `),( ol ` ),( neal ),( dave )) |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( 40's ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( art ),( clyde ),( karl ),( earl )) ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( junior_high ),( high_school ),( college )) ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( junior_high , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( art `),( al ` `),( clyde `),( mike `),( jim ), ` ` ` ` ` ` ` ` |
| ` `( john ),( lance ),( george ),( ralph ),( ike )) ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( high_school , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( greg ),( doug ),( pete ),( fred ),( nick ),` ` ` ` ` ` ` ` ` ` |
| ` `( karl ),( ken `),( earl ),( rick ),( neal ),( dave )) ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( college , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( sam ),( gene ),( phil ),( don ),( ned ),( ol ))` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( single ),( married ),( divorced )) ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( single ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( art `),( sam `),( clyde ),( mike ),( doug ),( pete ),` ` ` ` ` |
| ` `( fred ),( gene ),( ralph ),( ike `),( nick ),( ken `),( neal )) |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( married , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( al `),( greg ),( john ),( lance ),( phil ),` ` ` ` ` ` ` ` ` ` |
| ` `( don ),( ned `),( karl ),( earl `),( ol ` ))` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( divorced ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( jim ),( george ),( rick ),( dave ))` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( bookie ),( burglar ),( pusher )) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( bookie ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ( sam `),( clyde ),( mike ),( doug ), ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ( pete ),( ike ` ),( ned `),( karl ),( neal ))` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( burglar , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( al ` ` ),( jim ),( john ),( lance ), ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( george ),( don ),( ken `),( earl `),( rick ))` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( pusher ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( art ` ),( greg ),( fred ),( gene ),` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( ralph ),( phil ),( nick ),( ol ` ),( dave )) ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------------------------------------------------------------------o

We now apply Study, the program module that outputs all of the
satisfying interpretations of a propositional expression, to
the proposition defining the "Jets and Sharks" data base.

With a query on the name "ken" we obtain the following
output, giving all of the features associated with Ken:

o---------------------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ken ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `sharks ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` 20's` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `high_school` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` single` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `burglar` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------------------------------------------------------------------o

With a query on the two features "college" and "sharks" we obtain
the following outline of all features satisfying these constraints:

o---------------------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| college ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `sharks ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` 30's` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `married` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` bookie` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `ned` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` burglar ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `don` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` pusher` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `phil ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `ol ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------------------------------------------------------------------o

From this we discover that all college sharks are 30-something and married.
Further, we have a complete listing of their names broken down by occupation,
as no doubt all of them will be, eventually.

| Reference:
|
| McClelland, James L. & Rumelhart, David E.,
|'Explorations in Parallel Distributed Processing:
| A Handbook of Models, Programs, and Exercises',
| MIT Press, Cambridge, MA, 1988.

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 2

While I'm thinking about it, here is a brief explanation of
how I arrived at the boundary operator formalism by starting
with Peirce's logical graphs and applying another distinctively
Peircean idea, that of using logical operator variables, that is,
variables that stand in for logical connectives, and looking for
laws that are invariant over patterns of substitution for these.
This strategy is naturally suggested in contexts where one is
seeking what we might call "reflective programmability", the
ability to reflect critically on programming languages and
the programs that we run, perhaps unwittingly, also known
as habits of thought and action.

Spencer Brown noted that a variable -- say x in (x) --
stands for the contemplated absence or presence of
a constant () -- say (x) as () or (()) respectively.

What if you want to contemplate the absence or presence
of the operator (...) in (x), that is, a variation from
x to (x), respectively?

Long story short -- it turns out most convenient
to consider an expression like (x, y), letting it
reduce to (x) if y is blank and to x if y is ().

Iterating this idea is what leads to the cactus language.

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 3

I happened on the graphical syntax for propositional calculus that
I now call the "cactus language" while exploring the confluence of
three streams of thought. There was C.S. Peirce's use of operator
variables in logical forms and the operational representations of
logical concepts, there was George Spencer Brown's explanation of
a variable as the contemplated presence or absence of a constant,
and then there was the graph theory and group theory that I had
been picking up, bit by bit, since I first encountered them in
tandem in Frank Harary's foundations of math course, c. 1970.

More on that later, as the memories unthaw, but for the moment
I want very much to take care of some long-unfinished business,
and give a more detailed explanation of how I used this syntax
to represent a popular exercise from the PDP literature of the
late 1980's, McClelland's and Rumelhart's "Jets and Sharks".
Searching the Web on "Jets and Sharks", I found that this
golden oldie is still getting air time here and there:

http://www.itee.uq.edu.au/~cogs2010/cmc/chapters/IAC/
http://www.itee.uq.edu.au/~cogs2010...IAC/index3.html
http://www.psych.utoronto.ca/~reing.../brainwave.html

The knowledge base of the case can be expressed as a single proposition.
The following display presents it in the corresponding text file format.

File "jas.log". Jets and Sharks Example
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( art ` `),( al ` ),( sam `),( clyde ),( mike `),` ` ` |
| ` `( jim ` `),( greg ),( john ),( doug `),( lance ),` ` ` |
| ` `( george ),( pete ),( fred ),( gene `),( ralph ),` ` ` |
| ` `( phil ` ),( ike `),( nick ),( don ` ),( ned ` ),` ` ` |
| ` `( karl ` ),( ken `),( earl ),( rick `),( ol ` `),` ` ` |
| ` `( neal ` ),( dave )) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( jets , sharks ) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( jets , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` `( art ` `),( al ` ),( sam `),( clyde ),( mike `),` ` ` |
| ` `( jim ` `),( greg ),( john ),( doug `),( lance ),` ` ` |
| ` `( george ),( pete ),( fred ),( gene `),( ralph ))` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( sharks ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( phil ),( ike `),( nick ),( don ),( ned `),( karl ),` |
| ` `( ken `),( earl ),( rick ),( ol `),( neal ),( dave ))` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( 20's ),( 30's ),( 40's ))` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( 20's ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( sam ` `),( jim `),( greg ),( john ),( lance ), ` ` ` |
| ` `( george ),( pete ),( fred ),( gene ),( ken ` )) ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( 30's ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( al `),( mike ),( doug ),( ralph ),( phil ),` ` ` ` ` |
| ` `( ike ),( nick ),( don `),( ned ` ),( rick ),` ` ` ` ` |
| ` `( ol `),( neal ),( dave )) ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( 40's ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( art ),( clyde ),( karl ),( earl )) ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( junior_high ),( high_school ),( college )) ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( junior_high , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( art `),( al ` `),( clyde `),( mike `),( jim ), ` ` ` |
| ` `( john ),( lance ),( george ),( ralph ),( ike )) ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( high_school , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( greg ),( doug ),( pete ),( fred ), ` ` ` ` ` ` ` ` ` |
| ` `( nick ),( karl ),( ken `),( earl ), ` ` ` ` ` ` ` ` ` |
| ` `( rick ),( neal ),( dave )) ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( college , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( sam ),( gene ),( phil ),( don ),( ned ),( ol ))` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( single ),( married ),( divorced )) ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( single ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( art `),( sam `),( clyde ),( mike `),( doug ),` ` ` ` |
| ` `( pete ),( fred ),( gene `),( ralph ),( ike `),` ` ` ` |
| ` `( nick ),( ken `),( neal `)) ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( married , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( al `),( greg ),( john ),( lance ),( phil ),` ` ` ` ` |
| ` `( don ),( ned `),( karl ),( earl `),( ol ` ))` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( divorced ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( jim ),( george ),( rick ),( dave ))` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( bookie ),( burglar ),( pusher )) ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( bookie ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( sam `),( clyde ),( mike ),( doug ),` ` ` ` ` ` ` ` ` |
| ` `( pete ),( ike ` ),( ned `),( karl ),( neal )) ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( burglar , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( al ` ` ),( jim ),( john ),( lance ), ` ` ` ` ` ` ` ` |
| ` `( george ),( don ),( ken `),( earl `),( rick ))` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( pusher ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( art ` ),( greg ),( fred ),( gene ),` ` ` ` ` ` ` ` ` |
| ` `( ralph ),( phil ),( nick ),( ol ` ),( dave )) ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Let's start with the simplest clause of the conjoint proposition:

Drawn as the corresponding cactus graph, we have:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` jets` `sharks ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o-----o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 3.1. Cactus Graph for the Proposition "(jets, sharks)"

According to my earlier, if somewhat sketchy interpretive suggestions,
we are supposed to picture a quasi-neural pool that contains a couple
of quasi-neural agents or "units", that between the two of them stand
for the logical variables "jets" and "sharks", respectively. Further,
we imagine these agents to be mutually inhibitory, so that settlement
of the dynamic between them achieves equilibrium when just one of the
two is "active" or "changing" and the other is "stable" or "enduring".
This has the effect of an "exclusive or" connective between the two
logical variables "jets" and "sharks", partitioning the West Side
universe of discourse between the two ganglia.

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 4

We were focussing on a particular figure of syntax,
presented here in both graph and string renditions:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `x` ` `y` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o-----o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ( x , y ) ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 4.1. Cactus Graph for the Cactus Expression "(x, y)"

In traversing the cactus graph, in this case a cactus
of one rooted lobe, one starts at the root, reads off
a left parenthesis "(" on the ascent up the left side
of the lobe, reads off the variable "x", counts off a
comma "," as one transits the interior expanse of the
lobe, reads off the variable "y", and then sounds out
a right parenthesis ")" on the descent down the final
slope, bringing closure to this clause of the cactus.

According to the current story about how the abstract logical situation
is embodied in the concrete physical situation, the whole pool of units
that corresponds to this expression comes to its resting condition when
just one of the two units in {x, y} is resting and the other is charged.
We may think of the state of the whole pool as associated with the root
node of the cactus, here distinguished by an "amphora" or "at" sign "@",
but the root of the cactus is not represented by an individual agent of
the system, at least, not yet. We may summarize these facts in tabular
form, as shown in Table 4.2. Simply in order to use a comparative term,
let's count a single unit as a "pool of one".

Table 4.2. Dynamics of (x , y)
o---------o---------o---------o
| ` `x` ` | ` `y` ` | (x , y) |
o=========o=========o=========o
| resting | resting | charged |
o---------o---------o---------o
| resting | charged | resting |
o---------o---------o---------o
| charged | resting | resting |
o---------o---------o---------o
| charged | charged | charged |
o---------o---------o---------o

Table 4.2 shows the facts of the physical situation at equilibrium.
If we let B = {note, rest} = {moving, steady} = {charged, resting},
or whatever candidates you pick for the 2-membered set in question,
the Table shows a function f : B x B -> B, where f<x, y> = (x , y).

There are two ways that this physical function
might be taken to represent a logical function:

  1. If we make the identifications:

    charged = true (= "indicated"),

    resting = false (= "undicated"),

    then the physical function f : B x B -> B
    is tantamount to the logical function that
    is commonly known as "logical equivalence",
    or just plain "equality":

    Table 4.3. Equality Function
    o---------o---------o---------o
    | x ` ` ` | y ` ` ` | (x , y) |
    o=========o=========o=========o
    | false` `| false` `| true ` `|
    o---------o---------o---------o
    | false` `| true ` `| false` `|
    o---------o---------o---------o
    | true ` `| false` `| false` `|
    o---------o---------o---------o
    | true ` `| true ` `| true ` `|
    o---------o---------o---------o

  2. If we make the identifications:

    resting = true (= "indicated"),

    charged = false (= "undicated"),

    then the physical function f : B x B -> B
    is tantamount to the logical function that
    is commonly known as "logical difference",
    or "exclusive disjunction":

    Table 4.4. Difference Function
    o---------o---------o---------o
    | x ` ` ` | y ` ` ` | (x , y) |
    o=========o=========o=========o
    | true ` `| true ` `| false` `|
    o---------o---------o---------o
    | true ` `| false` `| true ` `|
    o---------o---------o---------o
    | false` `| true ` `| true ` `|
    o---------o---------o---------o
    | false` `| false` `| false` `|
    o---------o---------o---------o

Although the syntax of the cactus language modifies the
syntax of Peirce's graphical formalisms to some extent,
the first interpretation corresponds to what he called
the "entitative graphs" and the second interpretation
corresponds to what he called the "existential graphs".
In working through the present example, I have chosen
the existential interpretation of cactus expressions,
and so the form "(jets , sharks)" is interpreted as
saying that everything in the universe of discourse
is either a Jet or a Shark, but never both at once.

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 5

Before we tangle with the rest of the Jets and Sharks example,
let's look at a cactus expression that's next in the series
we just considered, this time a lobe with three variables.
For instance, let's analyze the cactus form whose graph
and string expressions are shown in the next display.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `x` y `z` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o--o--o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` (x, y, z) ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 5.1. Cactus Graph for the Cactus Expression "(x, y, z)"

As always in this competitive paradigm, we assume that
the units x, y, z are mutually inhibitory, so that the
only states that are possible at equilibrium are those
with exactly one unit charged and all the rest at rest.
Table 5.2 summarizes the lobal dynamics of "(x, y, z)".

Table 5.2. Lobal Dynamics of the Form "(x, y, z)"
o-----------o-----------o-----------o-----------o
| ` ` x ` ` | ` ` y ` ` | ` ` z ` ` | (x, y, z) |
o-----------o-----------o-----------o-----------o
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` |
| `charged` | `charged` | `charged` | `charged` |
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` |
| `charged` | `charged` | `resting` | `charged` |
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` |
| `charged` | `resting` | `charged` | `charged` |
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` |
| `charged` | `resting` | `resting` | `resting` |
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` |
| `resting` | `charged` | `charged` | `charged` |
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` |
| `resting` | `charged` | `resting` | `resting` |
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` |
| `resting` | `resting` | `charged` | `resting` |
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` |
| `resting` | `resting` | `resting` | `charged` |
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` |
o-----------o-----------o-----------o-----------o

Given B = {charged, resting} the Table presents the appearance
of a function f : B x B x B -> B, where f<x, y, z> = (x, y, z).

If we make the identifications, charged = false, resting = true,
in accord with the so-called "existential" interpretation, then
the physical function f : B^3 -> B is tantamount to the logical
function that is suggested by the phrase "just 1 of 3 is false".
Table 5.3 is the truth table for the pertinent logical function,
this time using 0 for false and 1 for true in the customary way.

Table 5.3. Existential Reading of "(x, y, z)"
o-----------o-----------o-----------o-----------o
| ` ` x ` ` | ` ` y ` ` | ` ` z ` ` | (x, y, z) |
o-----------o-----------o-----------o-----------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` |
| ` ` 0 ` ` ` ` ` 0 ` ` ` ` ` 0 ` ` | ` ` 0 ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` |
| ` ` 0 ` ` ` ` ` 0 ` ` ` ` ` 1 ` ` | ` ` 0 ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` |
| ` ` 0 ` ` ` ` ` 1 ` ` ` ` ` 0 ` ` | ` ` 0 ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` |
| ` ` 0 ` ` ` ` ` 1 ` ` ` ` ` 1 ` ` | ` ` 1 ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` |
| ` ` 1 ` ` ` ` ` 0 ` ` ` ` ` 0 ` ` | ` ` 0 ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` |
| ` ` 1 ` ` ` ` ` 0 ` ` ` ` ` 1 ` ` | ` ` 1 ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` |
| ` ` 1 ` ` ` ` ` 1 ` ` ` ` ` 0 ` ` | ` ` 1 ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` |
| ` ` 1 ` ` ` ` ` 1 ` ` ` ` ` 1 ` ` | ` ` 0 ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` |
o-----------------------------------o-----------o

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 6

I sometimes refer to the cactus lobe operators in the series
(), (x_1), (x_1, x_2), (x_1, x_2, x_3), ..., (x_1, ..., x_k)
as "boundary operators" and one of the reasons for this can
be seen most easily in the venn diagram for the k-argument
boundary operator (x_1, ..., x_k). Figure 6.1 shows the
venn diagram for the 3-fold boundary form "(x, y, z)".

o-----------------------------------------------------------o
| U ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o-------------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` X ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o--o----------o ` o----------o--o ` ` ` ` ` ` |
| ` ` ` ` ` `/` ` \%%%%%%%%%%\`/%%%%%%%%%%/ ` `\` ` ` ` ` ` |
| ` ` ` ` ` / ` ` `\%%%%%%%%%%o%%%%%%%%%%/` ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` ` ` ` \%%%%%%%%/ \%%%%%%%%/ ` ` ` `\` ` ` ` ` |
| ` ` ` ` / ` ` ` ` `\%%%%%%/ ` \%%%%%%/` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` \%%%%/ ` ` \%%%%/ ` ` ` ` ` `\` ` ` ` |
| ` ` ` o ` ` ` ` ` ` `o--o-------o--o` ` ` ` ` ` ` o ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` `Y` ` ` ` |%%%%%%%| ` ` ` `Z` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` o ` ` ` ` ` ` ` ` o%%%%%%%o ` ` ` ` ` ` ` ` o ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` `\%%%%%/` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` ` ` \%%%/ ` ` ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` `\%/` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` `/` ` ` ` ` ` |
| ` ` ` ` ` ` o-------------o ` o-------------o ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 6.1. Venn Diagram for the Boundary Operator (x, y, z)

In this picture, the "oval" (actually, octangular) regions that
are customarily said to be "indicated" by the basic propositions
x, y, z : B^3 -> B, that is, where the simple arguments x, y, z,
respectively, evaluate to true, are marked with the corresponding
capital letters X, Y, Z, respectively. The proposition (x, y, z)
comes out true in the region that is shaded with per cent signs.
Invoking various idioms of general usage, one may refer to this
region as the indicated region, truth set, or fibre of truth
of the proposition in question.

It is useful to consider the truth set of the proposition (x, y, z)
in relation to the logical conjunction xyz of its arguments x, y, z.

In relation to the central cell indicated by the conjunction xyz,
the region indicated by "(x, y, z)" is composed of the "adjacent"
or the "bordering" cells. Thus they are the cells that are just
across the boundary of the center cell, arrived at by taking all
of Leibniz's "minimal changes" from the given point of departure.

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 7

Any cell in a venn diagram has a well-defined set of nearest neighbors,
and so we can apply a boundary operator of the appropriate rank to the
list of signed features that, conjoined, indicate the cell in question.

For example, having computed the "boundary", or what is more properly
called the "point omitted neighborhood" (PON) of the center cell in a
3-dimensional universe of discourse, what is the PON of the cell that
is furthest from it, namely, the origin cell indicated by "(x)(y)(z)"?

The region bordering the origin cell, (x)(y)(z), can be computed by placing
its three signed conjuncts in a 3-place bracket like (__, __, __), arriving
at the cactus expression that is shown below in both graph and string forms.

o-----------------------------------------------------------o
| U ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o-------------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/%%%%%%%%%%%%%%%%%\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/%%%%%%%%%%%%%%%%%%%%%\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o%%%%%%%%%%%%%%%%%%%%%%%%%o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%% X %%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o--o----------o%%%o----------o--o ` ` ` ` ` ` |
| ` ` ` ` ` `/%%%%\ ` ` ` ` `\%/` ` ` ` ` /%%%%\` ` ` ` ` ` |
| ` ` ` ` ` /%%%%%%\` ` ` ` ` o ` ` ` ` `/%%%%%%\ ` ` ` ` ` |
| ` ` ` ` `/%%%%%%%%\ ` ` ` `/`\` ` ` ` /%%%%%%%%\` ` ` ` ` |
| ` ` ` ` /%%%%%%%%%%\` ` ` / ` \ ` ` `/%%%%%%%%%%\ ` ` ` ` |
| ` ` ` `/%%%%%%%%%%%%\ ` `/` ` `\` ` /%%%%%%%%%%%%\` ` ` ` |
| ` ` ` o%%%%%%%%%%%%%%o--o-------o--o%%%%%%%%%%%%%%o ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%% Y %%%%%%%| ` ` ` |%%%%%%% Z %%%%%%%| ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` o%%%%%%%%%%%%%%%%%o ` ` ` o%%%%%%%%%%%%%%%%%o ` ` ` |
| ` ` ` `\%%%%%%%%%%%%%%%%%\` ` `/%%%%%%%%%%%%%%%%%/` ` ` ` |
| ` ` ` ` \%%%%%%%%%%%%%%%%%\ ` /%%%%%%%%%%%%%%%%%/ ` ` ` ` |
| ` ` ` ` `\%%%%%%%%%%%%%%%%%\`/%%%%%%%%%%%%%%%%%/` ` ` ` ` |
| ` ` ` ` ` \%%%%%%%%%%%%%%%%%o%%%%%%%%%%%%%%%%%/ ` ` ` ` ` |
| ` ` ` ` ` `\%%%%%%%%%%%%%%%/ \%%%%%%%%%%%%%%%/` ` ` ` ` ` |
| ` ` ` ` ` ` o-------------o ` o-------------o ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 7.1. Cactus Graph for the Cactus Form ((x),(y),(z))

Figure 7.2 shows the venn diagram of this expression, whose meaning
is adequately suggested by the phrase "just one of x, y, z is true".

o-----------------------------------------------------------o
| U ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o-------------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/%%%%%%%%%%%%%%%%%\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/%%%%%%%%%%%%%%%%%%%%%\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o%%%%%%%%%%%%%%%%%%%%%%%%%o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%% X %%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o--o----------o%%%o----------o--o ` ` ` ` ` ` |
| ` ` ` ` ` `/%%%%\ ` ` ` ` `\%/` ` ` ` ` /%%%%\` ` ` ` ` ` |
| ` ` ` ` ` /%%%%%%\` ` ` ` ` o ` ` ` ` `/%%%%%%\ ` ` ` ` ` |
| ` ` ` ` `/%%%%%%%%\ ` ` ` `/`\` ` ` ` /%%%%%%%%\` ` ` ` ` |
| ` ` ` ` /%%%%%%%%%%\` ` ` / ` \ ` ` `/%%%%%%%%%%\ ` ` ` ` |
| ` ` ` `/%%%%%%%%%%%%\ ` `/` ` `\` ` /%%%%%%%%%%%%\` ` ` ` |
| ` ` ` o%%%%%%%%%%%%%%o--o-------o--o%%%%%%%%%%%%%%o ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%% Y %%%%%%%| ` ` ` |%%%%%% Z %%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` o%%%%%%%%%%%%%%%%%o ` ` ` o%%%%%%%%%%%%%%%%%o ` ` ` |
| ` ` ` `\%%%%%%%%%%%%%%%%%\` ` `/%%%%%%%%%%%%%%%%%/` ` ` ` |
| ` ` ` ` \%%%%%%%%%%%%%%%%%\ ` /%%%%%%%%%%%%%%%%%/ ` ` ` ` |
| ` ` ` ` `\%%%%%%%%%%%%%%%%%\`/%%%%%%%%%%%%%%%%%/` ` ` ` ` |
| ` ` ` ` ` \%%%%%%%%%%%%%%%%%o%%%%%%%%%%%%%%%%%/ ` ` ` ` ` |
| ` ` ` ` ` `\%%%%%%%%%%%%%%%/ \%%%%%%%%%%%%%%%/` ` ` ` ` ` |
| ` ` ` ` ` ` o-------------o ` o-------------o ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 7.2. Venn Diagram for the Cactus Form ((x),(y),(z))

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 8

Given the foregoing explanation of the k-fold boundary operator,
along with its use to express such forms of logical constraints
as "just 1 of k is false" and "just 1 of k is true", there will
be no trouble interpreting an expression of the following shape
from the Jets and Sharks example:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( art ` `),( al ` ),( sam `),( clyde ),( mike `),` ` ` |
| ` `( jim ` `),( greg ),( john ),( doug `),( lance ),` ` ` |
| ` `( george ),( pete ),( fred ),( gene `),( ralph ),` ` ` |
| ` `( phil ` ),( ike `),( nick ),( don ` ),( ned ` ),` ` ` |
| ` `( karl ` ),( ken `),( earl ),( rick `),( ol ` `),` ` ` |
| ` `( neal ` ),( dave )) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

This expression says that everything in the universe of discourse
is either Art, or Al, or ..., or Neal, or Dave, but never any two
of them at once. In effect, I've exploited the circumstance that
the universe contains but finitely many ostensible individuals to
dedicate its own predicate to each one of them, imposing only the
requirement that these predicates must be disjoint and exhaustive.

Likewise, each of the following clauses has the effect of
partitioning the universe of discourse among the factions
or features that are enumerated in the clause in question.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( jets , sharks ) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( 20's ),( 30's ),( 40's ))` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( junior_high ),( high_school ),( college )) ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( single ),( married ),( divorced )) ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (( bookie ),( burglar ),( pusher )) ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

We may note in passing that ( x , y ) = ((x),(y)),
but a rule of this form holds only in the case of
the 2-fold boundary operator.

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 9

Let's collect the various ways of representing the structure
of a universe of discourse that is described by the following
cactus formalisms, verbalized as "just 1 of x, y, z is true".

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `x` y `z` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o` o `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `|` | `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o--o--o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ((x),(y),(z)) ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 9.1. Cactus Graph for the Cactus Form ((x),(y),(z))

Table 9.2 shows the truth table for the existential
interpretation of the cactus formula ((x),(y),(z)).

Table 9.2. Existential Interpretation of ((x),(y),(z))
o-----------o-----------o-----------o-------------------o
| ` ` x ` ` | ` ` y ` ` | ` ` z ` ` | ` ((x),(y),(z)) ` |
o-----------o-----------o-----------o-------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` 0 ` ` ` ` ` 0 ` ` ` ` ` 0 ` ` | ` ` ` ` 0 ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` 0 ` ` ` ` ` 0 ` ` ` ` ` 1 ` ` | ` ` ` ` 1 ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` 0 ` ` ` ` ` 1 ` ` ` ` ` 0 ` ` | ` ` ` ` 1 ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` 0 ` ` ` ` ` 1 ` ` ` ` ` 1 ` ` | ` ` ` ` 0 ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` 1 ` ` ` ` ` 0 ` ` ` ` ` 0 ` ` | ` ` ` ` 1 ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` 1 ` ` ` ` ` 0 ` ` ` ` ` 1 ` ` | ` ` ` ` 0 ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` 1 ` ` ` ` ` 1 ` ` ` ` ` 0 ` ` | ` ` ` ` 0 ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` 1 ` ` ` ` ` 1 ` ` ` ` ` 1 ` ` | ` ` ` ` 0 ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-----------------------------------o-------------------o

Figure 9.3 shows the same data as a 2-colored 3-cube, coloring
a node with a zero (0) for "false" versus a one (1) for "true".

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `x `y `z ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` 0 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `/|\ ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` / | \ ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `/ `| `\ ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` / ` | ` \ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `/ ` `| ` `\ ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` / ` ` | ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/ ` x (y) z ` \ ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` `x `y (z) 0 ` ` ` 0 ` ` ` 0 (x) y `z ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` |\ ` ` / \ ` ` /| ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` | \ ` / ` \ ` / | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` | `\ / ` ` \ / `| ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` | ` \ ` ` ` / ` | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` | `/ \ ` ` / \ `| ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` | / ` \ ` / ` \ | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` |/ ` ` \ / ` ` \| ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `x (y)(z) 1 ` ` ` 1 ` ` ` 1 (x)(y) z ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` `\ `(x) y (z) `/ ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` \ ` ` | ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\ ` `| ` `/ ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` | ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\ `| `/ ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\|/ ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` 0 ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` (x)(y)(z) ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 9.3. Colored Cube Rendition of ((x),(y),(z))

Figure 9.4 repeats the venn diagram that we've already seen.

o-----------------------------------------------------------o
| U ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o-------------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/%%%%%%%%%%%%%%%%%\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/%%%%%%%%%%%%%%%%%%%%%\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o%%%%%%%%%%%%%%%%%%%%%%%%%o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%% X %%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o--o----------o%%%o----------o--o ` ` ` ` ` ` |
| ` ` ` ` ` `/%%%%\ ` ` ` ` `\%/` ` ` ` ` /%%%%\` ` ` ` ` ` |
| ` ` ` ` ` /%%%%%%\` ` ` ` ` o ` ` ` ` `/%%%%%%\ ` ` ` ` ` |
| ` ` ` ` `/%%%%%%%%\ ` ` ` `/`\` ` ` ` /%%%%%%%%\` ` ` ` ` |
| ` ` ` ` /%%%%%%%%%%\` ` ` / ` \ ` ` `/%%%%%%%%%%\ ` ` ` ` |
| ` ` ` `/%%%%%%%%%%%%\ ` `/` ` `\` ` /%%%%%%%%%%%%\` ` ` ` |
| ` ` ` o%%%%%%%%%%%%%%o--o-------o--o%%%%%%%%%%%%%%o ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` |%%%%%%% Y %%%%%%%| ` ` ` |%%%%%%% Z %%%%%%%| ` ` ` |
| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |%%%%%%%%%%%%%%%%%| ` ` ` |
| ` ` ` o%%%%%%%%%%%%%%%%%o ` ` ` o%%%%%%%%%%%%%%%%%o ` ` ` |
| ` ` ` `\%%%%%%%%%%%%%%%%%\` ` `/%%%%%%%%%%%%%%%%%/` ` ` ` |
| ` ` ` ` \%%%%%%%%%%%%%%%%%\ ` /%%%%%%%%%%%%%%%%%/ ` ` ` ` |
| ` ` ` ` `\%%%%%%%%%%%%%%%%%\`/%%%%%%%%%%%%%%%%%/` ` ` ` ` |
| ` ` ` ` ` \%%%%%%%%%%%%%%%%%o%%%%%%%%%%%%%%%%%/ ` ` ` ` ` |
| ` ` ` ` ` `\%%%%%%%%%%%%%%%/ \%%%%%%%%%%%%%%%/` ` ` ` ` ` |
| ` ` ` ` ` ` o-------------o ` o-------------o ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 9.4. Venn Diagram for the Cactus Form ((x),(y),(z))

Figure 9.5 is an alternative form of venn diagram for the same
proposition, where we collapse to a nullity all of the regions
on which the proposition in question evaluates to false. This
leaves a structure that partitions the universe into precisely
three parts. In mathematics, operations that identify diverse
elements are called "quotient operations". In this case, many
regions of the universe are being identified with the null set,
leaving only this 3-fold partition as the "quotient structure".

o-----------------------------------------------------------o
| \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / |
| ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` |
| ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` |
| ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` ` ` ` ` X ` ` ` ` ` ` ` ` / ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `Y` ` ` ` ` ` ` | ` ` ` ` ` ` `Z` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------o-----------------------------o
Figure 9.5. Quotient Structure Venn Diagram for ((x),(y),(z))

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 10

Let's now look at the last type of clause that we find in my
transcription of the Jets and Sharks data base, for instance,
as exemplified by the following couple of lobal expressions:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( jets , ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` `( art ` `),( al ` ),( sam `),( clyde ),( mike `),` ` ` |
| ` `( jim ` `),( greg ),( john ),( doug `),( lance ),` ` ` |
| ` `( george ),( pete ),( fred ),( gene `),( ralph ))` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( sharks ,` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `( phil ),( ike `),( nick ),( don ),( ned `),( karl ),` |
| ` `( ken `),( earl ),( rick ),( ol `),( neal ),( dave ))` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Each of these clauses exhibits a generic pattern whose logical properties
may be studied well enough in the form of the following schematic example.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` y `z` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `x` | `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o--o--o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ( x ,(y),(z)) ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 10.1. Cactus Graph for the Cactus Form ( x ,(y),(z))

The proposition (u, v, w) evaluates to true
if and only if just one of u, v, w is false.
In the same way, the proposition (x,(y),(z))
evaluates to true if and only if exactly one
of x, (y), (z) is false. Taking it by cases,
let us first suppose that x is true. Then it
has to be that just one of (y) or (z) is false,
which is tantamount to the proposition ((y),(z)),
which is equivalent to the proposition ( y , z ).
On the other hand, let us suppose that x is the
false one. Then both (y) and (z) must be true,
which is to say that y is false and z is false.

What we have just said here is that the region
where x is true is partitioned into the regions
where y and z are true, respectively, while the
region where x is false has both y and z false.
In other words, we have a "pie-chart" structure,
where the genus X is divided into the disjoint
and X-haustive couple of species Y and Z.

The same analysis applies to the generic form
(x, (x_1), ..., (x_k)), specifying a pie-chart
with a genus X and the k species X_1, ..., X_k.

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

JAS. Note 11

A cactus expression of the form (x, (x_1), ..., (x_k))
commonly arises when we need to describe a universe of
discourse where a genus X is partitioned into disjoint
and exhaustive species X_1, ..., X_k. For example, the
cactus form ( x ,(y),(z)) shown in Figure 11.1 gives the
ordinary venn diagram and quotient venn diagram that are
shown in Figures 11.2 and 11.3, respectively.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` y `z` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `x` | `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o--o--o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ( x ,(y),(z)) ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 11.1. Cactus Graph for the Cactus Form ( x ,(y),(z))

o-----------------------------------------------------------o
|% U %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%%%%%%%o-------------o%%%%%%%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%%%%%%/ ` ` ` ` ` ` ` \%%%%%%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%%%%%/` ` ` ` ` ` ` ` `\%%%%%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%%%%/ ` ` ` ` ` ` ` ` ` \%%%%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%%%/` ` ` ` ` ` ` ` ` ` `\%%%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%%/ ` ` ` ` ` ` ` ` ` ` ` \%%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%o` ` ` ` ` ` ` ` ` ` ` ` `o%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%|` ` ` ` ` ` X ` ` ` ` ` `|%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%|` ` ` ` ` ` ` ` ` ` ` ` `|%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%o--o----------o ` o----------o--o%%%%%%%%%%%%%|
|%%%%%%%%%%%%/` ` \%%%%%%%%%%\`/%%%%%%%%%%/ ` `\%%%%%%%%%%%%|
|%%%%%%%%%%%/ ` ` `\%%%%%%%%%%o%%%%%%%%%%/` ` ` \%%%%%%%%%%%|
|%%%%%%%%%%/` ` ` ` \%%%%%%%%/`\%%%%%%%%/ ` ` ` `\%%%%%%%%%%|
|%%%%%%%%%/ ` ` ` ` `\%%%%%%/ ` \%%%%%%/` ` ` ` ` \%%%%%%%%%|
|%%%%%%%%/` ` ` ` ` ` \%%%%/` ` `\%%%%/ ` ` ` ` ` `\%%%%%%%%|
|%%%%%%%o ` ` ` ` ` ` `o--o-------o--o` ` ` ` ` ` ` o%%%%%%%|
|%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%|
|%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%|
|%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%|
|%%%%%%%| ` ` ` `Y` ` ` ` | ` ` ` | ` ` ` `Z` ` ` ` |%%%%%%%|
|%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%|
|%%%%%%%o ` ` ` ` ` ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` o%%%%%%%|
|%%%%%%%%\` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` `/%%%%%%%%|
|%%%%%%%%%\ ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` /%%%%%%%%%|
|%%%%%%%%%%\` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` `/%%%%%%%%%%|
|%%%%%%%%%%%\ ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` /%%%%%%%%%%%|
|%%%%%%%%%%%%\` ` ` ` ` ` ` `/%\` ` ` ` ` ` ` `/%%%%%%%%%%%%|
|%%%%%%%%%%%%%o-------------o%%%o-------------o%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
|%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
o-----------------------------------------------------------o
Figure 11.2. Venn Diagram for the Cactus Form ( x ,(y),(z))

o-----------------------------------------------------------o
| U ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o------o------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` | ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` X ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` ` ` ` | ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` ` ` ` | ` ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` ` ` ` | ` ` ` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` | ` ` ` ` ` `o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` | ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` | ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` `Y` ` ` | ` ` `Z` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` | ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` | ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` | ` ` ` ` ` `o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` | ` ` ` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` | ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` | ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` ` | ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` ` | ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o------o------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 11.3. Quotient Venn Diagram for the Form ( x ,(y),(z))

Jon Awbrey

---------------------------------------------------------------------------------




Forum Sponsored by Wolfram Research

© 2004-2009 Wolfram Research, Inc. | Powered by vBulletin 2.3.0 © 2000-2002 Jelsoft Enterprises, Ltd. | Disclaimer
vB Easy Archive Final - Created by Xenon and modified/released by SkuZZy from the Job Openings