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import [notes found in a cactus patch]
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{{DISPLAYTITLE:Cactus Language}}
 
{{DISPLAYTITLE:Cactus Language}}
 +
 +
==Inquiry Driven Systems 1.3.10.8 – 1.3.10.13==
 +
 
<pre>
 
<pre>
 
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Line 3,153: Line 3,156:     
Which was to be shown.
 
Which was to be shown.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
</pre>
 +
 +
==Notes Found in a Cactus Patch==
 +
 +
<pre>
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
IDS, NKS -- CL
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
CL.  Cactus Language
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
CL.  Note 1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Table 13 illustrates the "existential interpretation"
 +
of cactus graphs and cactus expressions by providing
 +
English translations for a few of the most basic and
 +
commonly occurring forms.
 +
 +
Even though I do most of my thinking in the existential interpretation,
 +
I will continue to speak of these forms as "logical graphs", because
 +
I think it is an important fact about them that the formal validity
 +
of the axioms and theorems is not dependent on the choice between
 +
the entitative and the existential interpretations.
 +
 +
The first extension is the "reflective extension of logical graphs" (RefLog).
 +
It is obtained by generalizing the negation operator "(_)" in a certain way,
 +
calling "(_)" the "controlled", "moderated", or "reflective" negation operator
 +
of order 1, then adding another such operator for each finite k = 2, 3, ... .
 +
In sum, these operators are symbolized by bracketed argument lists as follows:
 +
"(_)", "(_,_)", "(_,_,_)", ..., where the number of slots is the order of the
 +
reflective negation operator in question.
 +
             
 +
The cactus graph and the cactus expression
 +
shown here are both described as a "spike".
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                  o                  |
 +
|                  |                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|                  ( )                  |
 +
o---------------------------------------o
 +
 +
The rule of reduction for a lobe is:
 +
 +
    x_1  x_2  ...  x_k
 +
    o-----o--- ... ---o
 +
      \              /
 +
      \            /
 +
        \          /
 +
        \        /
 +
          \      /
 +
          \    /
 +
            \  /
 +
            \ /
 +
              @            =      @
 +
 +
if and only if exactly one of the x_j is a spike.
 +
 +
In Ref Log, an expression of the form "(( e_1 ),( e_2 ),( ... ),( e_k ))"
 +
expresses the fact that "exactly one of the e_j is true, for j = 1 to k".
 +
Expressions of this form are called "universal partition" expressions, and
 +
they parse into a type of graph called a "painted and rooted cactus" (PARC):
 +
 +
    e_1  e_2  ...  e_k
 +
    o    o          o
 +
    |    |          |
 +
    o-----o--- ... ---o
 +
      \              /
 +
      \            /
 +
        \          /
 +
        \        /
 +
          \      /
 +
          \    /
 +
            \  /
 +
            \ /
 +
              @
 +
 +
 +
| ( x1, x2, ..., xk )  =  [blank]
 +
|
 +
| iff
 +
|
 +
| Just one of the arguments x1, x2, ..., xk  =  ()
 +
 +
The interpretation of these operators, read as assertions
 +
about the values of their listed arguments, is as follows:
 +
 +
1.  Existential Interpretation:  "Just one of the k argument is false."
 +
2.  Entitative  Interpretation:  "Not just one of the k arguments is true."
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
o-------------------o-------------------o-------------------o
 +
|      Graph      |      String      |    Translation    |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        @        |        " "        |      true.      |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |                  |
 +
|        @        |        ( )        |      untrue.      |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        r        |                  |                  |
 +
|        @        |        r        |        r.        |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        r        |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |                  |
 +
|        @        |        (r)        |      not r.      |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      r s t      |                  |                  |
 +
|        @        |      r s t      |  r and s and t.  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      r s t      |                  |                  |
 +
|      o o o      |                  |                  |
 +
|        \|/        |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |                  |
 +
|        @        |    ((r)(s)(t))    |    r or s or t.  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |    r implies s.  |
 +
|        r  s    |                  |                  |
 +
|        o---o    |                  |    if r then s.  |
 +
|        |        |                  |                  |
 +
|        @        |      (r (s))      |    no r sans s.  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      r  s      |                  |                  |
 +
|      o---o      |                  | r exclusive-or s. |
 +
|        \ /        |                  |                  |
 +
|        @        |      (r , s)      | r not equal to s. |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      r  s      |                  |                  |
 +
|      o---o      |                  |                  |
 +
|        \ /        |                  |                  |
 +
|        o        |                  | r if & only if s. |
 +
|        |        |                  |                  |
 +
|        @        |    ((r , s))    | r equates with s. |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      r  s  t      |                  |                  |
 +
|      o--o--o      |                  |                  |
 +
|      \  /      |                  |                  |
 +
|        \ /        |                  |  just one false  |
 +
|        @        |    (r , s , t)    |  out of r, s, t.  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      r  s  t      |                  |                  |
 +
|      o  o  o      |                  |                  |
 +
|      |  |  |      |                  |                  |
 +
|      o--o--o      |                  |                  |
 +
|      \  /      |                  |                  |
 +
|        \ /        |                  |  just one true  |
 +
|        @        |  ((r),(s),(t))  |  among r, s, t.  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |  genus t over    |
 +
|        r  s      |                  |  species r, s.  |
 +
|        o  o      |                  |                  |
 +
|      t  |  |      |                  |  partition t    |
 +
|      o--o--o      |                  |  among r & s.    |
 +
|      \  /      |                  |                  |
 +
|        \ /        |                  |  whole pie t:    |
 +
|        @        |  ( t ,(r),(s))  |  slices r, s.    |
 +
o-------------------o-------------------o-------------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Table 13.  The Existential Interpretation
 +
o-------------------o-------------------o-------------------o
 +
|  Cactus Graph    | Cactus Expression |    Existential    |
 +
|                  |                  |  Interpretation  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        @        |        " "        |      true.      |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |                  |
 +
|        @        |        ( )        |      untrue.      |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        a        |                  |                  |
 +
|        @        |        a        |        a.        |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        a        |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |                  |
 +
|        @        |        (a)        |      not a.      |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a b c      |                  |                  |
 +
|        @        |      a b c      |  a and b and c.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a b c      |                  |                  |
 +
|      o o o      |                  |                  |
 +
|        \|/        |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |                  |
 +
|        @        |    ((a)(b)(c))    |    a or b or c.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|                  |                  |    a implies b.  |
 +
|        a  b    |                  |                  |
 +
|        o---o    |                  |    if a then b.  |
 +
|        |        |                  |                  |
 +
|        @        |      (a (b))      |    no a sans b.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a  b      |                  |                  |
 +
|      o---o      |                  | a exclusive-or b. |
 +
|        \ /        |                  |                  |
 +
|        @        |      (a , b)      | a not equal to b. |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a  b      |                  |                  |
 +
|      o---o      |                  |                  |
 +
|        \ /        |                  |                  |
 +
|        o        |                  | a if & only if b. |
 +
|        |        |                  |                  |
 +
|        @        |    ((a , b))    | a equates with b. |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a  b  c      |                  |                  |
 +
|      o--o--o      |                  |                  |
 +
|      \  /      |                  |                  |
 +
|        \ /        |                  |  just one false  |
 +
|        @        |    (a , b , c)    |  out of a, b, c.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a  b  c      |                  |                  |
 +
|      o  o  o      |                  |                  |
 +
|      |  |  |      |                  |                  |
 +
|      o--o--o      |                  |                  |
 +
|      \  /      |                  |                  |
 +
|        \ /        |                  |  just one true  |
 +
|        @        |  ((a),(b),(c))  |  among a, b, c.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|                  |                  |  genus a over    |
 +
|        b  c      |                  |  species b, c.  |
 +
|        o  o      |                  |                  |
 +
|      a  |  |      |                  |  partition a    |
 +
|      o--o--o      |                  |  among b & c.    |
 +
|      \  /      |                  |                  |
 +
|        \ /        |                  |  whole pie a:    |
 +
|        @        |  ( a ,(b),(c))  |  slices b, c.    |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Table 14.  The Entitative Interpretation
 +
o-------------------o-------------------o-------------------o
 +
|  Cactus Graph    | Cactus Expression |    Entitative    |
 +
|                  |                  |  Interpretation  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        @        |        " "        |      untrue.      |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |                  |
 +
|        @        |        ( )        |      true.      |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        a        |                  |                  |
 +
|        @        |        a        |        a.        |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|        a        |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |                  |
 +
|        @        |        (a)        |      not a.      |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a b c      |                  |                  |
 +
|        @        |      a b c      |    a or b or c.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a b c      |                  |                  |
 +
|      o o o      |                  |                  |
 +
|        \|/        |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |                  |
 +
|        @        |    ((a)(b)(c))    |  a and b and c.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|                  |                  |    a implies b.  |
 +
|                  |                  |                  |
 +
|        o a      |                  |    if a then b.  |
 +
|        |        |                  |                  |
 +
|        @ b      |      (a) b        |    not a, or b.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a  b      |                  |                  |
 +
|      o---o      |                  | a if & only if b. |
 +
|        \ /        |                  |                  |
 +
|        @        |      (a , b)      | a equates with b. |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a  b      |                  |                  |
 +
|      o---o      |                  |                  |
 +
|        \ /        |                  |                  |
 +
|        o        |                  | a exclusive-or b. |
 +
|        |        |                  |                  |
 +
|        @        |    ((a , b))    | a not equal to b. |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a  b  c      |                  |                  |
 +
|      o--o--o      |                  |                  |
 +
|      \  /      |                  |                  |
 +
|        \ /        |                  | not just one true |
 +
|        @        |    (a , b , c)    | out of a, b, c.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a  b  c      |                  |                  |
 +
|      o--o--o      |                  |                  |
 +
|      \  /      |                  |                  |
 +
|        \ /        |                  |                  |
 +
|        o        |                  |                  |
 +
|        |        |                  |  just one true  |
 +
|        @        |  ((a , b , c))  |  among a, b, c.  |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
|                  |                  |                  |
 +
|      a            |                  |                  |
 +
|      o            |                  |  genus a over    |
 +
|      |  b  c      |                  |  species b, c.  |
 +
|      o--o--o      |                  |                  |
 +
|      \  /      |                  |  partition a    |
 +
|        \ /        |                  |  among b & c.    |
 +
|        o        |                  |                  |
 +
|        |        |                  |  whole pie a:    |
 +
|        @        |  ( a ,(b),(c))  |  slices b, c.    |
 +
|                  |                  |                  |
 +
o-------------------o-------------------o-------------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|      Graph      |    String      |  Entitative    |  Existential  |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|        @        |      " "      |    untrue.    |      true.      |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|        o        |                |                |                |
 +
|        |        |                |                |                |
 +
|        @        |      ( )      |      true.      |    untrue.    |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|        r        |                |                |                |
 +
|        @        |        r        |        r.      |        r.      |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|        r        |                |                |                |
 +
|        o        |                |                |                |
 +
|        |        |                |                |                |
 +
|        @        |      (r)      |      not r.    |      not r.    |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|      r s t      |                |                |                |
 +
|        @        |      r s t      |  r or s or t.  |  r and s and t. |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|      r s t      |                |                |                |
 +
|      o o o      |                |                |                |
 +
|      \|/      |                |                |                |
 +
|        o        |                |                |                |
 +
|        |        |                |                |                |
 +
|        @        |  ((r)(s)(t))  |  r and s and t. |  r or s or t.  |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |  r implies s.  |
 +
|                |                |                |                |
 +
|        o r      |                |                |  if r then s.  |
 +
|        |        |                |                |                |
 +
|        @ s      |      (r) s      |  not r, or s    |  no r sans s.  |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |  r implies s.  |
 +
|        r  s    |                |                |                |
 +
|        o---o    |                |                |  if r then s.  |
 +
|        |        |                |                |                |
 +
|        @        |    (r (s))    |                |  no r sans s.  |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|      r  s      |                |                |                |
 +
|      o---o      |                |                |r exclusive-or s.|
 +
|      \ /      |                |                |                |
 +
|        @        |    (r , s)    |                |r not equal to s.|
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|      r  s      |                |                |                |
 +
|      o---o      |                |                |                |
 +
|      \ /      |                |                |                |
 +
|        o        |                |                |r if & only if s.|
 +
|        |        |                |                |                |
 +
|        @        |    ((r , s))    |                |r equates with s.|
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|    r  s  t    |                |                |                |
 +
|    o--o--o    |                |                |                |
 +
|      \  /      |                |                |                |
 +
|      \ /      |                |                | just one false  |
 +
|        @        |  (r , s , t)  |                | out of r, s, t. |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |                |
 +
|    r  s  t    |                |                |                |
 +
|    o  o  o    |                |                |                |
 +
|    |  |  |    |                |                |                |
 +
|    o--o--o    |                |                |                |
 +
|      \  /      |                |                |                |
 +
|      \ /      |                |                |  just one true  |
 +
|        @        |  ((r),(s),(t))  |                |  among r, s, t. |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
|                |                |                |  genus t over  |
 +
|        r  s    |                |                |  species r, s.  |
 +
|        o  o    |                |                |                |
 +
|    t  |  |    |                |                |  partition t    |
 +
|    o--o--o    |                |                |  among r & s.  |
 +
|      \  /      |                |                |                |
 +
|      \ /      |                |                |  whole pie t:  |
 +
|        @        |  ( t ,(r),(s))  |                |  slices r, s.  |
 +
o-----------------o-----------------o-----------------o-----------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Differential Logic
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
One of the first things that you can do, once you
 +
have a really decent calculus for boolean functions
 +
or propositional logic, whatever you want to call it,
 +
is to compute the differentials of these functions or
 +
propositions.
 +
 +
Now there are many ways to dance around this idea,
 +
and I feel like I have tried them all, before one
 +
gets down to acting on it, and there many issues
 +
of interpretation and justification that we will
 +
have to clear up after the fact, that is, before
 +
we can be sure that it all really makes any sense,
 +
but I think this time I'll just jump in, and show
 +
you the form in which this idea first came to me.
 +
 +
Start with a proposition of the form x & y, which
 +
I graph as two labels attached to a root node, so:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                  x y                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|                x and y                |
 +
o---------------------------------------o
 +
 +
Written as a string, this is just the concatenation "x y".
 +
 +
The proposition xy may be taken as a boolean function f(x, y)
 +
having the abstract type f : B x B -> B, where B = {0, 1} is
 +
read in such a way that 0 means "false" and 1 means "true".
 +
 +
In this style of graphical representation,
 +
the value "true" looks like a blank label
 +
and the value "false" looks like an edge.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                                      |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|                true                  |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                  o                  |
 +
|                  |                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|                false                |
 +
o---------------------------------------o
 +
 +
Back to the proposition xy.  Imagine yourself standing
 +
in a fixed cell of the corresponding venn diagram, say,
 +
the cell where the proposition xy is true, as pictured:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                o    o                |
 +
|              / \  / \              |
 +
|              /  \ /  \              |
 +
|            /    ·    \            |
 +
|            /    /%\    \            |
 +
|          /    /%%%\    \          |
 +
|          /    /%%%%%\    \          |
 +
|        /    /%%%%%%%\    \        |
 +
|        /    /%%%%%%%%%\    \        |
 +
|      o  x  o%%%%%%%%%%%o  y  o      |
 +
|        \    \%%%%%%%%%/    /        |
 +
|        \    \%%%%%%%/    /        |
 +
|          \    \%%%%%/    /          |
 +
|          \    \%%%/    /          |
 +
|            \    \%/    /            |
 +
|            \    ·    /            |
 +
|              \  / \  /              |
 +
|              \ /  \ /              |
 +
|                o    o                |
 +
|                                      |
 +
o---------------------------------------o
 +
 +
Now ask yourself:  What is the value of the
 +
proposition xy at a distance of dx and dy
 +
from the cell xy where you are standing?
 +
 +
Don't think about it -- just compute:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|              dx o  o dy              |
 +
|                / \ / \                |
 +
|            x o---@---o y            |
 +
|                                      |
 +
o---------------------------------------o
 +
|        (x + dx) and (y + dy)        |
 +
o---------------------------------------o
 +
 +
To make future graphs easier to draw in Ascii land,
 +
I will use devices like @=@=@ and o=o=o to identify
 +
several nodes into one, as in this next redrawing:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|              x  dx y  dy              |
 +
|              o---o o---o              |
 +
|              \  | |  /              |
 +
|                \ | | /                |
 +
|                \| |/                |
 +
|                  @=@                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|        (x + dx) and (y + dy)        |
 +
o---------------------------------------o
 +
 +
However you draw it, these expressions follow because the
 +
expression x + dx, where the plus sign indicates (mod 2)
 +
addition in B, and thus corresponds to an exclusive-or
 +
in logic, parses to a graph of the following form:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                x    dx                |
 +
|                o---o                |
 +
|                  \ /                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|                x + dx                |
 +
o---------------------------------------o
 +
 +
Next question:  What is the difference between
 +
the value of the proposition xy "over there" and
 +
the value of the proposition xy where you are, all
 +
expressed as general formula, of course?  Here 'tis:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|        x  dx y  dy                    |
 +
|        o---o o---o                    |
 +
|        \  | |  /                    |
 +
|          \ | | /                      |
 +
|          \| |/        x y          |
 +
|            o=o-----------o            |
 +
|            \          /            |
 +
|              \        /              |
 +
|              \      /              |
 +
|                \    /                |
 +
|                \  /                |
 +
|                  \ /                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|      ((x + dx) & (y + dy)) - xy      |
 +
o---------------------------------------o
 +
 +
Oh, I forgot to mention:  Computed over B,
 +
plus and minus are the very same operation.
 +
This will make the relationship between the
 +
differential and the integral parts of the
 +
resulting calculus slightly stranger than
 +
usual, but never mind that now.
 +
 +
Last question, for now:  What is the value of this expression
 +
from your current standpoint, that is, evaluated at the point
 +
where xy is true?  Well, substituting 1 for x and 1 for y in
 +
the graph amounts to the same thing as erasing those labels:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|          dx    dy                    |
 +
|        o---o o---o                    |
 +
|        \  | |  /                    |
 +
|          \ | | /                      |
 +
|          \| |/                      |
 +
|            o=o-----------o            |
 +
|            \          /            |
 +
|              \        /              |
 +
|              \      /              |
 +
|                \    /                |
 +
|                \  /                |
 +
|                  \ /                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|      ((1 + dx) & (1 + dy)) - 1·1      |
 +
o---------------------------------------o
 +
 +
And this is equivalent to the following graph:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                dx  dy                |
 +
|                o  o                |
 +
|                  \ /                  |
 +
|                  o                  |
 +
|                  |                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
|              dx or dy                |
 +
o---------------------------------------o
 +
 +
Have to break here -- will explain later.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We have just met with the fact that
 +
the differential of the "and" is
 +
the "or" of the differentials.
 +
 +
x and y  --Diff-->  dx or dy.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                            dx  dy  |
 +
|                              o  o    |
 +
|                              \ /    |
 +
|                                o      |
 +
|      x y                      |      |
 +
|      @      --Diff-->        @      |
 +
|                                      |
 +
o---------------------------------------o
 +
|      x y      --Diff-->  ((dx)(dy))  |
 +
o---------------------------------------o
 +
 +
It will be necessary to develop a more refined analysis of
 +
this statement directly, but that is roughly the nub of it.
 +
 +
If the form of the above statement reminds you of DeMorgan's rule,
 +
it is no accident, as differentiation and negation turn out to be
 +
closely related operations.  Indeed, one can find discussions of
 +
logical difference calculus in the Boole-DeMorgan correspondence
 +
and Peirce also made use of differential operators in a logical
 +
context, but the exploration of these ideas has been hampered
 +
by a number of factors, not the least of which being a syntax
 +
adequate to handle the complexity of expressions that evolve.
 +
 +
For my part, it was definitely a case of the calculus being smarter
 +
than the calculator thereof.  The graphical pictures were catalytic
 +
in their power over my thinking process, leading me so quickly past
 +
so many obstructions that I did not have time to think about all of
 +
the difficulties that would otherwise have inhibited the derivation.
 +
It did eventually became necessary to write all this up in a linear
 +
script, and to deal with the various problems of interpretation and
 +
justification that I could imagine, but that took another 120 pages,
 +
and so, if you don't like this intuitive approach, then let that be
 +
your sufficient notice.
 +
 +
Let us run through the initial example again, this time attempting
 +
to interpret the formulas that develop at each stage along the way.
 +
 +
We begin with a proposition or a boolean function f(x, y) = xy.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                o    o                |
 +
|              / \  / \              |
 +
|              /  \ /  \              |
 +
|            /    ·    \            |
 +
|            /    /`\    \            |
 +
|          /    /```\    \          |
 +
|          /    /`````\    \          |
 +
|        /    /```````\    \        |
 +
|        /    /`````````\    \        |
 +
|      o  x  o`````f`````o  y  o      |
 +
|        \    \`````````/    /        |
 +
|        \    \```````/    /        |
 +
|          \    \`````/    /          |
 +
|          \    \```/    /          |
 +
|            \    \`/    /            |
 +
|            \    ·    /            |
 +
|              \  / \  /              |
 +
|              \ /  \ /              |
 +
|                o    o                |
 +
|                                      |
 +
o---------------------------------------o
 +
|                                      |
 +
|                  x y                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| f =              x y                  |
 +
o---------------------------------------o
 +
 +
A function like this has an abstract type and a concrete type.
 +
The abstract type is what we invoke when we write things like
 +
f : B x B -> B or f : B^2 -> B.  The concrete type takes into
 +
account the qualitative dimensions or the "units" of the case,
 +
which can be explained as follows.
 +
 +
1.  Let X be the set of values {(x), x} = {not x, x}.
 +
 +
2.  Let Y be the set of values {(y), y} = {not y, y}.
 +
 +
Then interpret the usual propositions about x, y
 +
as functions of the concrete type f : X x Y -> B.
 +
 +
We are going to consider various "operators" on these functions.
 +
Here, an operator F is a function that takes one function f into
 +
another function Ff.
 +
 +
The first couple of operators that we need to consider are logical analogues
 +
of those that occur in the classical "finite difference calculus", namely:
 +
 +
1.  The "difference" operator [capital Delta], written here as D.
 +
 +
2.  The "enlargement" operator [capital Epsilon], written here as E.
 +
 +
These days, E is more often called the "shift" operator.
 +
 +
In order to describe the universe in which these operators operate,
 +
it will be necessary to enlarge our original universe of discourse.
 +
We mount up from the space U = X x Y to its "differential extension",
 +
EU = U x dU = X x Y x dX x dY, with dX = {(dx), dx} and dY = {(dy), dy}.
 +
The interpretations of these new symbols can be diverse, but the easiest
 +
for now is just to say that dx means "change x" and dy means "change y".
 +
To draw the differential extension EU of our present universe U = X x Y
 +
as a venn diagram, it would take us four logical dimensions X, Y, dX, dY,
 +
but we can project a suggestion of what it's about on the universe X x Y
 +
by drawing arrows that cross designated borders, labeling the arrows as
 +
dx when crossing the border between x and (x) and as dy when crossing
 +
the border between y and (y), in either direction, in either case.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                o    o                |
 +
|              / \  / \              |
 +
|              /  \ /  \              |
 +
|            /    ·    \            |
 +
|            / dy  /`\  dx \            |
 +
|          /  ^ /```\ ^  \          |
 +
|          /    \`````/    \          |
 +
|        /    /`\```/`\    \        |
 +
|        /    /```\`/```\    \        |
 +
|      o  x  o`````o`````o  y  o      |
 +
|        \    \`````````/    /        |
 +
|        \    \```````/    /        |
 +
|          \    \`````/    /          |
 +
|          \    \```/    /          |
 +
|            \    \`/    /            |
 +
|            \    ·    /            |
 +
|              \  / \  /              |
 +
|              \ /  \ /              |
 +
|                o    o                |
 +
|                                      |
 +
o---------------------------------------o
 +
 +
We can form propositions from these differential variables in the same way
 +
that we would any other logical variables, for instance, interpreting the
 +
proposition (dx (dy)) to say "dx => dy", in other words, however you wish
 +
to take it, whether indicatively or injunctively, as saying something to
 +
the effect that there is "no change in x without a change in y".
 +
 +
Given the proposition f(x, y) in U = X x Y,
 +
the (first order) 'enlargement' of f is the
 +
proposition Ef in EU that is defined by the
 +
formula Ef(x, y, dx, dy) = f(x + dx, y + dy).
 +
 +
In the example f(x, y) = xy, we obtain:
 +
 +
Ef(x, y, dx, dy)  =  (x + dx)(y + dy).
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|              x  dx y  dy              |
 +
|              o---o o---o              |
 +
|              \  | |  /              |
 +
|                \ | | /                |
 +
|                \| |/                |
 +
|                  @=@                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Ef =      (x, dx) (y, dy)            |
 +
o---------------------------------------o
 +
 +
Given the proposition f(x, y) in U = X x Y,
 +
the (first order) 'difference' of f is the
 +
proposition Df in EU that is defined by the
 +
formula Df = Ef - f, or, written out in full,
 +
Df(x, y, dx, dy) = f(x + dx, y + dy) - f(x, y).
 +
 +
In the example f(x, y) = xy, the result is:
 +
 +
Df(x, y, dx, dy)  =  (x + dx)(y + dy) - xy.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|        x  dx y  dy                    |
 +
|        o---o o---o                    |
 +
|        \  | |  /                    |
 +
|          \ | | /                      |
 +
|          \| |/        x y          |
 +
|            o=o-----------o            |
 +
|            \          /            |
 +
|              \        /              |
 +
|              \      /              |
 +
|                \    /                |
 +
|                \  /                |
 +
|                  \ /                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Df =      ((x, dx)(y, dy), xy)      |
 +
o---------------------------------------o
 +
 +
We did not yet go through the trouble to interpret this (first order)
 +
"difference of conjunction" fully, but were happy simply to evaluate
 +
it with respect to a single location in the universe of discourse,
 +
namely, at the point picked out by the singular proposition xy,
 +
in as much as if to say, at the place where x = 1 and y = 1.
 +
This evaluation is written in the form Df|xy or Df|<1, 1>,
 +
and we arrived at the locally applicable law that states
 +
that f = xy = x & y  =>  Df|xy = ((dx)(dy)) = dx or dy.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                dx dy                |
 +
|                  ^                  |
 +
|                o  |  o                |
 +
|              / \ | / \              |
 +
|              /  \|/  \              |
 +
|            /dy  |  dx\            |
 +
|            /(dx) /|\ (dy)\            |
 +
|          /  ^ /`|`\ ^  \          |
 +
|          /    \``|``/    \          |
 +
|        /    /`\`|`/`\    \        |
 +
|        /    /```\|/```\    \        |
 +
|      o  x  o`````o`````o  y  o      |
 +
|        \    \`````````/    /        |
 +
|        \    \```````/    /        |
 +
|          \    \`````/    /          |
 +
|          \    \```/    /          |
 +
|            \    \`/    /            |
 +
|            \    ·    /            |
 +
|              \  / \  /              |
 +
|              \ /  \ /              |
 +
|                o    o                |
 +
|                                      |
 +
o---------------------------------------o
 +
|                                      |
 +
|                dx  dy                |
 +
|                o  o                |
 +
|                  \ /                  |
 +
|                  o                  |
 +
|                  |                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Df|xy =      ((dx)(dy))              |
 +
o---------------------------------------o
 +
 +
The picture illustrates the analysis of the inclusive disjunction ((dx)(dy))
 +
into the exclusive disjunction:  dx(dy) + dy(dx) + dx dy, a proposition that
 +
may be interpreted to say "change x or change y or both".  And this can be
 +
recognized as just what you need to do if you happen to find yourself in
 +
the center cell and desire a detailed description of ways to depart it.
 +
 +
Jon Awbrey --
 +
 +
Formerly Of:
 +
Center Cell,
 +
Chateau Dif.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Last time we computed what will variously be called
 +
the "difference map", the "difference proposition",
 +
or the "local proposition" Df_p for the proposition
 +
f(x, y) = xy at the point p where x = 1 and y = 1.
 +
 +
In the universe U = X x Y, the four propositions
 +
xy, x(y), (x)y, (x)(y) that indicate the "cells",
 +
or the smallest regions of the venn diagram, are
 +
called "singular propositions".  These serve as
 +
an alternative notation for naming the points
 +
<1, 1>, <1, 0>, <0, 1>, <0, 0>, respectively.
 +
 +
Thus, we can write Df_p = Df|p = Df|<1, 1> = Df|xy,
 +
so long as we know the frame of reference in force.
 +
 +
Sticking with the example f(x, y) = xy, let us compute the
 +
value of the difference proposition Df at all of the points.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|        x  dx y  dy                    |
 +
|        o---o o---o                    |
 +
|        \  | |  /                    |
 +
|          \ | | /                      |
 +
|          \| |/        x y          |
 +
|            o=o-----------o            |
 +
|            \          /            |
 +
|              \        /              |
 +
|              \      /              |
 +
|                \    /                |
 +
|                \  /                |
 +
|                  \ /                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Df =      ((x, dx)(y, dy), xy)        |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|          dx    dy                    |
 +
|        o---o o---o                    |
 +
|        \  | |  /                    |
 +
|          \ | | /                      |
 +
|          \| |/                      |
 +
|            o=o-----------o            |
 +
|            \          /            |
 +
|              \        /              |
 +
|              \      /              |
 +
|                \    /                |
 +
|                \  /                |
 +
|                  \ /                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Df|xy =      ((dx)(dy))              |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|              o                        |
 +
|          dx |  dy                    |
 +
|        o---o o---o                    |
 +
|        \  | |  /                    |
 +
|          \ | | /        o            |
 +
|          \| |/          |            |
 +
|            o=o-----------o            |
 +
|            \          /            |
 +
|              \        /              |
 +
|              \      /              |
 +
|                \    /                |
 +
|                \  /                |
 +
|                  \ /                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Df|x(y) =      (dx) dy                |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|        o                              |
 +
|        |  dx    dy                    |
 +
|        o---o o---o                    |
 +
|        \  | |  /                    |
 +
|          \ | | /        o            |
 +
|          \| |/          |            |
 +
|            o=o-----------o            |
 +
|            \          /            |
 +
|              \        /              |
 +
|              \      /              |
 +
|                \    /                |
 +
|                \  /                |
 +
|                  \ /                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Df|(x)y =      dx (dy)                |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|        o    o                        |
 +
|        |  dx |  dy                    |
 +
|        o---o o---o                    |
 +
|        \  | |  /                    |
 +
|          \ | | /      o  o          |
 +
|          \| |/        \ /          |
 +
|            o=o-----------o            |
 +
|            \          /            |
 +
|              \        /              |
 +
|              \      /              |
 +
|                \    /                |
 +
|                \  /                |
 +
|                  \ /                  |
 +
|                  @                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Df|(x)(y) =    dx dy                |
 +
o---------------------------------------o
 +
 +
The easy way to visualize the values of these graphical
 +
expressions is just to notice the following equivalents:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|  x                                    |
 +
|  o-o-o-...-o-o-o                      |
 +
|  \          /                      |
 +
|    \        /                        |
 +
|    \      /                        |
 +
|      \    /                x        |
 +
|      \  /                o        |
 +
|        \ /                  |        |
 +
|        @        =        @        |
 +
|                                      |
 +
o---------------------------------------o
 +
|  (x, , ... , , )  =        (x)        |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                o                      |
 +
| x_1 x_2  x_k  |                      |
 +
|  o---o-...-o---o                      |
 +
|  \          /                      |
 +
|    \        /                        |
 +
|    \      /                        |
 +
|      \    /                          |
 +
|      \  /                          |
 +
|        \ /            x_1 ... x_k    |
 +
|        @        =        @        |
 +
|                                      |
 +
o---------------------------------------o
 +
| (x_1, ..., x_k, ()) = x_1 · ... · x_k |
 +
o---------------------------------------o
 +
 +
Laying out the arrows on the augmented venn diagram,
 +
one gets a picture of a "differential vector field".
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                dx dy                |
 +
|                  ^                  |
 +
|                o  |  o                |
 +
|              / \ | / \              |
 +
|              /  \|/  \              |
 +
|            /dy  |  dx\            |
 +
|            /(dx) /|\ (dy)\            |
 +
|          /  ^ /`|`\ ^  \          |
 +
|          /    \``|``/    \          |
 +
|        /    /`\`|`/`\    \        |
 +
|        /    /```\|/```\    \        |
 +
|      o  x  o`````o`````o  y  o      |
 +
|        \    \`````````/    /        |
 +
|        \  o---->```<----o  /        |
 +
|          \  dy \``^``/ dx  /          |
 +
|          \(dx) \`|`/ (dy)/          |
 +
|            \    \|/    /            |
 +
|            \    |    /            |
 +
|              \  /|\  /              |
 +
|              \ / | \ /              |
 +
|                o  |  o                |
 +
|                  |                  |
 +
|                dx | dy                |
 +
|                  o                  |
 +
|                                      |
 +
o---------------------------------------o
 +
 +
This really just constitutes a depiction of
 +
the interpretations in EU = X x Y x dX x dY
 +
that satisfy the difference proposition Df,
 +
namely, these:
 +
 +
1.  x  y  dx  dy
 +
2.  x  y  dx (dy)
 +
3.  x  y (dx) dy
 +
4.  x (y)(dx) dy
 +
5.  (x) y  dx (dy)
 +
6.  (x)(y) dx  dy
 +
 +
By inspection, it is fairly easy to understand Df
 +
as telling you what you have to do from each point
 +
of U in order to change the value borne by f(x, y).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We have been studying the action of the difference operator D,
 +
also known as the "localization operator", on the proposition
 +
f : X x Y -> B that is commonly known as the conjunction x·y.
 +
We described Df as a (first order) differential proposition,
 +
that is, a proposition of the type Df : X x Y x dX x dY -> B.
 +
Abstracting from the augmented venn diagram that illustrates
 +
how the "models", or the "satisfying interpretations", of Df
 +
distribute within the extended universe EU = X x Y x dX x dY,
 +
we can depict Df in the form of a "digraph" or directed graph,
 +
one whose points are labeled with the elements of  U =  X x Y
 +
and whose arrows are labeled with the elements of dU = dX x dY.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                x · y                |
 +
|                                      |
 +
|                  o                  |
 +
|                  ^^^                  |
 +
|                / | \                |
 +
|      (dx)· dy  /  |  \  dx ·(dy)      |
 +
|              /  |  \              |
 +
|              /    |    \              |
 +
|            v    |    v            |
 +
|  x ·(y)  o      |      o  (x)· y  |
 +
|                  |                  |
 +
|                  |                  |
 +
|                dx · dy                |
 +
|                  |                  |
 +
|                  |                  |
 +
|                  v                  |
 +
|                  o                  |
 +
|                                      |
 +
|                (x)·(y)                |
 +
|                                      |
 +
o---------------------------------------o
 +
|                                      |
 +
|  f    =    x  y                      |
 +
|                                      |
 +
| Df    =    x  y  · ((dx)(dy))        |
 +
|                                      |
 +
|      +    x (y) ·  (dx) dy          |
 +
|                                      |
 +
|      +    (x) y  ·  dx (dy)        |
 +
|                                      |
 +
|      +    (x)(y) ·  dx  dy          |
 +
|                                      |
 +
o---------------------------------------o
 +
 +
Any proposition worth its salt, as they say,
 +
has many equivalent ways to look at it, any
 +
of which may reveal some unsuspected aspect
 +
of its meaning.  We will encounter more and
 +
more of these alternative readings as we go.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The enlargement operator E, also known as the "shift operator",
 +
has many interesting and very useful properties in its own right,
 +
so let us not fail to observe a few of the more salient features
 +
that play out on the surface of our simple example, f(x, y) = xy.
 +
 +
Introduce a suitably generic definition of the extended universe of discourse:
 +
 +
Let U = X_1 x ... x X_k and EU = U x dU = X_1 x ... x X_k x dX_1 x ... x dX_k.
 +
 +
For a proposition f : X_1 x ... x X_k -> B,
 +
the (first order) 'enlargement' of f is the
 +
proposition Ef : EU -> B that is defined by:
 +
 +
Ef(x_1, ..., x_k, dx_1, ..., dx_k)  =  f(x_1 + dx_1, ..., x_k + dx_k).
 +
 +
It should be noted that the so-called "differential variables" dx_j
 +
are really just the same kind of boolean variables as the other x_j.
 +
It is conventional to give the additional variables these brands of
 +
inflected names, but whatever extra connotations we might choose to
 +
attach to these syntactic conveniences are wholly external to their
 +
purely algebraic meanings.
 +
 +
For the example f(x, y) = xy, we obtain:
 +
 +
Ef(x, y, dx, dy)  =  (x + dx)(y + dy).
 +
 +
Given that this expression uses nothing more than the "boolean ring"
 +
operations of addition (+) and multiplication (·), it is permissible
 +
to "multiply things out" in the usual manner to arrive at the result:
 +
 +
Ef(x, y, dx, dy)  =  x·y  +  x·dy  +  y·dx  +  dx·dy.
 +
 +
To understand what this means in logical terms, for instance, as expressed
 +
in a boolean expansion or a "disjunctive normal form" (DNF), it is perhaps
 +
a little better to go back and analyze the expression the same way that we
 +
did for Df.  Thus, let us compute the value of the enlarged proposition Ef
 +
at each of the points in the universe of discourse U = X x Y.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|              x  dx y  dy              |
 +
|              o---o o---o              |
 +
|              \  | |  /              |
 +
|                \ | | /                |
 +
|                \| |/                |
 +
|                  @=@                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Ef =      (x, dx)·(y, dy)            |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                dx    dy              |
 +
|              o---o o---o              |
 +
|              \  | |  /              |
 +
|                \ | | /                |
 +
|                \| |/                |
 +
|                  @=@                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Ef|xy =      (dx)·(dy)              |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                    o                  |
 +
|                dx |  dy              |
 +
|              o---o o---o              |
 +
|              \  | |  /              |
 +
|                \ | | /                |
 +
|                \| |/                |
 +
|                  @=@                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Ef|x(y) =    (dx)· dy                |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|              o                        |
 +
|              |  dx    dy              |
 +
|              o---o o---o              |
 +
|              \  | |  /              |
 +
|                \ | | /                |
 +
|                \| |/                |
 +
|                  @=@                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Ef|(x)y =      dx ·(dy)              |
 +
o---------------------------------------o
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|              o    o                  |
 +
|              |  dx |  dy              |
 +
|              o---o o---o              |
 +
|              \  | |  /              |
 +
|                \ | | /                |
 +
|                \| |/                |
 +
|                  @=@                  |
 +
|                                      |
 +
o---------------------------------------o
 +
| Ef|(x)(y) =    dx · dy                |
 +
o---------------------------------------o
 +
 +
Given the sort of data that arises from this form of analysis,
 +
we can now fold the disjoined ingredients back into a boolean
 +
expansion or a DNF that is equivalent to the proposition Ef.
 +
 +
Ef  =  xy · Ef_xy  +  x(y) · Ef_x(y)  +  (x)y · Ef_(x)y  +  (x)(y) · Ef_(x)(y).
 +
 +
Here is a summary of the result, illustrated by means of a digraph picture,
 +
where the "no change" element (dx)(dy) is drawn as a loop at the point x·y.
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|                x · y                |
 +
|              (dx)·(dy)              |
 +
|                -->--                |
 +
|                \  /                |
 +
|                  \ /                  |
 +
|                  o                  |
 +
|                  ^^^                  |
 +
|                / | \                |
 +
|                /  |  \                |
 +
|    (dx)· dy  /  |  \  dx ·(dy)    |
 +
|              /    |    \              |
 +
|            /    |    \            |
 +
|  x ·(y)  o      |      o  (x)· y  |
 +
|                  |                  |
 +
|                  |                  |
 +
|                dx · dy                |
 +
|                  |                  |
 +
|                  |                  |
 +
|                  o                  |
 +
|                                      |
 +
|                (x)·(y)                |
 +
|                                      |
 +
o---------------------------------------o
 +
|                                      |
 +
|  f    =    x  y                      |
 +
|                                      |
 +
| Ef    =    x  y  · (dx)(dy)          |
 +
|                                      |
 +
|      +    x (y) · (dx) dy          |
 +
|                                      |
 +
|      +    (x) y  ·  dx (dy)          |
 +
|                                      |
 +
|      +    (x)(y) ·  dx  dy          |
 +
|                                      |
 +
o---------------------------------------o
 +
 +
We may understand the enlarged proposition Ef
 +
as telling us all the different ways to reach
 +
a model of f from any point of the universe U.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
To broaden our experience with simple examples, let us now contemplate the
 +
sixteen functions of concrete type X x Y -> B and abstract type B x B -> B.
 +
For future reference, I will set here a few tables that detail the actions
 +
of E and D and on each of these functions, allowing us to view the results
 +
in several different ways.
 +
 +
By way of initial orientation, Table 0 lists equivalent expressions for the
 +
sixteen functions in a number of different languages for zeroth order logic.
 +
 +
 +
Table 0.  Propositional Forms On Two Variables
 +
o---------o---------o---------o----------o------------------o----------o
 +
| L_1    | L_2    | L_3    | L_4      | L_5              | L_6      |
 +
|        |        |        |          |                  |          |
 +
| Decimal | Binary  | Vector  | Cactus  | English          | Vulgate  |
 +
o---------o---------o---------o----------o------------------o----------o
 +
|        |      x = 1 1 0 0 |          |                  |          |
 +
|        |      y = 1 0 1 0 |          |                  |          |
 +
o---------o---------o---------o----------o------------------o----------o
 +
|        |        |        |          |                  |          |
 +
| f_0    | f_0000  | 0 0 0 0 |    ()    | false            |    0    |
 +
|        |        |        |          |                  |          |
 +
| f_1    | f_0001  | 0 0 0 1 |  (x)(y)  | neither x nor y  | ~x & ~y  |
 +
|        |        |        |          |                  |          |
 +
| f_2    | f_0010  | 0 0 1 0 |  (x) y  | y and not x      | ~x &  y  |
 +
|        |        |        |          |                  |          |
 +
| f_3    | f_0011  | 0 0 1 1 |  (x)    | not x            | ~x      |
 +
|        |        |        |          |                  |          |
 +
| f_4    | f_0100  | 0 1 0 0 |  x (y)  | x and not y      |  x & ~y  |
 +
|        |        |        |          |                  |          |
 +
| f_5    | f_0101  | 0 1 0 1 |    (y)  | not y            |      ~y  |
 +
|        |        |        |          |                  |          |
 +
| f_6    | f_0110  | 0 1 1 0 |  (x, y)  | x not equal to y |  x +  y  |
 +
|        |        |        |          |                  |          |
 +
| f_7    | f_0111  | 0 1 1 1 |  (x  y)  | not both x and y | ~x v ~y  |
 +
|        |        |        |          |                  |          |
 +
| f_8    | f_1000  | 1 0 0 0 |  x  y  | x and y          |  x &  y  |
 +
|        |        |        |          |                  |          |
 +
| f_9    | f_1001  | 1 0 0 1 | ((x, y)) | x equal to y    |  x =  y  |
 +
|        |        |        |          |                  |          |
 +
| f_10    | f_1010  | 1 0 1 0 |      y  | y                |      y  |
 +
|        |        |        |          |                  |          |
 +
| f_11    | f_1011  | 1 0 1 1 |  (x (y)) | not x without y  |  x => y  |
 +
|        |        |        |          |                  |          |
 +
| f_12    | f_1100  | 1 1 0 0 |  x      | x                |  x      |
 +
|        |        |        |          |                  |          |
 +
| f_13    | f_1101  | 1 1 0 1 | ((x) y)  | not y without x  |  x <= y  |
 +
|        |        |        |          |                  |          |
 +
| f_14    | f_1110  | 1 1 1 0 | ((x)(y)) | x or y          |  x v  y  |
 +
|        |        |        |          |                  |          |
 +
| f_15    | f_1111  | 1 1 1 1 |  (())  | true            |    1    |
 +
|        |        |        |          |                  |          |
 +
o---------o---------o---------o----------o------------------o----------o
 +
 +
 +
The next four Tables expand the expressions of Ef and Df
 +
in two different ways, for each of the sixteen functions.
 +
Notice that the functions are given in a different order,
 +
here being collected into a set of seven natural classes.
 +
 +
 +
Table 1.  Ef Expanded Over Ordinary Features {x, y}
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
|      |    f      |  Ef | xy  | Ef | x(y)  | Ef | (x)y  | Ef | (x)(y)|
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_1  |  (x)(y)  |  dx  dy  |  dx (dy)  |  (dx) dy  |  (dx)(dy)  |
 +
|      |            |            |            |            |            |
 +
| f_2  |  (x) y    |  dx (dy)  |  dx  dy  |  (dx)(dy)  |  (dx) dy  |
 +
|      |            |            |            |            |            |
 +
| f_4  |    x (y)  |  (dx) dy  |  (dx)(dy)  |  dx  dy  |  dx (dy)  |
 +
|      |            |            |            |            |            |
 +
| f_8  |    x  y    |  (dx)(dy)  |  (dx) dy  |  dx (dy)  |  dx  dy  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_3  |  (x)      |  dx      |  dx      |  (dx)      |  (dx)      |
 +
|      |            |            |            |            |            |
 +
| f_12 |    x      |  (dx)      |  (dx)      |  dx      |  dx      |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_6  |  (x, y)  |  (dx, dy)  | ((dx, dy)) | ((dx, dy)) |  (dx, dy)  |
 +
|      |            |            |            |            |            |
 +
| f_9  |  ((x, y))  | ((dx, dy)) |  (dx, dy)  |  (dx, dy)  | ((dx, dy)) |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_5  |      (y)  |      dy  |      (dy)  |      dy  |      (dy)  |
 +
|      |            |            |            |            |            |
 +
| f_10 |      y    |      (dy)  |      dy  |      (dy)  |      dy  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_7  |  (x  y)  | ((dx)(dy)) | ((dx) dy)  |  (dx (dy)) |  (dx  dy)  |
 +
|      |            |            |            |            |            |
 +
| f_11 |  (x (y))  | ((dx) dy)  | ((dx)(dy)) |  (dx  dy)  |  (dx (dy)) |
 +
|      |            |            |            |            |            |
 +
| f_13 |  ((x) y)  |  (dx (dy)) |  (dx  dy)  | ((dx)(dy)) | ((dx) dy)  |
 +
|      |            |            |            |            |            |
 +
| f_14 |  ((x)(y))  |  (dx  dy)  |  (dx (dy)) | ((dx) dy)  | ((dx)(dy)) |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
 +
 +
Table 2.  Df Expanded Over Ordinary Features {x, y}
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
|      |    f      |  Df | xy  | Df | x(y)  | Df | (x)y  | Df | (x)(y)|
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_1  |  (x)(y)  |  dx  dy  |  dx (dy)  |  (dx) dy  | ((dx)(dy)) |
 +
|      |            |            |            |            |            |
 +
| f_2  |  (x) y    |  dx (dy)  |  dx  dy  | ((dx)(dy)) |  (dx) dy  |
 +
|      |            |            |            |            |            |
 +
| f_4  |    x (y)  |  (dx) dy  | ((dx)(dy)) |  dx  dy  |  dx (dy)  |
 +
|      |            |            |            |            |            |
 +
| f_8  |    x  y    | ((dx)(dy)) |  (dx) dy  |  dx (dy)  |  dx  dy  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_3  |  (x)      |  dx      |  dx      |  dx      |  dx      |
 +
|      |            |            |            |            |            |
 +
| f_12 |    x      |  dx      |  dx      |  dx      |  dx      |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_6  |  (x, y)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
 +
|      |            |            |            |            |            |
 +
| f_9  |  ((x, y))  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |  (dx, dy)  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_5  |      (y)  |      dy  |      dy  |      dy  |      dy  |
 +
|      |            |            |            |            |            |
 +
| f_10 |      y    |      dy  |      dy  |      dy  |      dy  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_7  |  (x  y)  | ((dx)(dy)) |  (dx) dy  |  dx (dy)  |  dx  dy  |
 +
|      |            |            |            |            |            |
 +
| f_11 |  (x (y))  |  (dx) dy  | ((dx)(dy)) |  dx  dy  |  dx (dy)  |
 +
|      |            |            |            |            |            |
 +
| f_13 |  ((x) y)  |  dx (dy)  |  dx  dy  | ((dx)(dy)) |  (dx) dy  |
 +
|      |            |            |            |            |            |
 +
| f_14 |  ((x)(y))  |  dx  dy  |  dx (dy)  |  (dx) dy  | ((dx)(dy)) |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_15 |    (())    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
 +
 +
Table 3.  Ef Expanded Over Differential Features {dx, dy}
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
|      |    f      |  T_11 f  |  T_10 f  |  T_01 f  |  T_00 f  |
 +
|      |            |            |            |            |            |
 +
|      |            | Ef| dx·dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_1  |  (x)(y)  |    x  y    |    x (y)  |  (x) y    |  (x)(y)  |
 +
|      |            |            |            |            |            |
 +
| f_2  |  (x) y    |    x (y)  |    x  y    |  (x)(y)  |  (x) y    |
 +
|      |            |            |            |            |            |
 +
| f_4  |    x (y)  |  (x) y    |  (x)(y)  |    x  y    |    x (y)  |
 +
|      |            |            |            |            |            |
 +
| f_8  |    x  y    |  (x)(y)  |  (x) y    |    x (y)  |    x  y    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_3  |  (x)      |    x      |    x      |  (x)      |  (x)      |
 +
|      |            |            |            |            |            |
 +
| f_12 |    x      |  (x)      |  (x)      |    x      |    x      |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_6  |  (x, y)  |  (x, y)  |  ((x, y))  |  ((x, y))  |  (x, y)  |
 +
|      |            |            |            |            |            |
 +
| f_9  |  ((x, y))  |  ((x, y))  |  (x, y)  |  (x, y)  |  ((x, y))  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_5  |      (y)  |      y    |      (y)  |      y    |      (y)  |
 +
|      |            |            |            |            |            |
 +
| f_10 |      y    |      (y)  |      y    |      (y)  |      y    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_7  |  (x  y)  |  ((x)(y))  |  ((x) y)  |  (x (y))  |  (x  y)  |
 +
|      |            |            |            |            |            |
 +
| f_11 |  (x (y))  |  ((x) y)  |  ((x)(y))  |  (x  y)  |  (x (y))  |
 +
|      |            |            |            |            |            |
 +
| f_13 |  ((x) y)  |  (x (y))  |  (x  y)  |  ((x)(y))  |  ((x) y)  |
 +
|      |            |            |            |            |            |
 +
| f_14 |  ((x)(y))  |  (x  y)  |  (x (y))  |  ((x) y)  |  ((x)(y))  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|                  |            |            |            |            |
 +
| Fixed Point Total |      4    |      4    |      4    |    16    |
 +
|                  |            |            |            |            |
 +
o-------------------o------------o------------o------------o------------o
 +
 +
 +
Table 4.  Df Expanded Over Differential Features {dx, dy}
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
|      |    f      | Df| dx·dy  | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_1  |  (x)(y)  |  ((x, y))  |    (y)    |    (x)    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_2  |  (x) y    |  (x, y)  |    y      |    (x)    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_4  |    x (y)  |  (x, y)  |    (y)    |    x      |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_8  |    x  y    |  ((x, y))  |    y      |    x      |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_3  |  (x)      |    (())    |    (())    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_12 |    x      |    (())    |    (())    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_6  |  (x, y)  |    ()    |    (())    |    (())    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_9  |  ((x, y))  |    ()    |    (())    |    (())    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_5  |      (y)  |    (())    |    ()    |    (())    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_10 |      y    |    (())    |    ()    |    (())    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_7  |  (x  y)  |  ((x, y))  |    y      |    x      |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_11 |  (x (y))  |  (x, y)  |    (y)    |    x      |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_13 |  ((x) y)  |  (x, y)  |    y      |    (x)    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_14 |  ((x)(y))  |  ((x, y))  |    (y)    |    (x)    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_15 |    (())    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
 +
 +
If the medium truly is the message,
 +
the blank slate is the innate idea.
 +
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 7
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
If you think that I linger in the realm of logical difference calculus
 +
out of sheer vacillation about getting down to the differential proper,
 +
it is probably out of a prior expectation that you derive from the art
 +
or the long-engrained practice of real analysis.  But the fact is that
 +
ordinary calculus only rushes on to the sundry orders of approximation
 +
because the strain of comprehending the full import of E and D at once
 +
whelm over its discrete and finite powers to grasp them.  But here, in
 +
the fully serene idylls of ZOL, we find ourselves fit with the compass
 +
of a wit that is all we'd ever wish to explore their effects with care.
 +
 +
So let us do just that.
 +
 +
I will first rationalize the novel grouping of propositional forms
 +
in the last set of Tables, as that will extend a gentle invitation
 +
to the mathematical subject of "group theory", and demonstrate its
 +
relevance to differential logic in a strikingly apt and useful way.
 +
The data for that account is contained in Table 3.
 +
 +
Table 3.  Ef Expanded Over Differential Features {dx, dy}
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
|      |    f      |  T_11 f  |  T_10 f  |  T_01 f  |  T_00 f  |
 +
|      |            |            |            |            |            |
 +
|      |            | Ef| dx·dy  | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_1  |  (x)(y)  |    x  y    |    x (y)  |  (x) y    |  (x)(y)  |
 +
|      |            |            |            |            |            |
 +
| f_2  |  (x) y    |    x (y)  |    x  y    |  (x)(y)  |  (x) y    |
 +
|      |            |            |            |            |            |
 +
| f_4  |    x (y)  |  (x) y    |  (x)(y)  |    x  y    |    x (y)  |
 +
|      |            |            |            |            |            |
 +
| f_8  |    x  y    |  (x)(y)  |  (x) y    |    x (y)  |    x  y    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_3  |  (x)      |    x      |    x      |  (x)      |  (x)      |
 +
|      |            |            |            |            |            |
 +
| f_12 |    x      |  (x)      |  (x)      |    x      |    x      |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_6  |  (x, y)  |  (x, y)  |  ((x, y))  |  ((x, y))  |  (x, y)  |
 +
|      |            |            |            |            |            |
 +
| f_9  |  ((x, y))  |  ((x, y))  |  (x, y)  |  (x, y)  |  ((x, y))  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_5  |      (y)  |      y    |      (y)  |      y    |      (y)  |
 +
|      |            |            |            |            |            |
 +
| f_10 |      y    |      (y)  |      y    |      (y)  |      y    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_7  |  (x  y)  |  ((x)(y))  |  ((x) y)  |  (x (y))  |  (x  y)  |
 +
|      |            |            |            |            |            |
 +
| f_11 |  (x (y))  |  ((x) y)  |  ((x)(y))  |  (x  y)  |  (x (y))  |
 +
|      |            |            |            |            |            |
 +
| f_13 |  ((x) y)  |  (x (y))  |  (x  y)  |  ((x)(y))  |  ((x) y)  |
 +
|      |            |            |            |            |            |
 +
| f_14 |  ((x)(y))  |  (x  y)  |  (x (y))  |  ((x) y)  |  ((x)(y))  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|                  |            |            |            |            |
 +
| Fixed Point Total |      4    |      4    |      4    |    16    |
 +
|                  |            |            |            |            |
 +
o-------------------o------------o------------o------------o------------o
 +
 +
The shift operator E can be understood as enacting a substitution operation
 +
on the proposition that is given as its argument.  In our immediate example,
 +
we have the following data and definition:
 +
 +
E : (U -> B)  ->  (EU -> B),
 +
 +
E :  f(x, y)  ->  Ef(x, y, dx, dy),
 +
 +
Ef(x, y, dx, dy)  =  f(x + dx, y + dy).
 +
 +
Therefore, if we evaluate Ef at particular values of dx and dy,
 +
for example, dx = i and dy = j, where i, j are in B, we obtain:
 +
 +
E_ij : (U -> B)  ->  (U -> B),
 +
 +
E_ij :    f      ->  E_ij f,
 +
 +
E_ij f  =  Ef | <dx = i, dy = j>  =  f(x + i, y + j).
 +
 +
The notation is a little bit awkward, but the data of the Table should
 +
make the sense clear.  The important thing to observe is that E_ij has
 +
the effect of transforming each proposition f : U -> B into some other
 +
proposition f' : U -> B.  As it happens, the action is one-to-one and
 +
onto for each E_ij, so the gang of four operators {E_ij : i, j in B}
 +
is an example of what is called a "transformation group" on the set
 +
of sixteen propositions.  Bowing to a longstanding local and linear
 +
tradition, I will therefore redub the four elements of this group
 +
as T_00, T_01, T_10, T_11, to bear in mind their transformative
 +
character, or nature, as the case may be.  Abstractly viewed,
 +
this group of order four has the following operation table:
 +
 +
o----------o----------o----------o----------o----------o
 +
|          %          |          |          |          |
 +
|    ·    %  T_00  |  T_01  |  T_10  |  T_11  |
 +
|          %          |          |          |          |
 +
o==========o==========o==========o==========o==========o
 +
|          %          |          |          |          |
 +
|  T_00  %  T_00  |  T_01  |  T_10  |  T_11  |
 +
|          %          |          |          |          |
 +
o----------o----------o----------o----------o----------o
 +
|          %          |          |          |          |
 +
|  T_01  %  T_01  |  T_00  |  T_11  |  T_10  |
 +
|          %          |          |          |          |
 +
o----------o----------o----------o----------o----------o
 +
|          %          |          |          |          |
 +
|  T_10  %  T_10  |  T_11  |  T_00  |  T_01  |
 +
|          %          |          |          |          |
 +
o----------o----------o----------o----------o----------o
 +
|          %          |          |          |          |
 +
|  T_11  %  T_11  |  T_10  |  T_01  |  T_00  |
 +
|          %          |          |          |          |
 +
o----------o----------o----------o----------o----------o
 +
 +
It happens that there are just two possible groups of 4 elements.
 +
One is the cyclic group Z_4 (German "Zyklus"), which this is not.
 +
The other is Klein's four-group V_4 (German "Vier"), which it is.
 +
 +
More concretely viewed, the group as a whole pushes the set
 +
of sixteen propositions around in such a way that they fall
 +
into seven natural classes, called "orbits".  One says that
 +
the orbits are preserved by the action of the group.  There
 +
is an "Orbit Lemma" of immense utility to "those who count"
 +
which, depending on your upbringing, you may associate with
 +
the names of Burnside, Cauchy, Frobenius, or some subset or
 +
superset of these three, vouching that the number of orbits
 +
is equal to the mean number of fixed points, in other words,
 +
the total number of points (in our case, propositions) that
 +
are left unmoved by the separate operations, divided by the
 +
order of the group.  In this instance, T_00 operates as the
 +
group identity, fixing all 16 propositions, while the other
 +
three group elements fix 4 propositions each, and so we get:
 +
Number of orbits  =  (4 + 4 + 4 + 16) / 4  =  7.  Amazing!
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 8
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We have been contemplating functions of the type f : U -> B
 +
studying the action of the operators E and D on this family.
 +
These functions, that we may identify for our present aims
 +
with propositions, inasmuch as they capture their abstract
 +
forms, are logical analogues of "scalar potential fields".
 +
These are the sorts of fields that are so picturesquely
 +
presented in elementary calculus and physics textbooks
 +
by images of snow-covered hills and parties of skiers
 +
who trek down their slopes like least action heroes.
 +
The analogous scene in propositional logic presents
 +
us with forms more reminiscent of plateaunic idylls,
 +
being all plains at one of two levels, the mesas of
 +
verity and falsity, as it were, with nary a niche
 +
to inhabit between them, restricting our options
 +
for a sporting gradient of downhill dynamics to
 +
just one of two, standing still on level ground
 +
or falling off a bluff.
 +
 +
We are still working well within the logical analogue of the
 +
classical finite difference calculus, taking in the novelties
 +
that the logical transmutation of familiar elements is able to
 +
bring to light.  Soon we will take up several different notions
 +
of approximation relationships that may be seen to organize the
 +
space of propositions, and these will allow us to define several
 +
different forms of differential analysis applying to propositions.
 +
In time we will find reason to consider more general types of maps,
 +
having concrete types of the form X_1 x ... x X_k -> Y_1 x ... x Y_n
 +
and abstract types B^k -> B^n.  We will think of these mappings as
 +
transforming universes of discourse into themselves or into others,
 +
in short, as "transformations of discourse".
 +
 +
Before we continue with this intinerary, however, I would like to highlight
 +
another sort of "differential aspect" that concerns the "boundary operator"
 +
or the "marked connective" that serves as one of the two basic connectives
 +
in the cactus language for ZOL.
 +
 +
For example, consider the proposition f of concrete type f : X x Y x Z -> B
 +
and abstract type f : B^3 -> B that is written "(x, y, z)" in cactus syntax.
 +
Taken as an assertion in what Peirce called the "existential interpretation",
 +
(x, y, z) says that just one of x, y, z is false.  It is useful to consider
 +
this assertion in relation to the conjunction xyz of the features that are
 +
engaged as its arguments.  A venn diagram of (x, y, z) looks like this:
 +
 +
o-----------------------------------------------------------o
 +
| U                                                        |
 +
|                                                          |
 +
|                      o-------------o                      |
 +
|                    /              \                    |
 +
|                    /                \                    |
 +
|                  /                  \                  |
 +
|                  /                    \                  |
 +
|                /                      \                |
 +
|                o            x            o                |
 +
|                |                        |                |
 +
|                |                        |                |
 +
|                |                        |                |
 +
|                |                        |                |
 +
|                |                        |                |
 +
|            o--o----------o  o----------o--o            |
 +
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
 +
|          /      \%%%%%%%%%%o%%%%%%%%%%/      \          |
 +
|          /        \%%%%%%%%/ \%%%%%%%%/        \          |
 +
|        /          \%%%%%%/  \%%%%%%/          \        |
 +
|        /            \%%%%/    \%%%%/            \        |
 +
|      o              o--o-------o--o              o      |
 +
|      |                |%%%%%%%|                |      |
 +
|      |                |%%%%%%%|                |      |
 +
|      |                |%%%%%%%|                |      |
 +
|      |                |%%%%%%%|                |      |
 +
|      |                |%%%%%%%|                |      |
 +
|      o        y        o%%%%%%%o        z        o      |
 +
|        \                \%%%%%/                /        |
 +
|        \                \%%%/                /        |
 +
|          \                \%/                /          |
 +
|          \                o                /          |
 +
|            \              / \              /            |
 +
|            o-------------o  o-------------o            |
 +
|                                                          |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
 +
In relation to the center cell indicated by the conjunction xyz,
 +
the region indicated by (x, y, z) is comprised of the "adjacent"
 +
or the "bordering" cells.  Thus they are the cells that are just
 +
across the boundary of the center cell, as if reached by way of
 +
Leibniz's "minimal changes" from the point of origin, here, xyz.
 +
 +
The same sort of boundary relationship holds for any cell of origin that
 +
one might elect to indicate, say, by means of the conjunction of positive
 +
or negative basis features u_1 · ... · u_k, with u_j = x_j or u_j = (x_j),
 +
for j = 1 to k.  The proposition (u_1, ..., u_k) indicates the disjunctive
 +
region consisting of the cells that are just next door to u_1 · ... · u_k.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 9
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might conceivably have
 +
| practical bearings you conceive the objects of your
 +
| conception to have.  Then, your conception of those
 +
| effects is the whole of your conception of the object.
 +
|
 +
| Charles Sanders Peirce, "The Maxim of Pragmatism, CP 5.438.
 +
 +
One other subject that it would be opportune to mention at this point,
 +
while we have an object example of a mathematical group fresh in mind,
 +
is the relationship between the pragmatic maxim and what are commonly
 +
known in mathematics as "representation principles".  As it turns out,
 +
with regard to its formal characteristics, the pragmatic maxim unites
 +
the aspects of a representation principle with the attributes of what
 +
would ordinarily be known as a "closure principle".  We will consider
 +
the form of closure that is invoked by the pragmatic maxim on another
 +
occasion, focusing here and now on the topic of group representations.
 +
 +
Let us return to the example of the so-called "four-group" V_4.
 +
We encountered this group in one of its concrete representations,
 +
namely, as a "transformation group" that acts on a set of objects,
 +
in this particular case a set of sixteen functions or propositions.
 +
Forgetting about the set of objects that the group transforms among
 +
themselves, we may take the abstract view of the group's operational
 +
structure, say, in the form of the group operation table copied here:
 +
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    ·    %    e    |    f    |    g    |    h    |
 +
|        %        |        |        |        |
 +
o=========o=========o=========o=========o=========o
 +
|        %        |        |        |        |
 +
|    e    %    e    |    f    |    g    |    h    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    f    %    f    |    e    |    h    |    g    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    g    %    g    |    h    |    e    |    f    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    h    %    h    |    g    |    f    |    e    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
 +
This table is abstractly the same as, or isomorphic to, the versions with
 +
the E_ij operators and the T_ij transformations that we discussed earlier.
 +
That is to say, the story is the same -- only the names have been changed.
 +
An abstract group can have a multitude of significantly and superficially
 +
different representations.  Even after we have long forgotten the details
 +
of the particular representation that we may have come in with, there are
 +
species of concrete representations, called the "regular representations",
 +
that are always readily available, as they can be generated from the mere
 +
data of the abstract operation table itself.
 +
 +
For example, select a group element from the top margin of the Table,
 +
and "consider its effects" on each of the group elements as they are
 +
listed along the left margin.  We may record these effects as Peirce
 +
usually did, as a logical "aggregate" of elementary dyadic relatives,
 +
that is to say, a disjunction or a logical sum whose terms represent
 +
the ordered pairs of <input : output> transactions that are produced
 +
by each group element in turn.  This yields what is usually known as
 +
one of the "regular representations" of the group, specifically, the
 +
"first", the "post-", or the "right" regular representation.  It has
 +
long been conventional to organize the terms in the form of a matrix:
 +
 +
Reading "+" as a logical disjunction:
 +
 +
G  =  e  +  f  +  g  + h,
 +
 +
And so, by expanding effects, we get:
 +
 +
G  =  e:e  +  f:f  +  g:g  +  h:h
 +
 +
  +  e:f  +  f:e  +  g:h  +  h:g
 +
 +
  +  e:g  +  f:h  +  g:e  +  h:f
 +
 +
  +  e:h  +  f:g  +  g:f  +  h:e
 +
 +
More on the pragmatic maxim as a representation principle later.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 10
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might conceivably have
 +
| practical bearings you conceive the objects of your
 +
| conception to have.  Then, your conception of those
 +
| effects is the whole of your conception of the object.
 +
|
 +
| Charles Sanders Peirce, "The Maxim of Pragmatism, CP 5.438.
 +
 +
The genealogy of this conception of pragmatic representation is very intricate.
 +
I will delineate some details that I presently fancy I remember clearly enough,
 +
subject to later correction.  Without checking historical accounts, I will not
 +
be able to pin down anything like a real chronology, but most of these notions
 +
were standard furnishings of the 19th Century mathematical study, and only the
 +
last few items date as late as the 1920's.
 +
 +
The idea about the regular representations of a group is universally known
 +
as "Cayley's Theorem", usually in the form:  "Every group is isomorphic to
 +
a subgroup of Aut(S), the group of automorphisms of an appropriate set S".
 +
There is a considerable generalization of these regular representations to
 +
a broad class of relational algebraic systems in Peirce's earliest papers.
 +
The crux of the whole idea is this:
 +
 +
| Consider the effects of the symbol, whose meaning you wish to investigate,
 +
| as they play out on "all" of the different stages of context on which you
 +
| can imagine that symbol playing a role.
 +
 +
This idea of contextual definition is basically the same as Jeremy Bentham's
 +
notion of "paraphrasis", a "method of accounting for fictions by explaining
 +
various purported terms away" (Quine, in Van Heijenoort, page 216).  Today
 +
we'd call these constructions "term models".  This, again, is the big idea
 +
behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus,
 +
and I reckon you know where that leads.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 11
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
Continuing to draw on the reduced example of group representations,
 +
I would like to draw out a few of the finer points and problems of
 +
regarding the maxim of pragmatism as a principle of representation.
 +
 +
Let us revisit the example of an abstract group that we had befour:
 +
 +
Table 1.  Klein Four-Group V_4
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    ·    %    e    |    f    |    g    |    h    |
 +
|        %        |        |        |        |
 +
o=========o=========o=========o=========o=========o
 +
|        %        |        |        |        |
 +
|    e    %    e    |    f    |    g    |    h    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    f    %    f    |    e    |    h    |    g    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    g    %    g    |    h    |    e    |    f    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    h    %    h    |    g    |    f    |    e    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
 +
I presented the regular post-representation
 +
of the four-group V_4 in the following form:
 +
 +
Reading "+" as a logical disjunction:
 +
 +
  G  =  e  +  f  +  g  + h
 +
 +
And so, by expanding effects, we get:
 +
 +
  G  =  e:e  +  f:f  +  g:g  +  h:h
 +
 +
      +  e:f  +  f:e  +  g:h  +  h:g
 +
 +
      +  e:g  +  f:h  +  g:e  +  h:f
 +
 +
      +  e:h  +  f:g  +  g:f  +  h:e
 +
 +
This presents the group in one big bunch,
 +
and there are occasions when one regards
 +
it this way, but that is not the typical
 +
form of presentation that we'd encounter.
 +
More likely, the story would go a little
 +
bit like this:
 +
 +
I cannot remember any of my math teachers
 +
ever invoking the pragmatic maxim by name,
 +
but it would be a very regular occurrence
 +
for such mentors and tutors to set up the
 +
subject in this wise:  Suppose you forget
 +
what a given abstract group element means,
 +
that is, in effect, 'what it is'.  Then a
 +
sure way to jog your sense of 'what it is'
 +
is to build a regular representation from
 +
the formal materials that are necessarily
 +
left lying about on that abstraction site.
 +
 +
Working through the construction for each
 +
one of the four group elements, we arrive
 +
at the following exegeses of their senses,
 +
giving their regular post-representations:
 +
 +
  e  =  e:e  +  f:f  +  g:g  +  h:h
 +
 +
  f  =  e:f  +  f:e  +  g:h  +  h:g
 +
 +
  g  =  e:g  +  f:h  +  g:e  +  h:f
 +
 +
  h  =  e:h  +  f:g  +  g:f  +  h:e
 +
 +
So if somebody asks you, say, "What is g?",
 +
you can say, "I don't know for certain but
 +
in practice its effects go a bit like this:
 +
Converting e to g, f to h, g to e, h to f".
 +
 +
I will have to check this out later on, but my impression is
 +
that Peirce tended to lean toward the other brand of regular,
 +
the "second", the "left", or the "ante-representation" of the
 +
groups that he treated in his earliest manuscripts and papers.
 +
I believe that this was because he thought of the actions on
 +
the pattern of dyadic relative terms like the "aftermath of".
 +
 +
Working through this alternative for each
 +
one of the four group elements, we arrive
 +
at the following exegeses of their senses,
 +
giving their regular ante-representations:
 +
 +
  e  =  e:e  +  f:f  +  g:g  +  h:h
 +
 +
  f  =  f:e  +  e:f  +  h:g  +  g:h
 +
 +
  g  =  g:e  +  h:f  +  e:g  +  f:h
 +
 +
  h  =  h:e  +  g:f  +  f:g  +  e:h
 +
 +
Your paraphrastic interpretation of what this all
 +
means would come out precisely the same as before.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 12
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Erratum
 +
 +
Oops!  I think that I have just confounded two entirely different issues:
 +
1.  The substantial difference between right and left regular representations.
 +
2.  The inessential difference between two conventions of presenting matrices.
 +
I will sort this out and correct it later, as need be.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 13
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
Let me return to Peirce's early papers on the algebra of relatives
 +
to pick up the conventions that he used there, and then rewrite my
 +
account of regular representations in a way that conforms to those.
 +
 +
Peirce expresses the action of an "elementary dual relative" like so:
 +
 +
| [Let] A:B be taken to denote
 +
| the elementary relative which
 +
| multiplied into B gives A.
 +
|
 +
| Peirce, 'Collected Papers', CP 3.123.
 +
 +
And though he is well aware that it is not at all necessary to arrange
 +
elementary relatives into arrays, matrices, or tables, when he does so
 +
he tends to prefer organizing dyadic relations in the following manner:
 +
 +
|  A:A  A:B  A:C  |
 +
|                  |
 +
|  B:A  B:B  B:C  |
 +
|                  |
 +
|  C:A  C:B  C:C  |
 +
 +
That conforms to the way that the last school of thought
 +
I matriculated into stipulated that we tabulate material:
 +
 +
|  e_11  e_12  e_13  |
 +
|                    |
 +
|  e_21  e_22  e_23  |
 +
|                    |
 +
|  e_31  e_32  e_33  |
 +
 +
So, for example, let us suppose that we have the small universe {A, B, C},
 +
and the 2-adic relation m = "mover of" that is represented by this matrix:
 +
 +
m  =
 +
 +
|  m_AA (A:A)  m_AB (A:B)  m_AC (A:C)  |
 +
|                                        |
 +
|  m_BA (B:A)  m_BB (B:B)  m_BC (B:C)  |
 +
|                                        |
 +
|  m_CA (C:A)  m_CB (C:B)  m_CC (C:C)  |
 +
 +
Also, let m be such that
 +
A is a mover of A and B,
 +
B is a mover of B and C,
 +
C is a mover of C and A.
 +
 +
In sum:
 +
 +
m  =
 +
 +
|  1 · (A:A)  1 · (A:B)  0 · (A:C)  |
 +
|                                    |
 +
|  0 · (B:A)  1 · (B:B)  1 · (B:C)  |
 +
|                                    |
 +
|  1 · (C:A)  0 · (C:B)  1 · (C:C)  |
 +
 +
For the sake of orientation and motivation,
 +
compare with Peirce's notation in CP 3.329.
 +
 +
I think that will serve to fix notation
 +
and set up the remainder of the account.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 14
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
I am beginning to see how I got confused.
 +
It is common in algebra to switch around
 +
between different conventions of display,
 +
as the momentary fancy happens to strike,
 +
and I see that Peirce is no different in
 +
this sort of shiftiness than anyone else.
 +
A changeover appears to occur especially
 +
whenever he shifts from logical contexts
 +
to algebraic contexts of application.
 +
 +
In the paper "On the Relative Forms of Quaternions" (CP 3.323),
 +
we observe Peirce providing the following sorts of explanation:
 +
 +
| If X, Y, Z denote the three rectangular components of a vector, and W denote
 +
| numerical unity (or a fourth rectangular component, involving space of four
 +
| dimensions), and (Y:Z) denote the operation of converting the Y component
 +
| of a vector into its Z component, then
 +
|
 +
|    1  =  (W:W) + (X:X) + (Y:Y) + (Z:Z)
 +
|
 +
|    i  =  (X:W) - (W:X) - (Y:Z) + (Z:Y)
 +
|
 +
|    j  =  (Y:W) - (W:Y) - (Z:X) + (X:Z)
 +
|
 +
|    k  =  (Z:W) - (W:Z) - (X:Y) + (Y:X)
 +
|
 +
| In the language of logic (Y:Z) is a relative term whose relate is
 +
| a Y component, and whose correlate is a Z component.  The law of
 +
| multiplication is plainly (Y:Z)(Z:X) = (Y:X), (Y:Z)(X:W) = 0,
 +
| and the application of these rules to the above values of
 +
| 1, i, j, k gives the quaternion relations
 +
|
 +
|    i^2  =  j^2  =  k^2  =  -1,
 +
|
 +
|    ijk  =  -1,
 +
|
 +
|    etc.
 +
|
 +
| The symbol a(Y:Z) denotes the changing of Y to Z and the
 +
| multiplication of the result by 'a'.  If the relatives be
 +
| arranged in a block
 +
|
 +
|    W:W    W:X    W:Y    W:Z
 +
|
 +
|    X:W    X:X    X:Y    X:Z
 +
|
 +
|    Y:W    Y:X    Y:Y    Y:Z
 +
|
 +
|    Z:W    Z:X    Z:Y    Z:Z
 +
|
 +
| then the quaternion w + xi + yj + zk
 +
| is represented by the matrix of numbers
 +
|
 +
|    w      -x      -y      -z
 +
|
 +
|    x        w      -z      y
 +
|
 +
|    y        z      w      -x
 +
|
 +
|    z      -y      x      w
 +
|
 +
| The multiplication of such matrices follows the same laws as the
 +
| multiplication of quaternions.  The determinant of the matrix =
 +
| the fourth power of the tensor of the quaternion.
 +
|
 +
| The imaginary x + y(-1)^(1/2) may likewise be represented by the matrix
 +
|
 +
|      x      y
 +
|
 +
|    -y      x
 +
|
 +
| and the determinant of the matrix = the square of the modulus.
 +
|
 +
| Charles Sanders Peirce, 'Collected Papers', CP 3.323.
 +
|'Johns Hopkins University Circulars', No. 13, p. 179, 1882.
 +
 +
This way of talking is the mark of a person who opts
 +
to multiply his matrices "on the rignt", as they say.
 +
Yet Peirce still continues to call the first element
 +
of the ordered pair (I:J) its "relate" while calling
 +
the second element of the pair (I:J) its "correlate".
 +
That doesn't comport very well, so far as I can tell,
 +
with his customary reading of relative terms, suited
 +
more to the multiplication of matrices "on the left".
 +
 +
So I still have a few wrinkles to iron out before
 +
I can give this story a smooth enough consistency.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 15
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
I have been planning for quite some time now to make my return to Peirce's
 +
skyshaking "Description of a Notation for the Logic of Relatives" (1870),
 +
and I can see that it's just about time to get down tuit, so let this
 +
current bit of rambling inquiry function as the preamble to that.
 +
All we need at the present, though, is a modus vivendi/operandi
 +
for telling what is substantial from what is inessential in
 +
the brook between symbolic conceits and dramatic actions
 +
that we find afforded by means of the pragmatic maxim.
 +
 +
Back to our "subinstance", the example in support of our first example.
 +
I will now reconstruct it in a way that may prove to be less confusing.
 +
 +
Let us make up the model universe $1$ = A + B + C and the 2-adic relation
 +
n = "noder of", as when "X is a data record that contains a pointer to Y".
 +
That interpretation is not important, it's just for the sake of intuition.
 +
In general terms, the 2-adic relation n can be represented by this matrix:
 +
 +
n  =
 +
 +
|  n_AA (A:A)  n_AB (A:B)  n_AC (A:C)  |
 +
|                                        |
 +
|  n_BA (B:A)  n_BB (B:B)  n_BC (B:C)  |
 +
|                                        |
 +
|  n_CA (C:A)  n_CB (C:B)  n_CC (C:C)  |
 +
 +
Also, let n be such that
 +
A is a noder of A and B,
 +
B is a noder of B and C,
 +
C is a noder of C and A.
 +
 +
Filling in the instantial values of the "coefficients" n_ij,
 +
as the indices i and j range over the universe of discourse:
 +
 +
n  =
 +
 +
|  1 · (A:A)  1 · (A:B)  0 · (A:C)  |
 +
|                                    |
 +
|  0 · (B:A)  1 · (B:B)  1 · (B:C)  |
 +
|                                    |
 +
|  1 · (C:A)  0 · (C:B)  1 · (C:C)  |
 +
 +
In Peirce's time, and even in some circles of mathematics today,
 +
the information indicated by the elementary relatives (I:J), as
 +
I, J range over the universe of discourse, would be referred to
 +
as the "umbral elements" of the algebraic operation represented
 +
by the matrix, though I seem to recall that Peirce preferred to
 +
call these terms the "ingredients".  When this ordered basis is
 +
understood well enough, one will tend to drop any mention of it
 +
from the matrix itself, leaving us nothing but these bare bones:
 +
 +
n  =
 +
 +
|  1  1  0  |
 +
|          |
 +
|  0  1  1  |
 +
|          |
 +
|  1  0  1  |
 +
 +
However the specification may come to be written, this
 +
is all just convenient schematics for stipulating that:
 +
 +
n  =  A:A  +  B:B  +  C:C  +  A:B  +  B:C  +  C:A
 +
 +
Recognizing !1! = A:A + B:B + C:C to be the identity transformation,
 +
the 2-adic relation n = "noder of" may be represented by an element
 +
!1! + A:B + B:C + C:A of the so-called "group ring", all of which
 +
just makes this element a special sort of linear transformation.
 +
 +
Up to this point, we are still reading the elementary relatives of
 +
the form I:J in the way that Peirce reads them in logical contexts:
 +
I is the relate, J is the correlate, and in our current example we
 +
read I:J, or more exactly, n_ij = 1, to say that I is a noder of J.
 +
This is the mode of reading that we call "multiplying on the left".
 +
 +
In the algebraic, permutational, or transformational contexts of
 +
application, however, Peirce converts to the alternative mode of
 +
reading, although still calling I the relate and J the correlate,
 +
the elementary relative I:J now means that I gets changed into J.
 +
In this scheme of reading, the transformation A:B + B:C + C:A is
 +
a permutation of the aggregate $1$ = A + B + C, or what we would
 +
now call the set {A, B, C}, in particular, it is the permutation
 +
that is otherwise notated as:
 +
 +
( A B C )
 +
<      >
 +
( B C A )
 +
 +
This is consistent with the convention that Peirce uses in
 +
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 16
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
We have been contemplating the virtues and the utilities of
 +
the pragmatic maxim as a hermeneutic heuristic, specifically,
 +
as a principle of interpretation that guides us in finding a
 +
clarifying representation for a problematic corpus of symbols
 +
in terms of their actions on other symbols or their effects on
 +
the syntactic contexts in which we conceive to distribute them.
 +
I started off considering the regular representations of groups
 +
as constituting what appears to be one of the simplest possible
 +
applications of this overall principle of representation.
 +
 +
There are a few problems of implementation that have to be worked out
 +
in practice, most of which are cleared up by keeping in mind which of
 +
several possible conventions we have chosen to follow at a given time.
 +
But there does appear to remain this rather more substantial question:
 +
 +
Are the effects we seek relates or correlates, or does it even matter?
 +
 +
I will have to leave that question as it is for now,
 +
in hopes that a solution will evolve itself in time.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 17
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
There a big reasons and little reasons for caring about this humble example.
 +
The little reasons we find all under our feet.  One big reason I can now
 +
quite blazonly enounce in the fashion of this not so subtle subtitle:
 +
 +
Obstacles to Applying the Pragmatic Maxim
 +
 +
No sooner do you get a good idea and try to apply it
 +
than you find that a motley array of obstacles arise.
 +
 +
It seems as if I am constantly lamenting the fact these days that people,
 +
and even admitted Peircean persons, do not in practice more consistently
 +
apply the maxim of pragmatism to the purpose for which it is purportedly
 +
intended by its author.  That would be the clarification of concepts, or
 +
intellectual symbols, to the point where their inherent senses, or their
 +
lacks thereof, would be rendered manifest to all and sundry interpreters.
 +
 +
There are big obstacles and little obstacles to applying the pragmatic maxim.
 +
In good subgoaling fashion, I will merely mention a few of the bigger blocks,
 +
as if in passing, and then get down to the devilish details that immediately
 +
obstruct our way.
 +
 +
Obstacle 1.  People do not always read the instructions very carefully.
 +
There is a tendency in readers of particular prior persuasions to blow
 +
the problem all out of proportion, to think that the maxim is meant to
 +
reveal the absolutely positive and the totally unique meaning of every
 +
preconception to which they might deign or elect to apply it.  Reading
 +
the maxim with an even minimal attention, you can see that it promises
 +
no such finality of unindexed sense, but ties what you conceive to you.
 +
I have lately come to wonder at the tenacity of this misinterpretation.
 +
Perhaps people reckon that nothing less would be worth their attention.
 +
I am not sure.  I can only say the achievement of more modest goals is
 +
the sort of thing on which our daily life depends, and there can be no
 +
final end to inquiry nor any ultimate community without a continuation
 +
of life, and that means life on a day to day basis.  All of which only
 +
brings me back to the point of persisting with local meantime examples,
 +
because if we can't apply the maxim there, we can't apply it anywhere.
 +
 +
And now I need to go out of doors and weed my garden for a time ...
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 18
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
Obstacles to Applying the Pragmatic Maxim
 +
 +
Obstacle 2.  Applying the pragmatic maxim, even with a moderate aim, can be hard.
 +
I think that my present example, deliberately impoverished as it is, affords us
 +
with an embarassing richness of evidence of just how complex the simple can be.
 +
 +
All the better reason for me to see if I can finish it up before moving on.
 +
 +
Expressed most simply, the idea is to replace the question of "what it is",
 +
which modest people know is far too difficult for them to answer right off,
 +
with the question of "what it does", which most of us know a modicum about.
 +
 +
In the case of regular representations of groups we found
 +
a non-plussing surplus of answers to sort our way through.
 +
So let us track back one more time to see if we can learn
 +
any lessons that might carry over to more realistic cases.
 +
 +
Here is is the operation table of V_4 once again:
 +
 +
Table 1.  Klein Four-Group V_4
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    ·    %    e    |    f    |    g    |    h    |
 +
|        %        |        |        |        |
 +
o=========o=========o=========o=========o=========o
 +
|        %        |        |        |        |
 +
|    e    %    e    |    f    |    g    |    h    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    f    %    f    |    e    |    h    |    g    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    g    %    g    |    h    |    e    |    f    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
|        %        |        |        |        |
 +
|    h    %    h    |    g    |    f    |    e    |
 +
|        %        |        |        |        |
 +
o---------o---------o---------o---------o---------o
 +
 +
A group operation table is really just a device for
 +
recording a certain 3-adic relation, to be specific,
 +
the set of triples of the form <x, y, z> satisfying
 +
the equation x·y = z where · is the group operation.
 +
 +
In the case of V_4 = (G, ·), where G is the "underlying set"
 +
{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
 +
whose triples are listed below:
 +
 +
|  <e, e, e>
 +
|  <e, f, f>
 +
|  <e, g, g>
 +
|  <e, h, h>
 +
|
 +
|  <f, e, f>
 +
|  <f, f, e>
 +
|  <f, g, h>
 +
|  <f, h, g>
 +
|
 +
|  <g, e, g>
 +
|  <g, f, h>
 +
|  <g, g, e>
 +
|  <g, h, f>
 +
|
 +
|  <h, e, h>
 +
|  <h, f, g>
 +
|  <h, g, f>
 +
|  <h, h, e>
 +
 +
It is part of the definition of a group that the 3-adic
 +
relation L c G^3 is actually a function L : G x G -> G.
 +
It is from this functional perspective that we can see
 +
an easy way to derive the two regular representations.
 +
Since we have a function of the type L : G x G -> G,
 +
we can define a couple of substitution operators:
 +
 +
1.  Sub(x, <_, y>) puts any specified x into
 +
    the empty slot of the rheme <_, y>, with
 +
    the effect of producing the saturated
 +
    rheme <x, y> that evaluates to x·y.
 +
 +
2.  Sub(x, <y, _>) puts any specified x into
 +
    the empty slot of the rheme <y, >, with
 +
    the effect of producing the saturated
 +
    rheme <y, x> that evaluates to y·x.
 +
 +
In (1), we consider the effects of each x in its
 +
practical bearing on contexts of the form <_, y>,
 +
as y ranges over G, and the effects are such that
 +
x takes <_, y> into x·y, for y in G, all of which
 +
is summarily notated as x = {(y : x·y) : y in G}.
 +
The pairs (y : x·y) can be found by picking an x
 +
from the left margin of the group operation table
 +
and considering its effects on each y in turn as
 +
these run across the top margin.  This aspect of
 +
pragmatic definition we recognize as the regular
 +
ante-representation:
 +
 +
    e  =  e:e  +  f:f  +  g:g  +  h:h
 +
 +
    f  =  e:f  +  f:e  +  g:h  +  h:g
 +
 +
    g  =  e:g  +  f:h  +  g:e  +  h:f
 +
 +
    h  =  e:h  +  f:g  +  g:f  +  h:e
 +
 +
In (2), we consider the effects of each x in its
 +
practical bearing on contexts of the form <y, _>,
 +
as y ranges over G, and the effects are such that
 +
x takes <y, _> into y·x, for y in G, all of which
 +
is summarily notated as x = {(y : y·x) : y in G}.
 +
The pairs (y : y·x) can be found by picking an x
 +
from the top margin of the group operation table
 +
and considering its effects on each y in turn as
 +
these run down the left margin.  This aspect of
 +
pragmatic definition we recognize as the regular
 +
post-representation:
 +
 +
    e  =  e:e  +  f:f  +  g:g  +  h:h
 +
 +
    f  =  e:f  +  f:e  +  g:h  +  h:g
 +
 +
    g  =  e:g  +  f:h  +  g:e  +  h:f
 +
 +
    h  =  e:h  +  f:g  +  g:f  +  h:e
 +
 +
If the ante-rep looks the same as the post-rep,
 +
now that I'm writing them in the same dialect,
 +
that is because V_4 is abelian (commutative),
 +
and so the two representations have the very
 +
same effects on each point of their bearing.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 19
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
So long as we're in the neighborhood, we might as well take in
 +
some more of the sights, for instance, the smallest example of
 +
a non-abelian (non-commutative) group.  This is a group of six
 +
elements, say, G = {e, f, g, h, i, j}, with no relation to any
 +
other employment of these six symbols being implied, of course,
 +
and it can be most easily represented as the permutation group
 +
on a set of three letters, say, X = {A, B, C}, usually notated
 +
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
 +
Here are the permutation (= substitution) operations in Sym(X):
 +
 +
Table 2.  Permutations or Substitutions in Sym_{A, B, C}
 +
o---------o---------o---------o---------o---------o---------o
 +
|        |        |        |        |        |        |
 +
|    e    |    f    |    g    |    h    |    i    |    j    |
 +
|        |        |        |        |        |        |
 +
o=========o=========o=========o=========o=========o=========o
 +
|        |        |        |        |        |        |
 +
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
 +
|        |        |        |        |        |        |
 +
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
 +
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
 +
|        |        |        |        |        |        |
 +
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
 +
|        |        |        |        |        |        |
 +
o---------o---------o---------o---------o---------o---------o
 +
 +
Here is the operation table for S_3, given in abstract fashion:
 +
 +
Table 3.  Symmetric Group S_3
 +
 +
|                        _
 +
|                    e / \ e
 +
|                      /  \
 +
|                    /  e  \
 +
|                  f / \  / \ f
 +
|                  /  \ /  \
 +
|                  /  f  \  f  \
 +
|              g / \  / \  / \ g
 +
|                /  \ /  \ /  \
 +
|              /  g  \  g  \  g  \
 +
|            h / \  / \  / \  / \ h
 +
|            /  \ /  \ /  \ /  \
 +
|            /  h  \  e  \  e  \  h  \
 +
|        i / \  / \  / \  / \  / \ i
 +
|          /  \ /  \ /  \ /  \ /  \
 +
|        /  i  \  i  \  f  \  j  \  i  \
 +
|      j / \  / \  / \  / \  / \  / \ j
 +
|      /  \ /  \ /  \ /  \ /  \ /  \
 +
|      (  j  \  j  \  j  \  i  \  h  \  j  )
 +
|      \  / \  / \  / \  / \  / \  /
 +
|        \ /  \ /  \ /  \ /  \ /  \ /
 +
|        \  h  \  h  \  e  \  j  \  i  /
 +
|          \  / \  / \  / \  / \  /
 +
|          \ /  \ /  \ /  \ /  \ /
 +
|            \  i  \  g  \  f  \  h  /
 +
|            \  / \  / \  / \  /
 +
|              \ /  \ /  \ /  \ /
 +
|              \  f  \  e  \  g  /
 +
|                \  / \  / \  /
 +
|                \ /  \ /  \ /
 +
|                  \  g  \  f  /
 +
|                  \  / \  /
 +
|                    \ /  \ /
 +
|                    \  e  /
 +
|                      \  /
 +
|                      \ /
 +
|                        ¯
 +
 +
By the way, we will meet with the symmetric group S_3 again
 +
when we return to take up the study of Peirce's early paper
 +
"On a Class of Multiple Algebras" (CP 3.324-327), and also
 +
his late unpublished work "The Simplest Mathematics" (1902)
 +
(CP 4.227-323), with particular reference to the section
 +
that treats of "Trichotomic Mathematics" (CP 4.307-323).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Work Area
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 20
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
By way of collecting a shot-term pay-off for all the work --
 +
not to mention the peirce-spiration -- that we sweated out
 +
over the regular representations of V_4 and S_3
 +
 +
Table 2.  Permutations or Substitutions in Sym_{A, B, C}
 +
o---------o---------o---------o---------o---------o---------o
 +
|        |        |        |        |        |        |
 +
|    e    |    f    |    g    |    h    |    i    |    j    |
 +
|        |        |        |        |        |        |
 +
o=========o=========o=========o=========o=========o=========o
 +
|        |        |        |        |        |        |
 +
|  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |  A B C  |
 +
|        |        |        |        |        |        |
 +
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
 +
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
 +
|        |        |        |        |        |        |
 +
|  A B C  |  C A B  |  B C A  |  A C B  |  C B A  |  B A C  |
 +
|        |        |        |        |        |        |
 +
o---------o---------o---------o---------o---------o---------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Note 21
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might 'conceivably'
 +
| have practical bearings you 'conceive' the
 +
| objects of your 'conception' to have.  Then,
 +
| your 'conception' of those effects is the
 +
| whole of your 'conception' of the object.
 +
|
 +
| Charles Sanders Peirce,
 +
| "Maxim of Pragmaticism", CP 5.438.
 +
 +
problem about writing
 +
 +
  e  =  e:e  +  f:f  +  g:g  +  h:h
 +
 +
no recursion intended
 +
need for a work-around
 +
ways way explaining it away
 +
 +
action on signs not objects
 +
 +
math def of rep
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Zeroth Order Logic
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Here is a scaled-down version of one of my very first applications,
 +
having to do with the demographic variables in a survey data base.
 +
 +
This Example illustrates the use of 2-variate logical forms
 +
for expressing and reasoning about the logical constraints
 +
that are involved in the following types of situations:
 +
 +
1.  Distinction:    A =/= B
 +
    Also known as:  logical inequality, exclusive disjunction
 +
    Represented as:  ( A , B )
 +
    Graphed as:
 +
    |
 +
    |  A  B
 +
    |  o---o
 +
    |    \ /
 +
    |    @
 +
 +
2.  Equality:        A = B
 +
    Also known as:  logical equivalence, if and only if, A <=> B
 +
    Represented as:  (( A , B ))
 +
    Graphed as:
 +
    |
 +
    |  A  B
 +
    |  o---o
 +
    |    \ /
 +
    |    o
 +
    |    |
 +
    |    @
 +
 +
3.  Implication:    A => B
 +
    Also known as:  entailment, if-then
 +
    Represented as:  ( A ( B ))
 +
    Graphed as:
 +
    |
 +
    |  A  B
 +
    |  o---o
 +
    |  |
 +
    |  @
 +
 +
Example of a proposition expressing a "zeroth order theory" (ZOT):
 +
 +
Consider the following text, written in what I am calling "Ref Log",
 +
also known as the "Cactus Language" synpropositional logic:
 +
 +
|  ( male  , female )
 +
|  (( boy  , male child ))
 +
|  (( girl , female child ))
 +
|  ( child ( human ))
 +
 +
Graphed as:
 +
 +
|                  boy  male    girl  female
 +
|                    o---o child    o---o child
 +
|  male  female      \ /            \ /          child  human
 +
|    o---o            o              o              o---o
 +
|      \ /              |              |              |
 +
|      @              @              @              @|
 +
 +
Nota Bene.  Due to graphic constraints -- no, the other
 +
kind of graphic constraints -- of the immediate medium,
 +
I am forced to string out the logical conjuncts of the
 +
actual cactus graph for this situation, one that might
 +
sufficiently be reasoned out from the exhibit supra by
 +
fusing together the four roots of the severed cactus.
 +
 +
Either of these expressions, text or graph, is equivalent to
 +
what would otherwise be written in a more ordinary syntax as:
 +
 +
|  male  =/=  female
 +
|  boy  <=>  male child
 +
|  girl  <=>  female child
 +
|  child  =>  human
 +
 +
This is a actually a single proposition, a conjunction of four lines:
 +
one distinction, two equations, and one implication.  Together these
 +
amount to a set of definitions conjointly constraining the logical
 +
compatibility of the six feature names that appear.  They may be
 +
thought of as sculpting out a space of models that is some subset
 +
of the 2^6 = 64 possible interpretations, and thereby shaping some
 +
universe of discourse.
 +
 +
Once this backdrop is defined, it is possible to "query" this universe,
 +
simply by conjoining additional propositions in further constraint of
 +
the underlying set of models.  This has many uses, as we shall see.
 +
 +
We are considering an Example of a propositional expression
 +
that is formed on the following "alphabet" or "lexicon" of
 +
six "logical features" or "boolean variables":
 +
 +
$A$  =  {"boy", "child", "female", "girl", "human", "male"}.
 +
 +
The expression is this:
 +
 +
|  ( male  , female )
 +
|  (( boy  , male child ))
 +
|  (( girl , female child ))
 +
|  ( child ( human ))
 +
 +
Putting it very roughly -- and putting off a better description
 +
of it till later -- we may think of this expression as notation
 +
for a boolean function f : %B%^6 -> %B%.  This is what we might
 +
call the "abstract type" of the function, but we will also find
 +
it convenient on many occasions to represent the points of this
 +
particular copy of the space %B%^6 in terms of the positive and
 +
negative versions of the features from $A$ that serve to encase
 +
them as logical "cells", as they are called in the venn diagram
 +
picture of the corresponding universe of discourse X = [$A$].
 +
 +
Just for concreteness, this form of representation begins and ends:
 +
 +
<0,0,0,0,0,0>  =  (boy)(child)(female)(girl)(human)(male),
 +
<0,0,0,0,0,1>  =  (boy)(child)(female)(girl)(human) male ,
 +
<0,0,0,0,1,0>  =  (boy)(child)(female)(girl) human (male),
 +
<0,0,0,0,1,1>  =  (boy)(child)(female)(girl) human  male ,
 +
...
 +
<1,1,1,1,0,0>  =  boy  child  female  girl (human)(male),
 +
<1,1,1,1,0,1>  =  boy  child  female  girl (human) male ,
 +
<1,1,1,1,1,0>  =  boy  child  female  girl  human (male),
 +
<1,1,1,1,1,1>  =  boy  child  female  girl  human  male .
 +
 +
I continue with the previous Example, that I bring forward and sum up here:
 +
 +
|                  boy  male          girl  female
 +
|                    o---o child          o---o child
 +
|  male  female      \ /                  \ /              child  human
 +
|    o---o            o                    o                    o--o
 +
|      \ /            |                    |                    |
 +
|      @              @                    @                    @
 +
|
 +
| (male , female)((boy , male child))((girl , female child))(child (human))
 +
 +
For my master's piece in Quantitative Psychology (Michigan State, 1989),
 +
I wrote a program, "Theme One" (TO) by name, that among its other duties
 +
operates to process the expressions of the cactus language in many of the
 +
most pressing ways that we need in order to be able to use it effectively
 +
as a propositional calculus.  The operational component of TO where one
 +
does the work of this logical modeling is called "Study", and the core
 +
of the logical calculator deep in the heart of this Study section is
 +
a suite of computational functions that evolve a particular species
 +
of "normal form", analogous to a "disjunctive normal form" (DNF),
 +
from whatever expression they are prebendered as their input.
 +
 +
This "canonical", "normal", or "stable" form of logical expression --
 +
I'll refine the distinctions among these subforms all in good time --
 +
permits succinct depiction as an "arboreal boolean expansion" (ABE).
 +
 +
Once again, the graphic limitations of this space prevail against
 +
any disposition that I might have to lay out a really substantial
 +
case before you, of the brand that might have a chance to impress
 +
you with the aptitude of this ilk of ABE in rooting out the truth
 +
of many a complexly obscurely subtly adamant whetstone of our wit.
 +
 +
So let me just illustrate the way of it with one conjunct of our Example.
 +
What follows will be a sequence of expressions, each one after the first
 +
being logically equal to the one that precedes it:
 +
 +
Step 1
 +
 +
|    g    fc
 +
|    o---o
 +
|      \ /
 +
|      o
 +
|      |
 +
|      @
 +
 +
Step 2
 +
 +
|                  o
 +
|        fc        |  fc
 +
|    o---o        o---o
 +
|      \ /          \ /
 +
|      o            o
 +
|      |            |
 +
|    g o-------------o--o g
 +
|        \          /
 +
|        \        /
 +
|          \      /
 +
|          \    /
 +
|            \  /
 +
|            \ /
 +
|              @
 +
 +
Step 3
 +
 +
|      f c
 +
|      o
 +
|      |            f c
 +
|      o            o
 +
|      |            |
 +
|    g o-------------o--o g
 +
|        \          /
 +
|        \        /
 +
|          \      /
 +
|          \    /
 +
|            \  /
 +
|            \ /
 +
|              @
 +
 +
Step 4
 +
 +
|          o
 +
|          |
 +
|    c o  o c          o
 +
|      |  |            |
 +
|      o  o      c o  o c
 +
|      |  |        |  |
 +
|    f o---o--o f  f o---o--o f
 +
|        \ /          \ /
 +
|      g o-------------o--o g
 +
|          \          /
 +
|          \        /
 +
|            \      /
 +
|            \    /
 +
|              \  /
 +
|              \ /
 +
|                @
 +
 +
Step 5
 +
 +
|          o      c o
 +
|      c  |        |
 +
|    f o---o--o f  f o---o--o f
 +
|        \ /          \ /
 +
|      g o-------------o--o g
 +
|          \          /
 +
|          \        /
 +
|            \      /
 +
|            \    /
 +
|              \  /
 +
|              \ /
 +
|                @
 +
 +
Step 6
 +
 +
|                                      o
 +
|                                      |
 +
|          o                      o  o
 +
|          |                      |  |
 +
|    c o---o--o c      o        c o---o--o c
 +
|        \ /            |            \ /
 +
|      f o-------------o--o f      f o-------------o--o f
 +
|          \          /              \          /
 +
|          \        /                \        /
 +
|            \      /                  \      /
 +
|            \    /                    \    /
 +
|              \  /                      \  /
 +
|              \ /                        \ /
 +
|              g o---------------------------o--o g
 +
|                \                        /
 +
|                  \                      /
 +
|                  \                    /
 +
|                    \                  /
 +
|                    \                /
 +
|                      \              /
 +
|                      \            /
 +
|                        \          /
 +
|                        \        /
 +
|                          \      /
 +
|                          \    /
 +
|                            \  /
 +
|                            \ /
 +
|                              @
 +
 +
Step 7
 +
 +
|          o                      o
 +
|          |                      |
 +
|    c o---o--o c      o        c o---o--o c
 +
|        \ /            |            \ /
 +
|      f o-------------o--o f      f o-------------o--o f
 +
|          \          /              \          /
 +
|          \        /                \        /
 +
|            \      /                  \      /
 +
|            \    /                    \    /
 +
|              \  /                      \  /
 +
|              \ /                        \ /
 +
|              g o---------------------------o--o g
 +
|                \                        /
 +
|                  \                      /
 +
|                  \                    /
 +
|                    \                  /
 +
|                    \                /
 +
|                      \              /
 +
|                      \            /
 +
|                        \          /
 +
|                        \        /
 +
|                          \      /
 +
|                          \    /
 +
|                            \  /
 +
|                            \ /
 +
|                              @
 +
 +
This last expression is the ABE of the input expression.
 +
It can be transcribed into ordinary logical language as:
 +
 +
| either girl and
 +
|        either female and
 +
|              either child and true
 +
|              or not child and false
 +
|        or not female and false
 +
| or not girl and
 +
|        either female and
 +
|              either child and false
 +
|              or not child and true
 +
|        or not female and true
 +
 +
The expression "((girl , female child))" is sufficiently evaluated
 +
by considering its logical values on the coordinate tuples of %B%^3,
 +
or its indications on the cells of the associated venn diagram that
 +
depicts the universe of discourse, namely, on these eight arguments:
 +
     
 +
<1, 1, 1>  =  girl  female  child ,
 +
<1, 1, 0>  =  girl  female (child),
 +
<1, 0, 1>  =  girl (female) child ,
 +
<1, 0, 0>  =  girl (female)(child),
 +
<0, 1, 1>  =  (girl) female  child ,
 +
<0, 1, 0>  =  (girl) female (child),
 +
<0, 0, 1>  =  (girl)(female) child ,
 +
<0, 0, 0>  =  (girl)(female)(child).
 +
 +
The ABE output expression tells us the logical values of
 +
the input expression on each of these arguments, doing so
 +
by attaching the values to the leaves of a tree, and acting
 +
as an "efficient" or "lazy" evaluator in the sense that the
 +
process that generates the tree follows each path only up to
 +
the point in the tree where it can determine the values on the
 +
entire subtree beyond that point.  Thus, the ABE tree tells us:
 +
 +
girl  female  child  -> 1
 +
girl  female (child)  -> 0
 +
girl (female) -> 0
 +
(girl) female  child  -> 0
 +
(girl) female (child)  -> 1
 +
(girl)(female) -> 1
 +
 +
Picking out the interpretations that yield the truth of the expression,
 +
and expanding the corresponding partial argument tuples, we arrive at
 +
the following interpretations that satisfy the input expression:
 +
 +
girl  female  child  -> 1
 +
(girl) female (child)  -> 1
 +
(girl)(female) child  -> 1
 +
(girl)(female)(child)  -> 1
 +
 +
In sum, if it's a female and a child, then it's a girl,
 +
and if it's either not a female or not a child or both,
 +
then it's not a girl.
 +
 +
Brief Automata
 +
 +
By way of providing a simple illustration of Cook's Theorem,
 +
that "Propositional Satisfiability is NP-Complete", here is
 +
an exposition of one way to translate Turing Machine set-ups
 +
into propositional expressions, employing the Ref Log Syntax
 +
for Prop Calc that I described in a couple of earlier notes:
 +
 +
Notation:
 +
 +
Stilt(k)  =  Space and Time Limited Turing Machine,
 +
            with k units of space and k units of time.
 +
 +
Stunt(k)  =  Space and Time Limited Turing Machine,
 +
            for computing the parity of a bit string,
 +
            with Number of Tape cells of input equal to k.
 +
 +
I will follow the pattern of the discussion in the book of
 +
Herbert Wilf, 'Algorithms & Complexity' (1986), pages 188-201,
 +
but translate into Ref Log, which is more efficient with respect
 +
to the number of propositional clauses that are required.
 +
 +
Parity Machine
 +
 +
|                    1/1/+1
 +
|                  ------->
 +
|              /\ /        \ /\
 +
|      0/0/+1  ^  0          1  ^  0/0/+1
 +
|              \/|\        /|\/
 +
|                | <------- |
 +
|        #/#/-1  |  1/1/+1  |  #/#/-1
 +
|                |          |
 +
|                v          v
 +
|                #          *
 +
 +
o-------o--------o-------------o---------o------------o
 +
| State | Symbol | Next Symbol | Ratchet | Next State |
 +
|  Q  |  S    |    S'      |  dR    |    Q'    |
 +
o-------o--------o-------------o---------o------------o
 +
|  0  |  0    |    0      |  +1    |    0      |
 +
|  0  |  1    |    1      |  +1    |    1      |
 +
|  0  |  #    |    #      |  -1    |    #      |
 +
|  1  |  0    |    0      |  +1    |    1      |
 +
|  1  |  1    |    1      |  +1    |    0      |
 +
|  1  |  #    |    #      |  -1    |    *      |
 +
o-------o--------o-------------o---------o------------o
 +
 +
The TM has a "finite automaton" (FA) as its component.
 +
Let us refer to this particular FA by the name of "M".
 +
 +
The "tape-head" (that is, the "read-unit") will be called "H".
 +
The "registers" are also called "tape-cells" or "tape-squares".
 +
 +
In order to consider how the finitely "stilted" rendition of this TM
 +
can be translated into the form of a purely propositional description,
 +
one now fixes k and limits the discussion to talking about a Stilt(k),
 +
which is really not a true TM anymore but a finite automaton in disguise.
 +
 +
In this example, for the sake of a minimal illustration, we choose k = 2,
 +
and discuss Stunt(2).  Since the zeroth tape cell and the last tape cell
 +
are occupied with bof and eof marks "#", this amounts to only one digit
 +
of significant computation.
 +
 +
To translate Stunt(2) into propositional form we use
 +
the following collection of propositional variables:
 +
 +
For the "Present State Function" QF : P -> Q,
 +
 +
{p0_q#, p0_q*, p0_q0, p0_q1,
 +
p1_q#, p1_q*, p1_q0, p1_q1,
 +
p2_q#, p2_q*, p2_q0, p2_q1,
 +
p3_q#, p3_q*, p3_q0, p3_q1}
 +
 +
The propositional expression of the form "pi_qj" says:
 +
 +
| At the point-in-time p_i,
 +
| the finite machine M is in the state q_j.
 +
 +
For the "Present Register Function" RF : P -> R,
 +
 +
{p0_r0, p0_r1, p0_r2, p0_r3,
 +
p1_r0, p1_r1, p1_r2, p1_r3,
 +
p2_r0, p2_r1, p2_r2, p2_r3,
 +
p3_r0, p3_r1, p3_r2, p3_r3}
 +
 +
The propositional expression of the form "pi_rj" says:
 +
 +
| At the point-in-time p_i,
 +
| the tape-head H is on the tape-cell r_j.
 +
 +
For the "Present Symbol Function" SF : P -> (R -> S),
 +
 +
{p0_r0_s#, p0_r0_s*, p0_r0_s0, p0_r0_s1,
 +
p0_r1_s#, p0_r1_s*, p0_r1_s0, p0_r1_s1,
 +
p0_r2_s#, p0_r2_s*, p0_r2_s0, p0_r2_s1,
 +
p0_r3_s#, p0_r3_s*, p0_r3_s0, p0_r3_s1,
 +
p1_r0_s#, p1_r0_s*, p1_r0_s0, p1_r0_s1,
 +
p1_r1_s#, p1_r1_s*, p1_r1_s0, p1_r1_s1,
 +
p1_r2_s#, p1_r2_s*, p1_r2_s0, p1_r2_s1,
 +
p1_r3_s#, p1_r3_s*, p1_r3_s0, p1_r3_s1,
 +
p2_r0_s#, p2_r0_s*, p2_r0_s0, p2_r0_s1,
 +
p2_r1_s#, p2_r1_s*, p2_r1_s0, p2_r1_s1,
 +
p2_r2_s#, p2_r2_s*, p2_r2_s0, p2_r2_s1,
 +
p2_r3_s#, p2_r3_s*, p2_r3_s0, p2_r3_s1,
 +
p3_r0_s#, p3_r0_s*, p3_r0_s0, p3_r0_s1,
 +
p3_r1_s#, p3_r1_s*, p3_r1_s0, p3_r1_s1,
 +
p3_r2_s#, p3_r2_s*, p3_r2_s0, p3_r2_s1,
 +
p3_r3_s#, p3_r3_s*, p3_r3_s0, p3_r3_s1}
 +
 +
The propositional expression of the form "pi_rj_sk" says:
 +
 +
| At the point-in-time p_i,
 +
| the tape-cell r_j bears the mark s_k.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~INPUTS~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Here are the Initial Conditions
 +
for the two possible inputs to the
 +
Ref Log redaction of this Parity TM:
 +
 +
o~~~~~~~~~o~~~~~~~~~o~INPUT~0~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Initial Conditions:
 +
 +
p0_q0
 +
 +
p0_r1
 +
 +
p0_r0_s#
 +
p0_r1_s0
 +
p0_r2_s#
 +
 +
The Initial Conditions are given by a logical conjunction
 +
that is composed of 5 basic expressions, altogether stating:
 +
 +
| At the point-in-time p_0, M is in the state q_0, and
 +
| At the point-in-time p_0, H is on the cell  r_1, and
 +
| At the point-in-time p_0, cell r_0 bears the mark "#", and
 +
| At the point-in-time p_0, cell r_1 bears the mark "0", and
 +
| At the point-in-time p_0, cell r_2 bears the mark "#".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~INPUT~1~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Initial Conditions:
 +
 +
p0_q0
 +
 +
p0_r1
 +
 +
p0_r0_s#
 +
p0_r1_s1
 +
p0_r2_s#
 +
 +
The Initial Conditions are given by a logical conjunction
 +
that is composed of 5 basic expressions, altogether stating:
 +
 +
| At the point-in-time p_0, M is in the state q_0, and
 +
| At the point-in-time p_0, H is on the cell  r_1, and
 +
| At the point-in-time p_0, cell r_0 bears the mark "#", and
 +
| At the point-in-time p_0, cell r_1 bears the mark "1", and
 +
| At the point-in-time p_0, cell r_2 bears the mark "#".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~PROGRAM~o~~~~~~~~~o~~~~~~~~~o
 +
 +
And here, yet again, just to store it nearby,
 +
is the logical rendition of the TM's program:
 +
 +
Mediate Conditions:
 +
 +
( p0_q#  ( p1_q# ))
 +
( p0_q*  ( p1_q* ))
 +
 +
( p1_q#  ( p2_q# ))
 +
( p1_q*  ( p2_q* ))
 +
 +
Terminal Conditions:
 +
 +
(( p2_q# )( p2_q* ))
 +
 +
State Partition:
 +
 +
(( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* ))
 +
(( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* ))
 +
(( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* ))
 +
 +
Register Partition:
 +
 +
(( p0_r0 ),( p0_r1 ),( p0_r2 ))
 +
(( p1_r0 ),( p1_r1 ),( p1_r2 ))
 +
(( p2_r0 ),( p2_r1 ),( p2_r2 ))
 +
 +
Symbol Partition:
 +
 +
(( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# ))
 +
(( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# ))
 +
(( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# ))
 +
 +
(( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# ))
 +
(( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# ))
 +
(( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# ))
 +
 +
(( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# ))
 +
(( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# ))
 +
(( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# ))
 +
 +
Interaction Conditions:
 +
 +
(( p0_r0 ) p0_r0_s0 ( p1_r0_s0 ))
 +
(( p0_r0 ) p0_r0_s1 ( p1_r0_s1 ))
 +
(( p0_r0 ) p0_r0_s# ( p1_r0_s# ))
 +
 +
(( p0_r1 ) p0_r1_s0 ( p1_r1_s0 ))
 +
(( p0_r1 ) p0_r1_s1 ( p1_r1_s1 ))
 +
(( p0_r1 ) p0_r1_s# ( p1_r1_s# ))
 +
 +
(( p0_r2 ) p0_r2_s0 ( p1_r2_s0 ))
 +
(( p0_r2 ) p0_r2_s1 ( p1_r2_s1 ))
 +
(( p0_r2 ) p0_r2_s# ( p1_r2_s# ))
 +
 +
(( p1_r0 ) p1_r0_s0 ( p2_r0_s0 ))
 +
(( p1_r0 ) p1_r0_s1 ( p2_r0_s1 ))
 +
(( p1_r0 ) p1_r0_s# ( p2_r0_s# ))
 +
 +
(( p1_r1 ) p1_r1_s0 ( p2_r1_s0 ))
 +
(( p1_r1 ) p1_r1_s1 ( p2_r1_s1 ))
 +
(( p1_r1 ) p1_r1_s# ( p2_r1_s# ))
 +
 +
(( p1_r2 ) p1_r2_s0 ( p2_r2_s0 ))
 +
(( p1_r2 ) p1_r2_s1 ( p2_r2_s1 ))
 +
(( p1_r2 ) p1_r2_s# ( p2_r2_s# ))
 +
 +
Transition Relations:
 +
 +
( p0_q0  p0_r1  p0_r1_s0  ( p1_q0  p1_r2  p1_r1_s0 ))
 +
( p0_q0  p0_r1  p0_r1_s1  ( p1_q1  p1_r2  p1_r1_s1 ))
 +
( p0_q0  p0_r1  p0_r1_s#  ( p1_q#  p1_r0  p1_r1_s# ))
 +
( p0_q0  p0_r2  p0_r2_s#  ( p1_q#  p1_r1  p1_r2_s# ))
 +
 +
( p0_q1  p0_r1  p0_r1_s0  ( p1_q1  p1_r2  p1_r1_s0 ))
 +
( p0_q1  p0_r1  p0_r1_s1  ( p1_q0  p1_r2  p1_r1_s1 ))
 +
( p0_q1  p0_r1  p0_r1_s#  ( p1_q*  p1_r0  p1_r1_s# ))
 +
( p0_q1  p0_r2  p0_r2_s#  ( p1_q*  p1_r1  p1_r2_s# ))
 +
 +
( p1_q0  p1_r1  p1_r1_s0  ( p2_q0  p2_r2  p2_r1_s0 ))
 +
( p1_q0  p1_r1  p1_r1_s1  ( p2_q1  p2_r2  p2_r1_s1 ))
 +
( p1_q0  p1_r1  p1_r1_s#  ( p2_q#  p2_r0  p2_r1_s# ))
 +
( p1_q0  p1_r2  p1_r2_s#  ( p2_q#  p2_r1  p2_r2_s# ))
 +
 +
( p1_q1  p1_r1  p1_r1_s0  ( p2_q1  p2_r2  p2_r1_s0 ))
 +
( p1_q1  p1_r1  p1_r1_s1  ( p2_q0  p2_r2  p2_r1_s1 ))
 +
( p1_q1  p1_r1  p1_r1_s#  ( p2_q*  p2_r0  p2_r1_s# ))
 +
( p1_q1  p1_r2  p1_r2_s#  ( p2_q*  p2_r1  p2_r2_s# ))
 +
 +
o~~~~~~~~~o~~~~~~~~~o~INTERPRETATION~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Interpretation of the Propositional Program:
 +
 +
Mediate Conditions:
 +
 +
( p0_q#  ( p1_q# ))
 +
( p0_q*  ( p1_q* ))
 +
 +
( p1_q#  ( p2_q# ))
 +
( p1_q*  ( p2_q* ))
 +
 +
In Ref Log, an expression of the form "( X ( Y ))"
 +
expresses an implication or an if-then proposition:
 +
"Not X without Y",  "If X then Y",  "X => Y",  etc.
 +
 +
A text string expression of the form "( X ( Y ))"
 +
parses to a graphical data-structure of the form:
 +
 +
    X  Y
 +
    o---o
 +
    |
 +
    @
 +
 +
All together, these Mediate Conditions state:
 +
 +
| If at p_0  M is in state q_#, then at p_1  M is in state q_#, and
 +
| If at p_0  M is in state q_*, then at p_1  M is in state q_*, and
 +
| If at p_1  M is in state q_#, then at p_2  M is in state q_#, and
 +
| If at p_1  M is in state q_*, then at p_2  M is in state q_*.
 +
 +
Terminal Conditions:
 +
 +
(( p2_q# )( p2_q* ))
 +
 +
In Ref Log, an expression of the form "(( X )( Y ))"
 +
expresses a disjunction "X or Y" and it parses into:
 +
 +
    X  Y
 +
    o  o
 +
    \ /
 +
      o
 +
      |
 +
      @
 +
 +
In effect, the Terminal Conditions state:
 +
 +
| At p_2,  M is in state q_#, or
 +
| At p_2,  M is in state q_*.
 +
 +
State Partition:
 +
 +
(( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* ))
 +
(( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* ))
 +
(( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* ))
 +
 +
In Ref Log, an expression of the form "(( e_1 ),( e_2 ),( ... ),( e_k ))"
 +
expresses the fact that "exactly one of the e_j is true, for j = 1 to k".
 +
Expressions of this form are called "universal partition" expressions, and
 +
they parse into a type of graph called a "painted and rooted cactus" (PARC):
 +
 +
    e_1  e_2  ...  e_k
 +
    o    o          o
 +
    |    |          |
 +
    o-----o--- ... ---o
 +
      \              /
 +
      \            /
 +
        \          /
 +
        \        /
 +
          \      /
 +
          \    /
 +
            \  /
 +
            \ /
 +
              @
 +
 +
The State Partition expresses the conditions that:
 +
 +
| At each of the points-in-time p_i, for i = 0 to 2,
 +
| M can be in exactly one state q_j, for j in the set {0, 1, #, *}.
 +
 +
Register Partition:
 +
 +
(( p0_r0 ),( p0_r1 ),( p0_r2 ))
 +
(( p1_r0 ),( p1_r1 ),( p1_r2 ))
 +
(( p2_r0 ),( p2_r1 ),( p2_r2 ))
 +
 +
The Register Partition expresses the conditions that:
 +
 +
| At each of the points-in-time p_i, for i = 0 to 2,
 +
| H can be on exactly one cell  r_j, for j = 0 to 2.
 +
 +
Symbol Partition:
 +
 +
(( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# ))
 +
(( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# ))
 +
(( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# ))
 +
 +
(( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# ))
 +
(( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# ))
 +
(( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# ))
 +
 +
(( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# ))
 +
(( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# ))
 +
(( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# ))
 +
 +
The Symbol Partition expresses the conditions that:
 +
 +
| At each of the points-in-time p_i, for i in {0, 1, 2},
 +
| in each of the tape-registers r_j, for j in {0, 1, 2},
 +
| there can be exactly one sign s_k, for k in {0, 1, #}.
 +
 +
Interaction Conditions:
 +
 +
(( p0_r0 ) p0_r0_s0 ( p1_r0_s0 ))
 +
(( p0_r0 ) p0_r0_s1 ( p1_r0_s1 ))
 +
(( p0_r0 ) p0_r0_s# ( p1_r0_s# ))
 +
 +
(( p0_r1 ) p0_r1_s0 ( p1_r1_s0 ))
 +
(( p0_r1 ) p0_r1_s1 ( p1_r1_s1 ))
 +
(( p0_r1 ) p0_r1_s# ( p1_r1_s# ))
 +
 +
(( p0_r2 ) p0_r2_s0 ( p1_r2_s0 ))
 +
(( p0_r2 ) p0_r2_s1 ( p1_r2_s1 ))
 +
(( p0_r2 ) p0_r2_s# ( p1_r2_s# ))
 +
 +
(( p1_r0 ) p1_r0_s0 ( p2_r0_s0 ))
 +
(( p1_r0 ) p1_r0_s1 ( p2_r0_s1 ))
 +
(( p1_r0 ) p1_r0_s# ( p2_r0_s# ))
 +
 +
(( p1_r1 ) p1_r1_s0 ( p2_r1_s0 ))
 +
(( p1_r1 ) p1_r1_s1 ( p2_r1_s1 ))
 +
(( p1_r1 ) p1_r1_s# ( p2_r1_s# ))
 +
 +
(( p1_r2 ) p1_r2_s0 ( p2_r2_s0 ))
 +
(( p1_r2 ) p1_r2_s1 ( p2_r2_s1 ))
 +
(( p1_r2 ) p1_r2_s# ( p2_r2_s# ))
 +
 +
In briefest terms, the Interaction Conditions merely express
 +
the circumstance that the sign in a tape-cell cannot change
 +
between two points-in-time unless the tape-head is over the
 +
cell in question at the initial one of those points-in-time.
 +
All that we have to do is to see how they manage to say this.
 +
 +
In Ref Log, an expression of the following form:
 +
 +
"(( p<i>_r<j> ) p<i>_r<j>_s<k> ( p<i+1>_r<j>_s<k> ))",
 +
 +
and which parses as the graph:
 +
 +
      p<i>_r<j> o  o  p<i+1>_r<j>_s<k>
 +
                  \ /
 +
    p<i>_r<j>_s<k> o
 +
                  |
 +
                  @
 +
 +
can be read in the form of the following implication:
 +
 +
| If
 +
| at the point-in-time p<i>, the tape-cell r<j> bears the mark s<k>,
 +
| but it is not the case that
 +
| at the point-in-time p<i>, the tape-head is on the tape-cell r<j>.
 +
| then
 +
| at the point-in-time p<i+1>, the tape-cell r<j> bears the mark s<k>.
 +
 +
Folks among us of a certain age and a peculiar manner of acculturation will
 +
recognize these as the "Frame Conditions" for the change of state of the TM.
 +
 +
Transition Relations:
 +
 +
( p0_q0  p0_r1  p0_r1_s0  ( p1_q0  p1_r2  p1_r1_s0 ))
 +
( p0_q0  p0_r1  p0_r1_s1  ( p1_q1  p1_r2  p1_r1_s1 ))
 +
( p0_q0  p0_r1  p0_r1_s#  ( p1_q#  p1_r0  p1_r1_s# ))
 +
( p0_q0  p0_r2  p0_r2_s#  ( p1_q#  p1_r1  p1_r2_s# ))
 +
 +
( p0_q1  p0_r1  p0_r1_s0  ( p1_q1  p1_r2  p1_r1_s0 ))
 +
( p0_q1  p0_r1  p0_r1_s1  ( p1_q0  p1_r2  p1_r1_s1 ))
 +
( p0_q1  p0_r1  p0_r1_s#  ( p1_q*  p1_r0  p1_r1_s# ))
 +
( p0_q1  p0_r2  p0_r2_s#  ( p1_q*  p1_r1  p1_r2_s# ))
 +
 +
( p1_q0  p1_r1  p1_r1_s0  ( p2_q0  p2_r2  p2_r1_s0 ))
 +
( p1_q0  p1_r1  p1_r1_s1  ( p2_q1  p2_r2  p2_r1_s1 ))
 +
( p1_q0  p1_r1  p1_r1_s#  ( p2_q#  p2_r0  p2_r1_s# ))
 +
( p1_q0  p1_r2  p1_r2_s#  ( p2_q#  p2_r1  p2_r2_s# ))
 +
 +
( p1_q1  p1_r1  p1_r1_s0  ( p2_q1  p2_r2  p2_r1_s0 ))
 +
( p1_q1  p1_r1  p1_r1_s1  ( p2_q0  p2_r2  p2_r1_s1 ))
 +
( p1_q1  p1_r1  p1_r1_s#  ( p2_q*  p2_r0  p2_r1_s# ))
 +
( p1_q1  p1_r2  p1_r2_s#  ( p2_q*  p2_r1  p2_r2_s# ))
 +
 +
The Transition Conditions merely serve to express,
 +
by means of 16 complex implication expressions,
 +
the data of the TM table that was given above.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~OUTPUTS~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
And here are the outputs of the computation,
 +
as emulated by its propositional rendition,
 +
and as actually generated within that form
 +
of transmogrification by the program that
 +
I wrote for finding all of the satisfying
 +
interpretations (truth-value assignments)
 +
of propositional expressions in Ref Log:
 +
 +
o~~~~~~~~~o~~~~~~~~~o~OUTPUT~0~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Output Conditions:
 +
 +
p0_q0
 +
  p0_r1
 +
  p0_r0_s#
 +
    p0_r1_s0
 +
    p0_r2_s#
 +
      p1_q0
 +
      p1_r2
 +
        p1_r2_s#
 +
        p1_r0_s#
 +
          p1_r1_s0
 +
          p2_q#
 +
            p2_r1
 +
            p2_r0_s#
 +
              p2_r1_s0
 +
              p2_r2_s#
 +
 +
The Output Conditions amount to the sole satisfying interpretation,
 +
that is, a "sequence of truth-value assignments" (SOTVA) that make
 +
the entire proposition come out true, and they state the following:
 +
 +
| At the point-in-time p_0, M is in the state q_0,      and
 +
| At the point-in-time p_0, H is on the cell  r_1,      and
 +
| At the point-in-time p_0, cell r_0 bears the mark "#", and
 +
| At the point-in-time p_0, cell r_1 bears the mark "0", and
 +
| At the point-in-time p_0, cell r_2 bears the mark "#", and
 +
|
 +
| At the point-in-time p_1, M is in the state q_0,      and
 +
| At the point-in-time p_1, H is on the cell  r_2,      and
 +
| At the point-in-time p_1, cell r_0 bears the mark "#", and
 +
| At the point-in-time p_1, cell r_1 bears the mark "0", and
 +
| At the point-in-time p_1, cell r_2 bears the mark "#", and
 +
|
 +
| At the point-in-time p_2, M is in the state q_#,      and
 +
| At the point-in-time p_2, H is on the cell  r_1,      and
 +
| At the point-in-time p_2, cell r_0 bears the mark "#", and
 +
| At the point-in-time p_2, cell r_1 bears the mark "0", and
 +
| At the point-in-time p_2, cell r_2 bears the mark "#".
 +
 +
In brief, the output for our sake being the symbol that rests
 +
under the tape-head H when the machine M gets to a rest state,
 +
we are now amazed by the remarkable result that Parity(0) = 0.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~OUTPUT~1~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Output Conditions:
 +
 +
p0_q0
 +
  p0_r1
 +
  p0_r0_s#
 +
    p0_r1_s1
 +
    p0_r2_s#
 +
      p1_q1
 +
      p1_r2
 +
        p1_r2_s#
 +
        p1_r0_s#
 +
          p1_r1_s1
 +
          p2_q*
 +
            p2_r1
 +
            p2_r0_s#
 +
              p2_r1_s1
 +
              p2_r2_s#
 +
 +
The Output Conditions amount to the sole satisfying interpretation,
 +
that is, a "sequence of truth-value assignments" (SOTVA) that make
 +
the entire proposition come out true, and they state the following:
 +
 +
| At the point-in-time p_0, M is in the state q_0,      and
 +
| At the point-in-time p_0, H is on the cell  r_1,      and
 +
| At the point-in-time p_0, cell r_0 bears the mark "#", and
 +
| At the point-in-time p_0, cell r_1 bears the mark "1", and
 +
| At the point-in-time p_0, cell r_2 bears the mark "#", and
 +
|
 +
| At the point-in-time p_1, M is in the state q_1,      and
 +
| At the point-in-time p_1, H is on the cell  r_2,      and
 +
| At the point-in-time p_1, cell r_0 bears the mark "#", and
 +
| At the point-in-time p_1, cell r_1 bears the mark "1", and
 +
| At the point-in-time p_1, cell r_2 bears the mark "#", and
 +
|
 +
| At the point-in-time p_2, M is in the state q_*,      and
 +
| At the point-in-time p_2, H is on the cell  r_1,      and
 +
| At the point-in-time p_2, cell r_0 bears the mark "#", and
 +
| At the point-in-time p_2, cell r_1 bears the mark "1", and
 +
| At the point-in-time p_2, cell r_2 bears the mark "#".
 +
 +
In brief, the output for our sake being the symbol that rests
 +
under the tape-head H when the machine M gets to a rest state,
 +
we are now amazed by the remarkable result that Parity(1) = 1.
 +
 +
I realized after sending that last bunch of bits that there is room
 +
for confusion about what is the input/output of the Study module of
 +
the Theme One program as opposed to what is the input/output of the
 +
"finitely approximated turing automaton" (FATA).  So here is better
 +
delineation of what's what.  The input to Study is a text file that
 +
is known as LogFile(Whatever) and the output of Study is a sequence
 +
of text files that summarize the various canonical and normal forms
 +
that it generates.  For short, let us call these NormFile(Whatelse).
 +
With that in mind, here are the actual IO's of Study, excluding the
 +
glosses in square brackets:
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~INPUT~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
[Input To Study = FATA Initial Conditions + FATA Program Conditions]
 +
 +
[FATA Initial Conditions For Input 0]
 +
 +
p0_q0
 +
 +
p0_r1
 +
 +
p0_r0_s#
 +
p0_r1_s0
 +
p0_r2_s#
 +
 +
[FATA Program Conditions For Parity Machine]
 +
 +
[Mediate Conditions]
 +
 +
( p0_q#  ( p1_q# ))
 +
( p0_q*  ( p1_q* ))
 +
 +
( p1_q#  ( p2_q# ))
 +
( p1_q*  ( p2_q* ))
 +
 +
[Terminal Conditions]
 +
 +
(( p2_q# )( p2_q* ))
 +
 +
[State Partition]
 +
 +
(( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* ))
 +
(( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* ))
 +
(( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* ))
 +
 +
[Register Partition]
 +
 +
(( p0_r0 ),( p0_r1 ),( p0_r2 ))
 +
(( p1_r0 ),( p1_r1 ),( p1_r2 ))
 +
(( p2_r0 ),( p2_r1 ),( p2_r2 ))
 +
 +
[Symbol Partition]
 +
 +
(( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# ))
 +
(( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# ))
 +
(( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# ))
 +
 +
(( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# ))
 +
(( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# ))
 +
(( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# ))
 +
 +
(( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# ))
 +
(( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# ))
 +
(( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# ))
 +
 +
[Interaction Conditions]
 +
 +
(( p0_r0 ) p0_r0_s0 ( p1_r0_s0 ))
 +
(( p0_r0 ) p0_r0_s1 ( p1_r0_s1 ))
 +
(( p0_r0 ) p0_r0_s# ( p1_r0_s# ))
 +
 +
(( p0_r1 ) p0_r1_s0 ( p1_r1_s0 ))
 +
(( p0_r1 ) p0_r1_s1 ( p1_r1_s1 ))
 +
(( p0_r1 ) p0_r1_s# ( p1_r1_s# ))
 +
 +
(( p0_r2 ) p0_r2_s0 ( p1_r2_s0 ))
 +
(( p0_r2 ) p0_r2_s1 ( p1_r2_s1 ))
 +
(( p0_r2 ) p0_r2_s# ( p1_r2_s# ))
 +
 +
(( p1_r0 ) p1_r0_s0 ( p2_r0_s0 ))
 +
(( p1_r0 ) p1_r0_s1 ( p2_r0_s1 ))
 +
(( p1_r0 ) p1_r0_s# ( p2_r0_s# ))
 +
 +
(( p1_r1 ) p1_r1_s0 ( p2_r1_s0 ))
 +
(( p1_r1 ) p1_r1_s1 ( p2_r1_s1 ))
 +
(( p1_r1 ) p1_r1_s# ( p2_r1_s# ))
 +
 +
(( p1_r2 ) p1_r2_s0 ( p2_r2_s0 ))
 +
(( p1_r2 ) p1_r2_s1 ( p2_r2_s1 ))
 +
(( p1_r2 ) p1_r2_s# ( p2_r2_s# ))
 +
 +
[Transition Relations]
 +
 +
( p0_q0  p0_r1  p0_r1_s0  ( p1_q0  p1_r2  p1_r1_s0 ))
 +
( p0_q0  p0_r1  p0_r1_s1  ( p1_q1  p1_r2  p1_r1_s1 ))
 +
( p0_q0  p0_r1  p0_r1_s#  ( p1_q#  p1_r0  p1_r1_s# ))
 +
( p0_q0  p0_r2  p0_r2_s#  ( p1_q#  p1_r1  p1_r2_s# ))
 +
 +
( p0_q1  p0_r1  p0_r1_s0  ( p1_q1  p1_r2  p1_r1_s0 ))
 +
( p0_q1  p0_r1  p0_r1_s1  ( p1_q0  p1_r2  p1_r1_s1 ))
 +
( p0_q1  p0_r1  p0_r1_s#  ( p1_q*  p1_r0  p1_r1_s# ))
 +
( p0_q1  p0_r2  p0_r2_s#  ( p1_q*  p1_r1  p1_r2_s# ))
 +
 +
( p1_q0  p1_r1  p1_r1_s0  ( p2_q0  p2_r2  p2_r1_s0 ))
 +
( p1_q0  p1_r1  p1_r1_s1  ( p2_q1  p2_r2  p2_r1_s1 ))
 +
( p1_q0  p1_r1  p1_r1_s#  ( p2_q#  p2_r0  p2_r1_s# ))
 +
( p1_q0  p1_r2  p1_r2_s#  ( p2_q#  p2_r1  p2_r2_s# ))
 +
 +
( p1_q1  p1_r1  p1_r1_s0  ( p2_q1  p2_r2  p2_r1_s0 ))
 +
( p1_q1  p1_r1  p1_r1_s1  ( p2_q0  p2_r2  p2_r1_s1 ))
 +
( p1_q1  p1_r1  p1_r1_s#  ( p2_q*  p2_r0  p2_r1_s# ))
 +
( p1_q1  p1_r2  p1_r2_s#  ( p2_q*  p2_r1  p2_r2_s# ))
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~OUTPUT~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
[Output Of Study = FATA Output For Input 0]
 +
 +
p0_q0
 +
  p0_r1
 +
  p0_r0_s#
 +
    p0_r1_s0
 +
    p0_r2_s#
 +
      p1_q0
 +
      p1_r2
 +
        p1_r2_s#
 +
        p1_r0_s#
 +
          p1_r1_s0
 +
          p2_q#
 +
            p2_r1
 +
            p2_r0_s#
 +
              p2_r1_s0
 +
              p2_r2_s#
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Turing automata, finitely approximated or not, make my head spin and
 +
my tape go loopy, and I still believe 'twere a far better thing I do
 +
if I work up to that level of complexity in a more gracile graduated
 +
manner.  So let us return to our Example in this gradual progress to
 +
that vastly more well-guarded grail of our long-term pilgrim's quest:
 +
 +
|                  boy  male          girl  female
 +
|                    o---o child          o---o child
 +
|  male  female      \ /                  \ /              child  human
 +
|    o---o            o                    o                    o--o
 +
|      \ /            |                    |                    |
 +
|      @              @                    @                    @
 +
|
 +
| (male , female)((boy , male child))((girl , female child))(child (human))
 +
 +
One section of the Theme One program has a suite of utilities that fall
 +
under the title "Theme One Study" ("To Study", or just "TOS" for short).
 +
To Study is to read and to parse a so-called and a generally so-suffixed
 +
"log" file, and then to conjoin what is called a "query", which is really
 +
just an additional propositional expression that imposes a further logical
 +
constraint on the input expression.
 +
 +
The Figure roughly sketches the conjuncts of the graph-theoretic
 +
data structure that the parser would commit to memory on reading
 +
the appropriate log file that contains the text along the bottom.
 +
 +
I will now explain the various sorts of things that the TOS utility
 +
can do with the log file that describes the universe of discourse in
 +
our present Example.
 +
 +
Theme One Study is built around a suite of four successive generators
 +
of "normal forms" for propositional expressions, just to use that term
 +
in a very approximate way.  The functions that compute these normal forms
 +
are called "Model", "Tenor", "Canon", and "Sense", and so we may refer to
 +
to their text-style outputs as the "mod", "ten", "can", and "sen" files.
 +
 +
Though it could be any propositional expression on the same vocabulary
 +
$A$ = {"boy", "child", "female", "girl", "human", "male"}, more usually
 +
the query is a simple conjunction of one or more positive features that
 +
we want to focus on or perhaps to filter out of the logical model space.
 +
On our first run through this Example, we take the log file proposition
 +
as it is, with no extra riders.
 +
 +
| Procedural Note.  TO Study Model displays a running tab of how much
 +
| free memory space it has left.  On some of the harder problems that
 +
| you may think of to give it, Model may run out of free memory and
 +
| terminate, abnormally exiting Theme One.  Sometimes it helps to:
 +
|
 +
| 1.  Rephrase the problem in logically equivalent
 +
|    but rhetorically increasedly felicitous ways.
 +
|
 +
| 2.  Think of additional facts that are taken for granted but not
 +
|    made explicit and that cannot be logically inferred by Model.
 +
 +
After Model has finished, it is ready to write out its mod file,
 +
which you may choose to show on the screen or save to a named file.
 +
Mod files are usually too long to see (or to care to see) all at once
 +
on the screen, so it is very often best to save them for later replay.
 +
In our Example the Model function yields a mod file that looks like so:
 +
 +
Model Output and
 +
Mod File Example
 +
o-------------------o
 +
| male              |
 +
|  female -        |  1
 +
|  (female )        |
 +
|  girl -          |  2
 +
|  (girl )        |
 +
|    child          |
 +
|    boy          |
 +
|      human *      |  3 *
 +
|      (human ) -  |  4
 +
|    (boy ) -      |  5
 +
|    (child )      |
 +
|    boy -        |  6
 +
|    (boy ) *      |  7 *
 +
| (male )          |
 +
|  female          |
 +
|  boy -          |  8
 +
|  (boy )          |
 +
|    child          |
 +
|    girl          |
 +
|      human *      |  9 *
 +
|      (human ) -  | 10
 +
|    (girl ) -    | 11
 +
|    (child )      |
 +
|    girl -        | 12
 +
|    (girl ) *    | 13 *
 +
|  (female ) -      | 14
 +
o-------------------o
 +
 +
Counting the stars "*" that indicate true interpretations
 +
and the bars "-" that indicate false interpretations of
 +
the input formula, we can see that the Model function,
 +
out of the 64 possible interpretations, has actually
 +
gone through the work of making just 14 evaluations,
 +
all in order to find the 4 models that are allowed
 +
by the input definitions.
 +
 +
To be clear about what this output means, the starred paths
 +
indicate all of the complete specifications of objects in the
 +
universe of discourse, that is, all of the consistent feature
 +
conjunctions of maximum length, as permitted by the definitions
 +
that are given in the log file.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Let's take a little break from the Example in progress
 +
and look at where we are and what we have been doing from
 +
computational, logical, and semiotic perspectives.  Because,
 +
after all, as is usually the case, we should not let our focus
 +
and our fascination with this particular Example prevent us from
 +
recognizing it, and all that we do with it, as just an Example of
 +
much broader paradigms and predicaments and principles, not to say
 +
but a glimmer of ultimately more concernful and fascinating objects.
 +
 +
I chart the progression that we have just passed through in this way:
 +
 +
|                    Parse
 +
|      Sign A  o-------------->o  Sign 1
 +
|            ^                |
 +
|            /                |
 +
|          /                  |
 +
|          /                  |
 +
| Object  o                    |  Transform
 +
|          ^                  |
 +
|          \                  |
 +
|            \                |
 +
|            \                v
 +
|      Sign B  o<--------------o  Sign 2
 +
|                    Verse
 +
|
 +
| Figure.  Computation As Sign Transformation
 +
 +
In the present case, the Object is an objective situation
 +
or a state of affairs, in effect, a particular pattern of
 +
feature concurrences occurring to us in that world through
 +
which we find ourselves most frequently faring, wily nily,
 +
and the Signs are different tokens and different types of
 +
data structures that we somehow or other find it useful
 +
to devise or to discover for the sake of representing
 +
current objects to ourselves on a recurring basis.
 +
 +
But not all signs, not even signs of a single object, are alike
 +
in every other respect that one might name, not even with respect
 +
to their powers of relating, significantly, to that common object.
 +
 +
And that is what our whole business of computation busies itself about,
 +
when it minds its business best, that is, transmuting signs into signs
 +
in ways that augment their powers of relating significantly to objects.
 +
 +
We have seen how the Model function and the mod output format
 +
indicate all of the complete specifications of objects in the
 +
universe of discourse, that is, all of the consistent feature
 +
conjunctions of maximal specificity that are permitted by the
 +
constraints or the definitions that are given in the log file.
 +
 +
To help identify these specifications of particular cells in
 +
the universe of discourse, the next function and output format,
 +
called "Tenor", edits the mod file to give only the true paths,
 +
in effect, the "positive models", that are by default what we
 +
usually mean when we say "models", and not the "anti-models"
 +
or the "negative models" that fail to satisfy the formula
 +
in question.
 +
 +
In the present Example the Tenor function
 +
generates a Ten file that looks like this:
 +
 +
Tenor Output and
 +
Ten File Example
 +
o-------------------o
 +
| male              |
 +
|  (female )        |
 +
|  (girl )        |
 +
|    child          |
 +
|    boy          |
 +
|      human *      | <1>
 +
|    (child )      |
 +
|    (boy ) *      | <2>
 +
| (male )          |
 +
|  female          |
 +
|  (boy )          |
 +
|    child          |
 +
|    girl          |
 +
|      human *      | <3>
 +
|    (child )      |
 +
|    (girl ) *    | <4>
 +
o-------------------o
 +
 +
As I said, the Tenor function just abstracts a transcript of the models,
 +
that is, the satisfying interpretations, that were already interspersed
 +
throughout the complete Model output.  These specifications, or feature
 +
conjunctions, with the positive and the negative features listed in the
 +
order of their actual budding on the "arboreal boolean expansion" twigs,
 +
may be gathered and arranged in this antherypulogical flowering bouquet:
 +
 +
1.  male  (female ) (girl )  child    boy    human  *
 +
2.  male  (female ) (girl ) (child ) (boy  )          *
 +
3.  (male )  female  (boy  )  child    girl    human  *
 +
4.  (male )  female  (boy  ) (child ) (girl )          *
 +
 +
Notice that Model, as reflected in this abstract, did not consider
 +
the six positive features in the same order along each path.  This
 +
is because the algorithm was designed to proceed opportunistically
 +
in its attempt to reduce the original proposition through a series
 +
of case-analytic considerations and the resulting simplifications.
 +
 +
Notice, too, that Model is something of a lazy evaluator, quitting work
 +
when and if a value is determined by less than the full set of variables.
 +
This is the reason why paths <2> and <4> are not ostensibly of the maximum
 +
length.  According to this lazy mode of understanding, any path that is not
 +
specified on a set of features really stands for the whole bundle of paths
 +
that are derived by freely varying those features.  Thus, specifications
 +
<2> and <4> summarize four models altogether, with the logical choice
 +
between "human" and "not human" being left open at the point where
 +
they leave off their branches in the releavent deciduous tree.
 +
 +
The last two functions in the Study section, "Canon" and "Sense",
 +
extract further derivatives of the normal forms that are produced
 +
by Model and Tenor.  Both of these functions take the set of model
 +
paths and simply throw away the negative labels.  You may think of
 +
these as the "rose colored glasses" or "job interview" normal forms,
 +
in that they try to say everything that's true, so long as it can be
 +
expressed in positive terms.  Generally, this would mean losing a lot
 +
of information, and the result could no longer be expected to have the
 +
property of remaining logically equivalent to the original proposition.
 +
 +
Fortunately, however, it seems that this type of positive projection of
 +
the whole truth is just what is possible, most needed, and most clear in
 +
many of the "natural" examples, that is, in examples that arise from the
 +
domains of natural language and natural conceptual kinds.  In these cases,
 +
where most of the logical features are redundantly coded, for example, in
 +
the way that "adult" = "not child" and "child" = "not adult", the positive
 +
feature bearing redacts are often sufficiently expressive all by themselves.
 +
 +
Canon merely censors its printing of the negative labels as it traverses the
 +
model tree.  This leaves the positive labels in their original columns of the
 +
outline form, giving it a slightly skewed appearance.  This can be misleading
 +
unless you already know what you are looking for.  However, this Canon format
 +
is computationally quick, and frequently suffices, especially if you already
 +
have a likely clue about what to expect in the way of a question's outcome.
 +
 +
In the present Example the Canon function
 +
generates a Can file that looks like this:
 +
 +
Canon Output and
 +
Can File Example
 +
o-------------------o
 +
| male              |
 +
|    child          |
 +
|    boy          |
 +
|      human        |
 +
|  female          |
 +
|    child          |
 +
|    girl          |
 +
|      human        |
 +
o-------------------o
 +
 +
The Sense function does the extra work that is required
 +
to place the positive labels of the model tree at their
 +
proper level in the outline.
 +
 +
In the present Example the Sense function
 +
generates a Sen file that looks like this:
 +
 +
Sense Output and
 +
Sen File Example
 +
o-------------------o
 +
| male              |
 +
|  child            |
 +
|  boy            |
 +
|    human          |
 +
| female            |
 +
|  child            |
 +
|  girl            |
 +
|    human          |
 +
o-------------------o
 +
 +
The Canon and Sense outlines for this Example illustrate a certain
 +
type of general circumstance that needs to be noted at this point.
 +
Recall the model paths or the feature specifications that were
 +
numbered <2> and <4> in the listing of the output for Tenor.
 +
These paths, in effect, reflected Model's discovery that
 +
the venn diagram cells for male or female non-children
 +
and male or female non-humans were not excluded by
 +
the definitions that were given in the Log file.
 +
In the abstracts given by Canon and Sense, the
 +
specifications <2> and <4> have been subsumed,
 +
or absorbed unmarked, under the general topics
 +
of their respective genders, male or female.
 +
This happens because no purely positive
 +
features were supplied to distinguish
 +
the non-child and non-human cases.
 +
 +
That completes the discussion of
 +
this six-dimensional Example.
 +
 +
Nota Bene, for possible future use.  In the larger current of work
 +
with respect to which this meander of a conduit was initially both
 +
diversionary and tributary, before those high and dry regensquirm
 +
years when it turned into an intellectual interglacial oxbow lake,
 +
I once had in mind a scape in which expressions in a definitional
 +
lattice were ordered according to their simplicity on some scale
 +
or another, and in this setting the word "sense" was actually an
 +
acronym for "semantically equivalent next-simplest expression".
 +
 +
| If this is starting to sound a little bit familiar,
 +
| it may be because the relationship between the two
 +
| kinds of pictures of propositions, namely:
 +
|
 +
| 1.  Propositions about things in general, here,
 +
|    about the times when certain facts are true,
 +
|    having the form of functions f : X -> B,
 +
|
 +
| 2.  Propositions about binary codes, here, about
 +
|    the bit-vector labels on venn diagram cells,
 +
|    having the form of functions f' : B^k -> B,
 +
|
 +
| is an epically old story, one that I, myself,
 +
| have related one or twice upon a time before,
 +
| to wit, at least, at the following two cites:
 +
|
 +
| http://suo.ieee.org/email/msg01251.html
 +
| http://suo.ieee.org/email/msg01293.html
 +
|
 +
| There, and now here, once more, and again, it may be observed
 +
| that the relation is one whereby the proposition f : X -> B,
 +
| the one about things and times and mores in general, factors
 +
| into a coding function c : X -> B^k, followed by a derived
 +
| proposition f' : B^k -> B that judges the resulting codes.
 +
|
 +
|                        f
 +
|                  X o------>o B
 +
|                      \    ^
 +
|  c = <x_1, ..., x_k> \  / f'
 +
|                        v /
 +
|                        o
 +
|                        B^k
 +
|
 +
| You may remember that this was supposed to illustrate
 +
| the "factoring" of a proposition f : X -> B = {0, 1}
 +
| into the composition f'(c(x)), where c : X -> B^k is
 +
| the "coding" of each x in X as an k-bit string in B^k,
 +
| and where f' is the mapping of codes into a co-domain
 +
| that we interpret as t-f-values, B = {0, 1} = {F, T}.
 +
 +
In short, there is the standard equivocation ("systematic ambiguity"?) as to
 +
whether we are talking about the "applied" and concretely typed proposition
 +
f : X -> B or the "pure" and abstractly typed proposition f' : B^k -> B.
 +
Or we can think of the latter object as the approximate code icon of
 +
the former object.
 +
 +
Anyway, these types of formal objects are the sorts of things that
 +
I take to be the denotational objects of propositional expressions.
 +
These objects, along with their invarious and insundry mathematical
 +
properties, are the orders of things that I am talking about when
 +
I refer to the "invariant structures in these objects themselves".
 +
 +
"Invariant" means "invariant under a suitable set of transformations",
 +
in this case the translations between various languages that preserve
 +
the objects and the structures in question.  In extremest generality,
 +
this is what universal constructions in category theory are all about.
 +
 +
In summation, the functions f : X -> B and f' : B* -> B have invariant, formal,
 +
mathematical, objective properties that any adequate language might eventually
 +
evolve to express, only some languages express them more obscurely than others.
 +
 +
To be perfectly honest, I continue to be surprised that anybody in this group
 +
has trouble with this.  There are perfectly apt and familiar examples in the
 +
contrast between roman numerals and arabic numerals, or the contrast between
 +
redundant syntaxes, like those that use the pentalphabet {~, &, v, =>, <=>},
 +
and trimmer syntaxes, like those used in existential and conceptual graphs.
 +
Every time somebody says "Let's take {~, &, v, =>, <=>} as an operational
 +
basis for logic" it's just like that old joke that mathematicians tell on
 +
engineers where the ingenue in question says "1 is a prime, 2 is a prime,
 +
3 is a prime, 4 is a prime, ..." -- and I know you think that I'm being
 +
hyperbolic, but I'm really only up to parabolas here ...
 +
 +
I have already refined my criticism so that it does not apply to
 +
the spirit of FOL or KIF or whatever, but only to the letters of
 +
specific syntactic proposals.  There is a fact of the matter as
 +
to whether a concrete language provides a clean or a cluttered
 +
basis for representing the identified set of formal objects.
 +
And it shows up in pragmatic realities like the efficiency
 +
of real time concept formation, concept use, learnability,
 +
reasoning power, and just plain good use of real time.
 +
These are the dire consequences that I learned in my
 +
very first tries at mathematically oriented theorem
 +
automation, and the only factor that has obscured
 +
them in mainstream work since then is the speed
 +
with which folks can now do all of the same
 +
old dumb things that they used to do on
 +
their way to kludging out the answers.
 +
 +
It seems to be darn near impossible to explain to the
 +
centurion all of the neat stuff that he's missing by
 +
sticking to his old roman numerals.  He just keeps
 +
on reckoning that what he can't count must be of
 +
no account at all.  There is way too much stuff
 +
that these original syntaxes keep us from even
 +
beginning to discuss, like differential logic,
 +
just for starters.
 +
 +
Our next Example illustrates the use of the Cactus Language
 +
for representing "absolute" and "relative" partitions, also
 +
known as "complete" and "contingent" classifications of the
 +
universe of discourse, all of which amounts to divvying it
 +
up into mutually exclusive regions, exhaustive or not, as
 +
one frequently needs in situations involving a genus and
 +
its sundry species, and frequently pictures in the form
 +
of a venn diagram that looks just like a "pie chart".
 +
 +
Example.  Partition, Genus & Species
 +
 +
The idea that one needs for expressing partitions
 +
in cactus expressions can be summed up like this:
 +
 +
| If the propositional expression
 +
|
 +
| "( p , q , r , ... )"
 +
|
 +
| means that just one of
 +
|
 +
| p, q, r, ... is false,
 +
|
 +
| then the propositional expression
 +
|
 +
| "((p),(q),(r), ... )"
 +
|
 +
| must mean that just one of
 +
|
 +
| (p), (q), (r), ... is false,
 +
|
 +
| in other words, that just one of
 +
|
 +
| p, q, r, ... is true.
 +
 +
Thus we have an efficient means to express and to enforce
 +
a partition of the space of models, in effect, to maintain
 +
the condition that a number of features or propositions are
 +
to be held in mutually exclusive and exhaustive disjunction.
 +
This supplies a much needed bridge between the binary domain
 +
of two values and any other domain with a finite number of
 +
feature values.
 +
 +
Another variation on this theme allows one to maintain the
 +
subsumption of many separate species under an explicit genus.
 +
To see this, let us examine the following form of expression:
 +
 +
( q , ( q_1 ) , ( q_2 ) , ( q_3 ) ).
 +
 +
Now consider what it would mean for this to be true.  We see two cases:
 +
 +
1.  If the proposition q is true, then exactly one of the
 +
    propositions (q_1), (q_2), (q_3) must be false, and so
 +
    just one of the propositions q_1, q_2, q_3 must be true.
 +
 +
2.  If the proposition q is false, then every one of the
 +
    propositions (q_1), (q_2), (q_2) must be true, and so
 +
    each one of the propositions q_1, q_2, q_3 must be false.
 +
    In short, if q is false then all of the other q's are also.
 +
 +
Figures 1 and 2 illustrate this type of situation.
 +
 +
Figure 1 is the venn diagram of a 4-dimensional universe of discourse
 +
X = [q, q_1, q_2, q_3], conventionally named after the gang of four
 +
logical features that generate it.  Strictly speaking, X is made up
 +
of two layers, the position space X of abstract type %B%^4, and the
 +
proposition space X^ = (X -> %B%) of abstract type %B%^4 -> %B%,
 +
but it is commonly lawful enough to sign the signature of both
 +
spaces with the same X, and thus to give the power of attorney
 +
for the propositions to the so-indicted position space thereof.
 +
 +
Figure 1 also makes use of the convention whereby the regions
 +
or the subsets of the universe of discourse that correspond
 +
to the basic features q, q_1, q_2, q_3 are labelled with
 +
the parallel set of upper case letters Q, Q_1, Q_2, Q_3.
 +
 +
|                        o
 +
|                      / \
 +
|                      /  \
 +
|                    /    \
 +
|                    /      \
 +
|                  o        o
 +
|                  /%\      /%\
 +
|                /%%%\    /%%%\
 +
|                /%%%%%\  /%%%%%\
 +
|              /%%%%%%%\ /%%%%%%%\
 +
|              o%%%%%%%%%o%%%%%%%%%o
 +
|            / \%%%%%%%/ \%%%%%%%/ \
 +
|            /  \%%%%%/  \%%%%%/  \
 +
|          /    \%%%/    \%%%/    \
 +
|          /      \%/      \%/      \
 +
|        o        o        o        o
 +
|        / \      /%\      / \      / \
 +
|      /  \    /%%%\    /  \    /  \
 +
|      /    \  /%%%%%\  /    \  /    \
 +
|    /      \ /%%%%%%%\ /      \ /      \
 +
|    o        o%%%%%%%%%o        o        o
 +
|    ·\      / \%%%%%%%/ \      / \      /·
 +
|    · \    /  \%%%%%/  \    /  \    / ·
 +
|    ·  \  /    \%%%/    \  /    \  /  ·
 +
|    ·  \ /      \%/      \ /      \ /  ·
 +
|    ·    o        o        o        o    ·
 +
|    ·    ·\      / \      / \      /·    ·
 +
|    ·    · \    /  \    /  \    / ·    ·
 +
|    ·    ·  \  /    \  /    \  /  ·    ·
 +
|    · Q  ·  \ /      \ /      \ /  ·Q_3 ·
 +
|    ··········o        o        o··········
 +
|        ·    \      /%\      /    ·
 +
|        ·      \    /%%%\    /      ·
 +
|        ·      \  /%%%%%\  /      ·
 +
|        · Q_1    \ /%%%%%%%\ /    Q_2 ·
 +
|        ··········o%%%%%%%%%o··········
 +
|                    \%%%%%%%/
 +
|                    \%%%%%/
 +
|                      \%%%/
 +
|                      \%/
 +
|                        o
 +
|
 +
| Figure 1.  Genus Q and Species Q_1, Q_2, Q_3
 +
 +
Figure 2 is another form of venn diagram that one often uses,
 +
where one collapses the unindited cells and leaves only the
 +
models of the proposition in question.  Some people would
 +
call the transformation that changes from the first form
 +
to the next form an operation of "taking the quotient",
 +
but I tend to think of it as the "soap bubble picture"
 +
or more exactly the "wire & thread & soap film" model
 +
of the universe of discourse, where one pops out of
 +
consideration the sections of the soap film that
 +
stretch across the anti-model regions of space.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|  X                                              |
 +
|                                                |
 +
|                        o                        |
 +
|                      / \                      |
 +
|                      /  \                      |
 +
|                    /    \                    |
 +
|                    /      \                    |
 +
|                  /        \                  |
 +
|                  o    Q_1    o                  |
 +
|                / \        / \                |
 +
|                /  \      /  \                |
 +
|              /    \    /    \              |
 +
|              /      \  /      \              |
 +
|            /        \ /        \            |
 +
|            /          Q          \            |
 +
|          /            |            \          |
 +
|          /            |            \          |
 +
|        /      Q_2    |    Q_3      \        |
 +
|        /              |              \        |
 +
|      /                |                \      |
 +
|      o-----------------o-----------------o      |
 +
|                                                |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
Figure 2.  Genus Q and Species Q_1, Q_2, Q_3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Example.  Partition, Genus & Species (cont.)
 +
 +
Last time we considered in general terms how the forms
 +
of complete partition and contingent partition operate
 +
to maintain mutually disjoint and possibly exhaustive
 +
categories of positions in a universe of discourse.
 +
 +
This time we contemplate another concrete Example of
 +
near minimal complexity, designed to demonstrate how
 +
the forms of partition and subsumption can interact
 +
in structuring a space of feature specifications.
 +
 +
In this Example, we describe a universe of discourse
 +
in terms of the following vocabulary of five features:
 +
 +
| L.  living_thing
 +
|
 +
| N.  non_living
 +
|
 +
| A.  animal
 +
|
 +
| V.  vegetable
 +
|
 +
| M.  mineral
 +
 +
Let us construe these features as being subject to four constraints:
 +
 +
| 1.  Everything is either a living_thing or non_living, but not both.
 +
|
 +
| 2.  Everything is either animal, vegetable, or mineral,
 +
|    but no two of these together.
 +
|
 +
| 3.  A living_thing is either animal or vegetable, but not both,
 +
|    and everything animal or vegetable is a living_thing.
 +
|
 +
| 4.  Everything mineral is non_living.
 +
 +
These notions and constructions are expressed in the Log file shown below:
 +
 +
Logical Input File
 +
o-------------------------------------------------o
 +
|                                                |
 +
|  ( living_thing , non_living )                |
 +
|                                                |
 +
|  (( animal ),( vegetable ),( mineral ))        |
 +
|                                                |
 +
|  ( living_thing ,( animal ),( vegetable ))    |
 +
|                                                |
 +
|  ( mineral ( non_living ))                    |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
The cactus expression in this file is the expression
 +
of a "zeroth order theory" (ZOT), one that can be
 +
paraphrased in more ordinary language to say:
 +
 +
Translation
 +
o-------------------------------------------------o
 +
|                                                |
 +
|  living_thing  =/=  non_living                |
 +
|                                                |
 +
|  par : all -> {animal, vegetable, mineral}    |
 +
|                                                |
 +
|  par : living_thing -> {animal, vegetable}    |
 +
|                                                |
 +
|  mineral => non_living                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
Here, "par : all -> {p, q, r}" is short for an assertion
 +
that the universe as a whole is partitioned into subsets
 +
that correspond to the features p, q, r.
 +
 +
Also, "par : q -> {r, s}" asserts that "Q partitions into R and S.
 +
 +
It is probably enough just to list the outputs of Model, Tenor, and Sense
 +
when run on the preceding Log file.  Using the same format and labeling as
 +
before, we may note that Model has, from 2^5 = 32 possible interpretations,
 +
made 11 evaluations, and found 3 models answering the generic descriptions
 +
that were imposed by the logical input file.
 +
 +
Model Outline
 +
o------------------------o
 +
| living_thing          |
 +
|  non_living -          |  1
 +
|  (non_living )        |
 +
|  mineral -            |  2
 +
|  (mineral )          |
 +
|    animal              |
 +
|    vegetable -        |  3
 +
|    (vegetable ) *    |  4 *
 +
|    (animal )          |
 +
|    vegetable *        |  5 *
 +
|    (vegetable ) -    |  6
 +
| (living_thing )        |
 +
|  non_living            |
 +
|  animal -            |  7
 +
|  (animal )            |
 +
|    vegetable -        |  8
 +
|    (vegetable )        |
 +
|    mineral *          |  9 *
 +
|    (mineral ) -      | 10
 +
|  (non_living ) -      | 11
 +
o------------------------o
 +
 +
Tenor Outline
 +
o------------------------o
 +
| living_thing          |
 +
|  (non_living )        |
 +
|  (mineral )          |
 +
|    animal              |
 +
|    (vegetable ) *    | <1>
 +
|    (animal )          |
 +
|    vegetable *        | <2>
 +
| (living_thing )        |
 +
|  non_living            |
 +
|  (animal )            |
 +
|    (vegetable )        |
 +
|    mineral *          | <3>
 +
o------------------------o
 +
 +
Sense Outline
 +
o------------------------o
 +
| living_thing          |
 +
|  animal                |
 +
|  vegetable            |
 +
| non_living            |
 +
|  mineral              |
 +
o------------------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Example.  Molly's World
 +
 +
I think that we are finally ready to tackle a more respectable example.
 +
The Example known as "Molly's World" is borrowed from the literature on
 +
computational learning theory, adapted with a few changes from the example
 +
called "Molly’s Problem" in the paper "Learning With Hints" by Dana Angluin.
 +
By way of setting up the problem, I quote Angluin's motivational description:
 +
 +
| Imagine that you have become acquainted with an alien named Molly from the
 +
| planet Ornot, who is currently employed in a day-care center.  She is quite
 +
| good at propositional logic, but a bit weak on knowledge of Earth.  So you
 +
| decide to formulate the beginnings of a propositional theory to help her
 +
| label things in her immediate environment.
 +
|
 +
| Angluin, Dana, "Learning With Hints", pages 167-181, in:
 +
| David Haussler & Leonard Pitt (eds.), 'Proceedings of the 1988 Workshop
 +
| on Computational Learning Theory', Morgan Kaufmann, San Mateo, CA, 1989.
 +
 +
The purpose of this quaint pretext is, of course, to make sure that the
 +
reader appreciates the constraints of the problem:  that no extra savvy
 +
is fair, all facts must be presumed or deduced on the immediate premises.
 +
 +
My use of this example is not directly relevant to the purposes of the
 +
discussion from which it is taken, so I simply give my version of it
 +
without comment on those issues.
 +
 +
Here is my rendition of the initial knowledge base delimiting Molly’s World:
 +
 +
Logical Input File:  Molly.Log
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
| ( object ,( toy ),( vehicle ))                                      |
 +
| (( small_size ),( medium_size ),( large_size ))                    |
 +
| (( two_wheels ),( three_wheels ),( four_wheels ))                  |
 +
| (( no_seat ),( one_seat ),( few_seats ),( many_seats ))            |
 +
| ( object ,( scooter ),( bike ),( trike ),( car ),( bus ),( wagon )) |
 +
| ( two_wheels    no_seat            ,( scooter ))                    |
 +
| ( two_wheels    one_seat    pedals ,( bike ))                      |
 +
| ( three_wheels  one_seat    pedals ,( trike ))                      |
 +
| ( four_wheels  few_seats  doors  ,( car ))                        |
 +
| ( four_wheels  many_seats  doors  ,( bus ))                        |
 +
| ( four_wheels  no_seat    handle ,( wagon ))                      |
 +
| ( scooter          ( toy  small_size ))                            |
 +
| ( wagon            ( toy  small_size ))                            |
 +
| ( trike            ( toy  small_size ))                            |
 +
| ( bike  small_size  ( toy ))                                        |
 +
| ( bike  medium_size ( vehicle ))                                    |
 +
| ( bike  large_size  )                                              |
 +
| ( car              ( vehicle  large_size ))                        |
 +
| ( bus              ( vehicle  large_size ))                        |
 +
| ( toy              ( object ))                                    |
 +
| ( vehicle          ( object ))                                    |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
 +
All of the logical forms that are used in the preceding Log file
 +
will probably be familiar from earlier discussions.  The purpose
 +
of one or two constructions may, however, be a little obscure,
 +
so I will insert a few words of additional explanation here:
 +
 +
The rule "( bike large_size )", for example, merely
 +
says that nothing can be both a bike and large_size.
 +
 +
The rule "( three_wheels one_seat pedals ,( trike ))" says that anything
 +
with all the features of three_wheels, one_seat, and pedals is excluded
 +
from being anything but a trike.  In short, anything with just those
 +
three features is equivalent to a trike.
 +
 +
Recall that the form "( p , q )" may be interpreted to assert either
 +
the exclusive disjunction or the logical inequivalence of p and q.
 +
 +
The rules have been stated in this particular way simply
 +
to imitate the style of rules in the reference example.
 +
 +
This last point does bring up an important issue, the question
 +
of "rhetorical" differences in expression and their potential
 +
impact on the "pragmatics" of computation.  Unfortunately,
 +
I will have to abbreviate my discussion of this topic for
 +
now, and only mention in passing the following facts.
 +
 +
Logically equivalent expressions, even though they must lead
 +
to logically equivalent normal forms, may have very different
 +
characteristics when it comes to the efficiency of processing.
 +
 +
For instance, consider the following four forms:
 +
 +
| 1.  (( p , q ))
 +
|
 +
| 2.  ( p ,( q ))
 +
|
 +
| 3.  (( p ), q )
 +
|
 +
| 4.  (( p , q ))
 +
 +
All of these are equally succinct ways of maintaining that
 +
p is logically equivalent to q, yet each can have different
 +
effects on the route that Model takes to arrive at an answer.
 +
Apparently, some equalities are more equal than others.
 +
 +
These effects occur partly because the algorithm chooses to make cases
 +
of variables on a basis of leftmost shallowest first, but their impact
 +
can be complicated by the interactions that each expression has with
 +
the context that it occupies.  The main lesson to take away from all
 +
of this, at least, for the time being, is that it is probably better
 +
not to bother too much about these problems, but just to experiment
 +
with different ways of expressing equivalent pieces of information
 +
until you get a sense of what works best in various situations.
 +
 +
I think that you will be happy to see only the
 +
ultimate Sense of Molly’s World, so here it is:
 +
 +
Sense Outline:  Molly.Sen
 +
o------------------------o
 +
| object                |
 +
|  two_wheels            |
 +
|  no_seat              |
 +
|    scooter            |
 +
|    toy                |
 +
|      small_size        |
 +
|  one_seat            |
 +
|    pedals              |
 +
|    bike              |
 +
|      small_size        |
 +
|      toy              |
 +
|      medium_size      |
 +
|      vehicle          |
 +
|  three_wheels          |
 +
|  one_seat            |
 +
|    pedals              |
 +
|    trike              |
 +
|      toy              |
 +
|      small_size      |
 +
|  four_wheels          |
 +
|  few_seats            |
 +
|    doors              |
 +
|    car                |
 +
|      vehicle          |
 +
|      large_size      |
 +
|  many_seats          |
 +
|    doors              |
 +
|    bus                |
 +
|      vehicle          |
 +
|      large_size      |
 +
|  no_seat              |
 +
|    handle              |
 +
|    wagon              |
 +
|      toy              |
 +
|      small_size      |
 +
o------------------------o
 +
 +
This outline is not the Sense of the unconstrained Log file,
 +
but the result of running Model with a query on the single
 +
feature "object".  Using this focus helps the Modeler
 +
to make more relevant Sense of Molly’s World.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DM = Douglas McDavid
 +
 +
DM: This, again, is an example of how real issues of ontology are
 +
    so often trivialized at the expense of technicalities.  I just
 +
    had a burger, some fries, and a Coke.  I would say all that was
 +
    non-living and non-mineral.  A virus, I believe is non-animal,
 +
    non-vegetable, but living (and non-mineral).  Teeth, shells,
 +
    and bones are virtually pure mineral, but living.  These are
 +
    the kinds of issues that are truly "ontological," in my
 +
    opinion.  You are not the only one to push them into
 +
    the background as of lesser importance.  See the
 +
    discussion of "18-wheelers" in John Sowa's book.
 +
 +
it's not my example, and from you say, it's not your example either.
 +
copied it out of a book or a paper somewhere, too long ago to remember.
 +
i am assuming that the author or tardition from which it came must have
 +
seen some kind of sense in it.  tell you what, write out your own theory
 +
of "what is" in so many variables, more or less, publish it in a book or
 +
a paper, and then folks will tell you that they dispute each and every
 +
thing that you have just said, and it won't really matter all that much
 +
how complex it is or how subtle you are.  that has been the way of all
 +
ontology for about as long as anybody can remember or even read about.
 +
me?  i don't have sufficient arrogance to be an ontologist, and you
 +
know that's saying a lot, as i can't even imagine a way to convince
 +
myself that i believe i know "what is", really and truly for sure
 +
like some folks just seem to do.  so i am working to improve our
 +
technical ability to do logic, which is mostly a job of shooting
 +
down the more serious delusions that we often get ourselves into.
 +
can i be of any use to ontologists?  i dunno.  i guess it depends
 +
on how badly they are attached to some of the delusions of knowing
 +
what their "common" sense tells them everybody ought to already know,
 +
but that every attempt to check that out in detail tells them it just
 +
ain't so.  a problem for which denial was just begging to be invented,
 +
and so it was.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Example.  Molly's World (cont.)
 +
 +
In preparation for a contingently possible future discussion,
 +
I need to attach a few parting thoughts to the case workup
 +
of Molly's World that may not seem terribly relevant to
 +
the present setting, but whose pertinence I hope will
 +
become clearer in time.
 +
 +
The logical paradigm from which this Example was derived is that
 +
of "Zeroth Order Horn Clause Theories".  The clauses at issue
 +
in these theories are allowed to be of just three kinds:
 +
 +
| 1.  p & q & r & ... => z
 +
|
 +
| 2.  z
 +
|
 +
| 3.  ~[p & q & r & ...]
 +
 +
Here, the proposition letters "p", "q", "r", ..., "z"
 +
are restricted to being single positive features, not
 +
themselves negated or otherwise complex expressions.
 +
 +
In the Cactus Language or Existential Graph syntax
 +
these forms would take on the following appearances:
 +
 +
| 1.  ( p q r ... ( z ))
 +
|
 +
| 2.    z
 +
|
 +
| 3.  ( p q r ... )
 +
 +
The style of deduction in Horn clause logics is essentially
 +
proof-theoretic in character, with the main burden of proof
 +
falling on implication relations ("=>") and on "projective"
 +
forms of inference, that is, information-losing inferences
 +
like modus ponens and resolution.  Cf. [Llo], [MaW].
 +
 +
In contrast, the method used here is substantially model-theoretic,
 +
the stress being to start from more general forms of expression for
 +
laying out facts (for example, distinctions, equations, partitions)
 +
and to work toward results that maintain logical equivalence with
 +
their origins.
 +
 +
What all of this has to do with the output above is this:
 +
>From the perspective that is adopted in the present work,
 +
almost any theory, for example, the one that is founded
 +
on the postulates of Molly's World, will have far more
 +
models than the implicational and inferential mode of
 +
reasoning is designed to discover.  We will be forced
 +
to confront them, however, if we try to run Model on
 +
a large set of implications.
 +
 +
The typical Horn clause interpreter gets around this
 +
difficulty only by a stratagem that takes clauses to
 +
mean something other than what they say, that is, by
 +
distorting the principles of semantics in practice.
 +
Our Model, on the other hand, has no such finesse.
 +
 +
This explains why it was necessary to impose the
 +
prerequisite "object" constraint on the Log file
 +
for Molly's World.  It supplied no more than what
 +
we usually take for granted, in order to obtain
 +
a set of models that we would normally think of
 +
as being the intended import of the definitions.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Example.  Jets & Sharks
 +
 +
The propositional calculus based on the boundary operator, that is,
 +
the multigrade logical connective of the form "( , , , ... )" 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 p, q, r by means
 +
of the expression "( p , q , r )".  To illustrate this possibility,
 +
I transcribe into cactus language expressions a notorious example
 +
from the "parallel distributed processing" (PDP) paradigm [McR]
 +
and work through two of the associated exercises as portrayed
 +
in this format.
 +
 +
Logical Input File:  JAS  =  ZOT(Jets And Sharks)
 +
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 to the proposition that
 +
defines the Jets and Sharks knowledge base,
 +
that is to say, the knowledge that we are
 +
given about the Jets and Sharks, not the
 +
knowledge that the Jets and Sharks have.
 +
 +
With a query on the name "ken" we obtain the following
 +
output, giving all of the features associated with Ken:
 +
 +
Sense Outline:  JAS & 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 of the features
 +
that satisfy these constraints:
 +
 +
Sense Outline:  JAS & College & Sharks
 +
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.  Furthermore,
 +
we have a complete listing of their names
 +
broken down by occupation, as I have no
 +
doubt that all of them will be in time.
 +
 +
| Reference:
 +
|
 +
| McClelland, James L. & Rumelhart, David E.,
 +
|'Explorations in Parallel Distributed Processing:
 +
| A Handbook of Models, Programs, and Exercises',
 +
| MIT Press, Cambridge, MA, 1988.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
One of the issues that my pondering weak and weary over
 +
has caused me to burn not a few barrels of midnight oil
 +
over the past elventeen years or so is the relationship
 +
among divers and sundry "styles of inference", by which
 +
I mean particular choices of inference paradigms, rules,
 +
or schemata.  The chief breakpoint seems to lie between
 +
information-losing and information-maintaining modes of
 +
inference, also called "implicational" and "equational",
 +
or "projective" and "preservative" brands, respectively.
 +
 +
Since it appears to be mostly the implicational and projective
 +
styles of inference that are more familiar to folks hereabouts,
 +
I will start off this subdiscussion by introducing a number of
 +
risibly simple but reasonably manageable examples of the other
 +
brand of inference, treated as equational reasoning approaches
 +
to problems about satisfying "zeroth order constraints" (ZOC's).
 +
 +
Applications of a Propositional Calculator:
 +
Constraint Satisfaction Problems.
 +
Jon Awbrey, April 24, 1995.
 +
 +
The Four Houses Puzzle
 +
 +
Constructed on the model of the "Five Houses Puzzle" in [VaH, 132-136].
 +
 +
Problem Statement.  Four people with different nationalities live in the
 +
first four houses of a street.  They practice four distinct professions,
 +
and each of them has a favorite animal, all of them different.  The four
 +
houses are painted different colors.  The following facts are known:
 +
 +
|  1.  The Englander lives in the first house on the left.
 +
|  2.  The doctor lives in the second house.
 +
|  3.  The third house is painted red.
 +
|  4.  The zebra is a favorite in the fourth house.
 +
|  5.  The person in the first house has a dog.
 +
|  6.  The Japanese lives in the third house.
 +
|  7.  The red house is on the left of the yellow one.
 +
|  8.  They breed snails in the house to right of the doctor.
 +
|  9.  The Englander lives next to the green house.
 +
| 10.  The fox is in the house next to to the diplomat.
 +
| 11.  The Spaniard likes zebras.
 +
| 12.  The Japanese is a painter.
 +
| 13.  The Italian lives in the green house.
 +
| 14.  The violinist lives in the yellow house.
 +
| 15.  The dog is a pet in the blue house.
 +
| 16.  The doctor keeps a fox.
 +
 +
The problem is to find all of the assignments of
 +
features to houses that satisfy these requirements.
 +
 +
Logical Input File:  House^4.Log
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|  eng_1  doc_2  red_3  zeb_4  dog_1  jap_3                          |
 +
|                                                                    |
 +
|  (( red_1  yel_2 ),( red_2  yel_3 ),( red_3  yel_4 ))              |
 +
|  (( doc_1  sna_2 ),( doc_2  sna_3 ),( doc_3  sna_4 ))              |
 +
|                                                                    |
 +
|  (( eng_1  gre_2 ),                                                |
 +
|  ( eng_2  gre_3 ),( eng_2  gre_1 ),                                |
 +
|  ( eng_3  gre_4 ),( eng_3  gre_2 ),                                |
 +
|                    ( eng_4  gre_3 ))                                |
 +
|                                                                    |
 +
|  (( dip_1  fox_2 ),                                                |
 +
|  ( dip_2  fox_3 ),( dip_2  fox_1 ),                                |
 +
|  ( dip_3  fox_4 ),( dip_3  fox_2 ),                                |
 +
|                    ( dip_4  fox_3 ))                                |
 +
|                                                                    |
 +
|  (( spa_1 zeb_1 ),( spa_2 zeb_2 ),( spa_3 zeb_3 ),( spa_4 zeb_4 ))  |
 +
|  (( jap_1 pai_1 ),( jap_2 pai_2 ),( jap_3 pai_3 ),( jap_4 pai_4 ))  |
 +
|  (( ita_1 gre_1 ),( ita_2 gre_2 ),( ita_3 gre_3 ),( ita_4 gre_4 ))  |
 +
|                                                                    |
 +
|  (( yel_1 vio_1 ),( yel_2 vio_2 ),( yel_3 vio_3 ),( yel_4 vio_4 ))  |
 +
|  (( blu_1 dog_1 ),( blu_2 dog_2 ),( blu_3 dog_3 ),( blu_4 dog_4 ))  |
 +
|                                                                    |
 +
|  (( doc_1 fox_1 ),( doc_2 fox_2 ),( doc_3 fox_3 ),( doc_4 fox_4 ))  |
 +
|                                                                    |
 +
|  ((                                                                |
 +
|                                                                    |
 +
|  (( eng_1 ),( eng_2 ),( eng_3 ),( eng_4 ))                          |
 +
|  (( spa_1 ),( spa_2 ),( spa_3 ),( spa_4 ))                          |
 +
|  (( jap_1 ),( jap_2 ),( jap_3 ),( jap_4 ))                          |
 +
|  (( ita_1 ),( ita_2 ),( ita_3 ),( ita_4 ))                          |
 +
|                                                                    |
 +
|  (( eng_1 ),( spa_1 ),( jap_1 ),( ita_1 ))                          |
 +
|  (( eng_2 ),( spa_2 ),( jap_2 ),( ita_2 ))                          |
 +
|  (( eng_3 ),( spa_3 ),( jap_3 ),( ita_3 ))                          |
 +
|  (( eng_4 ),( spa_4 ),( jap_4 ),( ita_4 ))                          |
 +
|                                                                    |
 +
|  (( gre_1 ),( gre_2 ),( gre_3 ),( gre_4 ))                          |
 +
|  (( red_1 ),( red_2 ),( red_3 ),( red_4 ))                          |
 +
|  (( yel_1 ),( yel_2 ),( yel_3 ),( yel_4 ))                          |
 +
|  (( blu_1 ),( blu_2 ),( blu_3 ),( blu_4 ))                          |
 +
|                                                                    |
 +
|  (( gre_1 ),( red_1 ),( yel_1 ),( blu_1 ))                          |
 +
|  (( gre_2 ),( red_2 ),( yel_2 ),( blu_2 ))                          |
 +
|  (( gre_3 ),( red_3 ),( yel_3 ),( blu_3 ))                          |
 +
|  (( gre_4 ),( red_4 ),( yel_4 ),( blu_4 ))                          |
 +
|                                                                    |
 +
|  (( pai_1 ),( pai_2 ),( pai_3 ),( pai_4 ))                          |
 +
|  (( dip_1 ),( dip_2 ),( dip_3 ),( dip_4 ))                          |
 +
|  (( vio_1 ),( vio_2 ),( vio_3 ),( vio_4 ))                          |
 +
|  (( doc_1 ),( doc_2 ),( doc_3 ),( doc_4 ))                          |
 +
|                                                                    |
 +
|  (( pai_1 ),( dip_1 ),( vio_1 ),( doc_1 ))                          |
 +
|  (( pai_2 ),( dip_2 ),( vio_2 ),( doc_2 ))                          |
 +
|  (( pai_3 ),( dip_3 ),( vio_3 ),( doc_3 ))                          |
 +
|  (( pai_4 ),( dip_4 ),( vio_4 ),( doc_4 ))                          |
 +
|                                                                    |
 +
|  (( dog_1 ),( dog_2 ),( dog_3 ),( dog_4 ))                          |
 +
|  (( zeb_1 ),( zeb_2 ),( zeb_3 ),( zeb_4 ))                          |
 +
|  (( fox_1 ),( fox_2 ),( fox_3 ),( fox_4 ))                          |
 +
|  (( sna_1 ),( sna_2 ),( sna_3 ),( sna_4 ))                          |
 +
|                                                                    |
 +
|  (( dog_1 ),( zeb_1 ),( fox_1 ),( sna_1 ))                          |
 +
|  (( dog_2 ),( zeb_2 ),( fox_2 ),( sna_2 ))                          |
 +
|  (( dog_3 ),( zeb_3 ),( fox_3 ),( sna_3 ))                          |
 +
|  (( dog_4 ),( zeb_4 ),( fox_4 ),( sna_4 ))                          |
 +
|                                                                    |
 +
|  ))                                                                |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
 +
Sense Outline:  House^4.Sen
 +
o-----------------------------o
 +
| eng_1                      |
 +
|  doc_2                      |
 +
|  red_3                    |
 +
|    zeb_4                    |
 +
|    dog_1                  |
 +
|      jap_3                  |
 +
|      yel_4                |
 +
|        sna_3                |
 +
|        gre_2              |
 +
|          dip_1              |
 +
|          fox_2            |
 +
|            spa_4            |
 +
|            pai_3          |
 +
|              ita_2          |
 +
|              vio_4        |
 +
|                blu_1        |
 +
o-----------------------------o
 +
 +
Table 1.  Solution to the Four Houses Puzzle
 +
o------------o------------o------------o------------o------------o
 +
|            | House 1    | House 2    | House 3    | House 4    |
 +
o------------o------------o------------o------------o------------o
 +
| Nation    | England    | Italy      | Japan      | Spain      |
 +
| Color      | blue      | green      | red        | yellow    |
 +
| Profession | diplomat  | doctor    | painter    | violinist  |
 +
| Animal    | dog        | fox        | snails    | zebra      |
 +
o------------o------------o------------o------------o------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
First off, I do not trivialize the "real issues of ontology", indeed,
 +
it is precisely my estimate of the non-trivial difficulty of this task,
 +
of formulating the types of "generic ontology" that we propose to do here,
 +
that forces me to choose and to point out the inescapability of the approach
 +
that I am currently taking, which is to enter on the necessary preliminary of
 +
building up the logical tools that we need to tackle the ontology task proper.
 +
And I would say, to the contrary, that it is those who think we can arrive at
 +
a working general ontology by sitting on the porch shooting the breeze about
 +
"what it is" until the cows come home -- that is, the method for which it
 +
has become cliche to indict the Ancient Greeks, though, if truth be told,
 +
we'd have to look to the pre-socratics and the pre-stoics to find a good
 +
match for the kinds of revelation that are common hereabouts -- I would
 +
say that it's those folks who trivialize the "real issues of ontology".
 +
 +
A person, living in our times, who is serious about knowing the being of things,
 +
really only has one choice -- to pick what tiny domain of things he or she just
 +
has to know about the most, thence to hie away to the adept gurus of the matter
 +
in question, forgeting the rest, cause "general ontology" is a no-go these days.
 +
It is presently in a state like astronomy before telescopes, and that means not
 +
entirely able to discern itself from astrology and other psychically projective
 +
exercises of wishful and dreadful thinking like that.
 +
 +
So I am busy grinding lenses ...
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DM = Douglas McDavid
 +
 +
DM: Thanks for both the original and additional response.  I'm not trying to
 +
    single you out, as I have been picking  on various postings in a similar
 +
    manner ever since I started contributing to this discussion.  I agree with
 +
    you that the task of this working group is non-trivially difficult.  In fact,
 +
    I believe we are still a long way from a clear and useful agreement about what
 +
    constitutes "upper" ontology, and what it would mean to standardize it.  However,
 +
    I don't agree that the only place to make progress is in tiny domains of things.
 +
    I've contributed the thought that a fundamental, upper-level concept is the
 +
    concept of system, and that that would be a good place to begin.  And I'll
 +
    never be able to refrain from evaluating the content as well as the form
 +
    of any examples presented for consideration here.  Probably should
 +
    accompany these comments with a ;-)
 +
 +
There will never be a standard universal ontology
 +
of the absolute essential impertubable monolithic
 +
variety that some people still dream of in their
 +
fantasies of spectating on and speculating about
 +
a pre-relativistically non-participatory universe
 +
from their singular but isolated gods'eye'views.
 +
The bells tolled for that one many years ago,
 +
but some of the more blithe of the blissful
 +
islanders have just not gotten the news yet.
 +
 +
But there is still a lot to do that would be useful
 +
under the banner of a "standard upper ontology",
 +
if only we stay loose in our interpretation
 +
of what that implies in practical terms.
 +
 +
One likely approach to the problem would be to take
 +
a hint from the afore-allusioned history of physics --
 +
to inquire for whom, else, the bell tolls -- and to
 +
see if there are any bits of wisdom from that prior
 +
round of collective experience that can be adapted
 +
by dint of analogy to our present predicament.
 +
I happen to think that there are.
 +
 +
And there the answer was, not to try and force a return,
 +
though lord knows they all gave it their very best shot,
 +
to an absolute and imperturbable framework of existence,
 +
but to see the reciprocal participant relation that all
 +
partakers have to the constitution of that framing, yes,
 +
even unto those who would abdictators and abstainees be.
 +
 +
But what does that imply about some shred of a standard?
 +
It means that we are better off seeking, not a standard,
 +
one-size-fits-all ontology, but more standard resources
 +
for trying to interrelate diverse points of view and to
 +
transform the data that's gathered from one perspective
 +
in ways that it can most appropriately be compared with
 +
the data that is gathered from other standpoints on the
 +
splendorous observational scenes and theorematic stages.
 +
 +
That is what I am working on.
 +
And it hasn't been merely
 +
for a couple of years.
 +
 +
As to this bit:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|  ( living_thing , non_living )                |
 +
|                                                |
 +
|  (( animal ),( vegetable ),( mineral ))        |
 +
|                                                |
 +
|  ( living_thing ,( animal ),( vegetable ))    |
 +
|                                                |
 +
|  ( mineral ( non_living ))                    |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
My 5-dimensional Example, that I borrowed from some indifferent source
 +
of what is commonly recognized as "common sense" -- and I think rather
 +
obviously designed more for the classification of pre-modern species
 +
of whole critters and pure matters of natural substance than the
 +
motley mixture of un/natural and in/organic conglouterites that
 +
we find served up on the menu of modernity -- was not intended
 +
even so much as a toy ontology, but simply as an expository
 +
example, concocted for the sake of illustrating the sorts
 +
of logical interaction that occur among four different
 +
patterns of logical constraint, all of which types
 +
arise all the time no matter what the domain, and
 +
which I believe that my novel forms of expression,
 +
syntactically speaking, express quite succinctly,
 +
especially when you contemplate the complexities
 +
of the computation that may flow and must follow
 +
from even these meagre propositional expressions.
 +
 +
Yes, systems -- but -- even here usage differs in significant ways.
 +
I have spent ten years now trying to integrate my earlier efforts
 +
under an explicit systems banner, but even within the bounds of
 +
a systems engineering programme at one site there is a wide
 +
semantic dispersion that issues from this word "system".
 +
I am committed, and in writing, to taking what we so
 +
glibly and prospectively call "intelligent systems"
 +
seriously as dynamical systems.  That has many
 +
consequences, and I have to pick and choose
 +
which of those I may be suited to follow.
 +
 +
But that is too long a story for now ...
 +
 +
";-)"?
 +
 +
Somehow that has always looked like
 +
the Chesshire Cat's grin to me ...
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
By way of catering to popular demand, I have decided to
 +
render this symposium a bit more à la carte, and thus to
 +
serve up as faster food than heretofore a choice selection
 +
of the more sumptuous bits that I have in my logical larder,
 +
not yet full fare, by any means, but a sample of what might
 +
one day approach to being an abundantly moveable feast of
 +
ontological contents and general metaphysical delights.
 +
I'll leave it to you to name your poison, as it were.
 +
 +
Applications of a Propositional Calculator:
 +
Constraint Satisfaction Problems.
 +
Jon Awbrey, April 24, 1995.
 +
 +
Fabric Knowledge Base
 +
Based on the example in [MaW, pages 8-16].
 +
 +
Logical Input File:  Fab.Log
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
| (has_floats , plain_weave )                                        |
 +
| (has_floats ,(twill_weave ),(satin_weave ))                        |
 +
|                                                                    |
 +
| (plain_weave ,                                                      |
 +
|  (plain_weave  one_color ),                                        |
 +
|  (color_groups  ),                                                  |
 +
|  (grouped_warps ),                                                  |
 +
|  (some_thicker  ),                                                  |
 +
|  (crossed_warps ),                                                  |
 +
|  (loop_threads  ),                                                  |
 +
|  (plain_weave  flannel ))                                          |
 +
|                                                                    |
 +
| (plain_weave  one_color  cotton  balanced  smooth  ,(percale ))    |
 +
| (plain_weave  one_color  cotton            sheer  ,(organdy ))    |
 +
| (plain_weave  one_color  silk              sheer  ,(organza ))    |
 +
|                                                                    |
 +
| (plain_weave  color_groups  warp_stripe  fill_stripe ,(plaid  ))  |
 +
| (plaid        equal_stripe                          ,(gingham ))  |
 +
|                                                                    |
 +
| (plain_weave  grouped_warps ,(basket_weave ))                      |
 +
|                                                                    |
 +
| (basket_weave  typed ,                                              |
 +
|  (type_2_to_1 ),                                                    |
 +
|  (type_2_to_2 ),                                                    |
 +
|  (type_4_to_4 ))                                                    |
 +
|                                                                    |
 +
| (basket_weave  typed  type_2_to_1  thicker_fill  ,(oxford      )) |
 +
| (basket_weave  typed (type_2_to_2 ,                                |
 +
|                      type_4_to_4 ) same_thickness ,(monks_cloth )) |
 +
| (basket_weave (typed )              rough  open    ,(hopsacking  )) |
 +
|                                                                    |
 +
| (typed (basket_weave ))                                            |
 +
|                                                                    |
 +
| (basket_weave ,(oxford ),(monks_cloth ),(hopsacking ))              |
 +
|                                                                    |
 +
| (plain_weave  some_thicker ,(ribbed_weave ))                      |
 +
|                                                                    |
 +
| (ribbed_weave ,(small_rib ),(medium_rib ),(heavy_rib ))            |
 +
| (ribbed_weave ,(flat_rib  ),(round_rib ))                          |
 +
|                                                                    |
 +
| (ribbed_weave  thicker_fill          ,(cross_ribbed ))              |
 +
| (cross_ribbed  small_rib  flat_rib  ,(faille      ))              |
 +
| (cross_ribbed  small_rib  round_rib ,(grosgrain    ))              |
 +
| (cross_ribbed  medium_rib  round_rib ,(bengaline    ))              |
 +
| (cross_ribbed  heavy_rib  round_rib ,(ottoman      ))              |
 +
|                                                                    |
 +
| (cross_ribbed ,(faille ),(grosgrain ),(bengaline ),(ottoman ))      |
 +
|                                                                    |
 +
| (plain_weave  crossed_warps ,(leno_weave  ))                        |
 +
| (leno_weave  open          ,(marquisette ))                        |
 +
| (plain_weave  loop_threads  ,(pile_weave ))                        |
 +
|                                                                    |
 +
| (pile_weave ,(fill_pile ),(warp_pile ))                            |
 +
| (pile_weave ,(cut ),(uncut ))                                      |
 +
|                                                                    |
 +
| (pile_weave  warp_pile  cut                  ,(velvet    ))        |
 +
| (pile_weave  fill_pile  cut    aligned_pile  ,(corduroy  ))        |
 +
| (pile_weave  fill_pile  cut    staggered_pile ,(velveteen ))        |
 +
| (pile_weave  fill_pile  uncut  reversible    ,(terry    ))        |
 +
|                                                                    |
 +
| (pile_weave  fill_pile  cut ( (aligned_pile , staggered_pile ) ))  |
 +
|                                                                    |
 +
| (pile_weave ,(velvet ),(corduroy ),(velveteen ),(terry ))          |
 +
|                                                                    |
 +
| (plain_weave ,                                                      |
 +
|  (percale    ),(organdy    ),(organza    ),(plaid  ),            |
 +
|  (oxford    ),(monks_cloth ),(hopsacking ),                        |
 +
|  (faille    ),(grosgrain  ),(bengaline  ),(ottoman ),            |
 +
|  (leno_weave ),(pile_weave  ),(plain_weave  flannel ))            |
 +
|                                                                    |
 +
| (twill_weave ,                                                      |
 +
|  (warp_faced ),                                                    |
 +
|  (filling_faced ),                                                  |
 +
|  (even_twill ),                                                    |
 +
|  (twill_weave  flannel ))                                          |
 +
|                                                                    |
 +
| (twill_weave  warp_faced  colored_warp  white_fill ,(denim ))      |
 +
| (twill_weave  warp_faced  one_color                ,(drill ))      |
 +
| (twill_weave  even_twill  diagonal_rib            ,(serge ))      |
 +
|                                                                    |
 +
| (twill_weave  warp_faced (                                          |
 +
|  (one_color ,                                                      |
 +
|  ((colored_warp )(white_fill )) )                                  |
 +
| ))                                                                  |
 +
|                                                                    |
 +
| (twill_weave  warp_faced ,(denim ),(drill ))                        |
 +
| (twill_weave  even_twill ,(serge ))                                |
 +
|                                                                    |
 +
| ((                                                                  |
 +
|    (  ((plain_weave )(twill_weave ))                              |
 +
|        ((cotton      )(wool        )) napped ,(flannel ))          |
 +
| ))                                                                  |
 +
|                                                                    |
 +
| (satin_weave ,(warp_floats ),(fill_floats ))                        |
 +
|                                                                    |
 +
| (satin_weave ,(satin_weave smooth ),(satin_weave napped ))          |
 +
| (satin_weave ,(satin_weave cotton ),(satin_weave silk  ))          |
 +
|                                                                    |
 +
| (satin_weave  warp_floats  smooth        ,(satin    ))            |
 +
| (satin_weave  fill_floats  smooth        ,(sateen  ))            |
 +
| (satin_weave              napped  cotton ,(moleskin ))            |
 +
|                                                                    |
 +
| (satin_weave ,(satin ),(sateen ),(moleskin ))                      |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
 +
| Reference [MaW]
 +
|
 +
| Maier, David & Warren, David S.,
 +
|'Computing with Logic:  Logic Programming with Prolog',
 +
| Benjamin/Cummings, Menlo Park, CA, 1988.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
I think that it might be a good idea to go back to a simpler example
 +
of a constraint satisfaction problem, and to discuss the elements of
 +
its expression as a ZOT in a less cluttered setting before advancing
 +
onward once again to problems on the order of the Four Houses Puzzle.
 +
 +
| Applications of a Propositional Calculator:
 +
| Constraint Satisfaction Problems.
 +
| Jon Awbrey, April 24, 1995.
 +
 +
Graph Coloring
 +
 +
Based on the discussion in [Wil, page 196].
 +
 +
One is given three colors, say, orange, silver, indigo,
 +
and a graph on four nodes that has the following shape:
 +
 +
|          1
 +
|          o
 +
|        / \
 +
|        /  \
 +
|    4 o-----o 2
 +
|        \  /
 +
|        \ /
 +
|          o
 +
|          3
 +
 +
The problem is to color the nodes of the graph
 +
in such a way that no pair of nodes that are
 +
adjacent in the graph, that is, linked by
 +
an edge, get the same color.
 +
 +
The objective situation that is to be achieved can be represented
 +
in a so-called "declarative" fashion, in effect, by employing the
 +
cactus language as a very simple sort of declarative programming
 +
language, and depicting the prospective solution to the problem
 +
as a ZOT.
 +
 +
To do this, begin by declaring the following set of
 +
twelve boolean variables or "zeroth order features":
 +
 +
{1_orange, 1_silver, 1_indigo,
 +
2_orange, 2_silver, 2_indigo,
 +
3_orange, 3_silver, 3_indigo,
 +
4_orange, 4_silver, 4_indigo}
 +
 +
The interpretation to keep in mind will be such that
 +
the feature name of the form "<node i>_<color j>"
 +
says that the node i is assigned the color j.
 +
 +
Logical Input File:  Color.Log
 +
o----------------------------------------------------------------------o
 +
|                                                                      |
 +
|  (( 1_orange ),( 1_silver ),( 1_indigo ))                            |
 +
|  (( 2_orange ),( 2_silver ),( 2_indigo ))                            |
 +
|  (( 3_orange ),( 3_silver ),( 3_indigo ))                            |
 +
|  (( 4_orange ),( 4_silver ),( 4_indigo ))                            |
 +
|                                                                      |
 +
|  ( 1_orange  2_orange )( 1_silver  2_silver )( 1_indigo  2_indigo )  |
 +
|  ( 1_orange  4_orange )( 1_silver  4_silver )( 1_indigo  4_indigo )  |
 +
|  ( 2_orange  3_orange )( 2_silver  3_silver )( 2_indigo  3_indigo )  |
 +
|  ( 2_orange  4_orange )( 2_silver  4_silver )( 2_indigo  4_indigo )  |
 +
|  ( 3_orange  4_orange )( 3_silver  4_silver )( 3_indigo  4_indigo )  |
 +
|                                                                      |
 +
o----------------------------------------------------------------------o
 +
 +
The first stanza of verses declares that
 +
every node is assigned exactly one color.
 +
 +
The second stanza of verses declares that
 +
no adjacent nodes get the very same color.
 +
 +
Each satisfying interpretation of this ZOT
 +
that is also a program corresponds to what
 +
graffitists call a "coloring" of the graph.
 +
 +
Theme One's Model interpreter, when we set
 +
it to work on this ZOT, will array  before
 +
our eyes all of the colorings of the graph.
 +
 +
Sense Outline:  Color.Sen
 +
o-----------------------------o
 +
| 1_orange                    |
 +
|  2_silver                  |
 +
|  3_orange                  |
 +
|    4_indigo                |
 +
|  2_indigo                  |
 +
|  3_orange                  |
 +
|    4_silver                |
 +
| 1_silver                    |
 +
|  2_orange                  |
 +
|  3_silver                  |
 +
|    4_indigo                |
 +
|  2_indigo                  |
 +
|  3_silver                  |
 +
|    4_orange                |
 +
| 1_indigo                    |
 +
|  2_orange                  |
 +
|  3_indigo                  |
 +
|    4_silver                |
 +
|  2_silver                  |
 +
|  3_indigo                  |
 +
|    4_orange                |
 +
o-----------------------------o
 +
 +
| Reference [Wil]
 +
|
 +
| Wilf, Herbert S.,
 +
|'Algorithms and Complexity',
 +
| Prentice-Hall, Englewood Cliffs, NJ, 1986.
 +
|
 +
| Nota Bene.  There is a wrong Figure in some
 +
| printings of the book, that does not match
 +
| the description of the Example that is
 +
| given in the text.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Let us continue to examine the properties of the cactus language
 +
as a minimal style of declarative programming language.  Even in
 +
the likes of this zeroth order microcosm one can observe, and on
 +
a good day still more clearly for the lack of other distractions,
 +
many of the buzz worlds that will spring into full bloom, almost
 +
as if from nowhere, to become the first order of business in the
 +
latter day logical organa, plus combinators, plus lambda calculi.
 +
 +
By way of homage to the classics of the art, I can hardly pass
 +
this way without paying my dues to the next sample of examples.
 +
 +
N Queens Problem
 +
 +
I will give the ZOT that describes the N Queens Problem for N = 5,
 +
since that is the most that I and my old 286 could do when last I
 +
wrote up this Example.
 +
 +
The problem is now to write a "zeroth order program" (ZOP) that
 +
describes the following objective:  To place 5 chess queens on
 +
a 5 by 5 chessboard so that no queen attacks any other queen.
 +
 +
It is clear that there can be at most one queen on each row
 +
of the board and so by dint of regal necessity, exactly one
 +
queen in each row of the desired array.  This gambit allows
 +
us to reduce the problem to one of picking a permutation of
 +
five things in fives places, and this affords us sufficient
 +
clue to begin down a likely path toward the intended object,
 +
by recruiting the following phalanx of 25 logical variables:
 +
 +
Literal Input File:  Q5.Lit
 +
o---------------------------------------o
 +
|                                      |
 +
|  q1_r1, q1_r2, q1_r3, q1_r4, q1_r5,  |
 +
|  q2_r1, q2_r2, q2_r3, q2_r4, q2_r5,  |
 +
|  q3_r1, q3_r2, q3_r3, q3_r4, q3_r5,  |
 +
|  q4_r1, q4_r2, q4_r3, q4_r4, q4_r5,  |
 +
|  q5_r1, q5_r2, q5_r3, q5_r4, q5_r5.  |
 +
|                                      |
 +
o---------------------------------------o
 +
 +
Thus we seek to define a function, of abstract type f : %B%^25 -> %B%,
 +
whose fibre of truth f^(-1)(%1%) is a set of interpretations, each of
 +
whose elements bears the abstract type of a point in the space %B%^25,
 +
and whose reading will inform us of our desired set of configurations.
 +
 +
Logical Input File:  Q5.Log
 +
o------------------------------------------------------------o
 +
|                                                            |
 +
|  ((q1_r1 ),(q1_r2 ),(q1_r3 ),(q1_r4 ),(q1_r5 ))            |
 +
|  ((q2_r1 ),(q2_r2 ),(q2_r3 ),(q2_r4 ),(q2_r5 ))            |
 +
|  ((q3_r1 ),(q3_r2 ),(q3_r3 ),(q3_r4 ),(q3_r5 ))            |
 +
|  ((q4_r1 ),(q4_r2 ),(q4_r3 ),(q4_r4 ),(q4_r5 ))            |
 +
|  ((q5_r1 ),(q5_r2 ),(q5_r3 ),(q5_r4 ),(q5_r5 ))            |
 +
|                                                            |
 +
|  ((q1_r1 ),(q2_r1 ),(q3_r1 ),(q4_r1 ),(q5_r1 ))            |
 +
|  ((q1_r2 ),(q2_r2 ),(q3_r2 ),(q4_r2 ),(q5_r2 ))            |
 +
|  ((q1_r3 ),(q2_r3 ),(q3_r3 ),(q4_r3 ),(q5_r3 ))            |
 +
|  ((q1_r4 ),(q2_r4 ),(q3_r4 ),(q4_r4 ),(q5_r4 ))            |
 +
|  ((q1_r5 ),(q2_r5 ),(q3_r5 ),(q4_r5 ),(q5_r5 ))            |
 +
|                                                            |
 +
|  ((                                                        |
 +
|                                                            |
 +
|  (q1_r1 q2_r2 )(q1_r1 q3_r3 )(q1_r1 q4_r4 )(q1_r1 q5_r5 )  |
 +
|                (q2_r2 q3_r3 )(q2_r2 q4_r4 )(q2_r2 q5_r5 )  |
 +
|                              (q3_r3 q4_r4 )(q3_r3 q5_r5 )  |
 +
|                                            (q4_r4 q5_r5 )  |
 +
|                                                            |
 +
|  (q1_r2 q2_r3 )(q1_r2 q3_r4 )(q1_r2 q4_r5 )                |
 +
|                (q2_r3 q3_r4 )(q2_r3 q4_r5 )                |
 +
|                              (q3_r4 q4_r5 )                |
 +
|                                                            |
 +
|  (q1_r3 q2_r4 )(q1_r3 q3_r5 )                              |
 +
|                (q2_r4 q3_r5 )                              |
 +
|                                                            |
 +
|  (q1_r4 q2_r5 )                                            |
 +
|                                                            |
 +
|  (q2_r1 q3_r2 )(q2_r1 q4_r3 )(q2_r1 q5_r4 )                |
 +
|                (q3_r2 q4_r3 )(q3_r2 q5_r4 )                |
 +
|                              (q4_r3 q5_r4 )                |
 +
|                                                            |
 +
|  (q3_r1 q4_r2 )(q3_r1 q5_r3 )                              |
 +
|                (q4_r2 q5_r3 )                              |
 +
|                                                            |
 +
|  (q4_r1 q5_r2 )                                            |
 +
|                                                            |
 +
|  (q1_r5 q2_r4 )(q1_r5 q3_r3 )(q1_r5 q4_r2 )(q1_r5 q5_r1 )  |
 +
|                (q2_r4 q3_r3 )(q2_r4 q4_r2 )(q2_r4 q5_r1 )  |
 +
|                              (q3_r3 q4_r2 )(q3_r3 q5_r1 )  |
 +
|                                            (q4_r2 q5_r1 )  |
 +
|                                                            |
 +
|  (q2_r5 q3_r4 )(q2_r5 q4_r3 )(q2_r5 q5_r2 )                |
 +
|                (q3_r4 q4_r3 )(q3_r4 q5_r2 )                |
 +
|                              (q4_r3 q5_r2 )                |
 +
|                                                            |
 +
|  (q3_r5 q4_r4 )(q3_r5 q5_r3 )                              |
 +
|                (q4_r4 q5_r3 )                              |
 +
|                                                            |
 +
|  (q4_r5 q5_r4 )                                            |
 +
|                                                            |
 +
|  (q1_r4 q2_r3 )(q1_r4 q3_r2 )(q1_r4 q4_r1 )                |
 +
|                (q2_r3 q3_r2 )(q2_r3 q4_r1 )                |
 +
|                              (q3_r2 q4_r1 )                |
 +
|                                                            |
 +
|  (q1_r3 q2_r2 )(q1_r3 q3_r1 )                              |
 +
|                (q2_r2 q3_r1 )                              |
 +
|                                                            |
 +
|  (q1_r2 q2_r1 )                                            |
 +
|                                                            |
 +
|  ))                                                        |
 +
|                                                            |
 +
o------------------------------------------------------------o
 +
 +
The vanguard of this logical regiment consists of two
 +
stock'a'block platoons, the pattern of whose features
 +
is the usual sort of array for conveying permutations.
 +
Between the stations of their respective offices they
 +
serve to warrant that all of the interpretations that
 +
are left standing on the field of valor at the end of
 +
the day will be ones that tell of permutations 5 by 5.
 +
The rest of the ruck and the runt of the mill in this
 +
regimental logos are there to cover the diagonal bias
 +
against attacking queens that is our protocol to suit.
 +
 +
And here is the issue of the day:
 +
 +
Sense Output:  Q5.Sen
 +
o-------------------o
 +
| q1_r1            |
 +
|  q2_r3            |
 +
|  q3_r5          |
 +
|    q4_r2          |
 +
|    q5_r4        | <1>
 +
|  q2_r4            |
 +
|  q3_r2          |
 +
|    q4_r5          |
 +
|    q5_r3        | <2>
 +
| q1_r2            |
 +
|  q2_r4            |
 +
|  q3_r1          |
 +
|    q4_r3          |
 +
|    q5_r5        | <3>
 +
|  q2_r5            |
 +
|  q3_r3          |
 +
|    q4_r1          |
 +
|    q5_r4        | <4>
 +
| q1_r3            |
 +
|  q2_r1            |
 +
|  q3_r4          |
 +
|    q4_r2          |
 +
|    q5_r5        | <5>
 +
|  q2_r5            |
 +
|  q3_r2          |
 +
|    q4_r4          |
 +
|    q5_r1        | <6>
 +
| q1_r4            |
 +
|  q2_r1            |
 +
|  q3_r3          |
 +
|    q4_r5          |
 +
|    q5_r2        | <7>
 +
|  q2_r2            |
 +
|  q3_r5          |
 +
|    q4_r3          |
 +
|    q5_r1        | <8>
 +
| q1_r5            |
 +
|  q2_r2            |
 +
|  q3_r4          |
 +
|    q4_r1          |
 +
|    q5_r3        | <9>
 +
|  q2_r3            |
 +
|  q3_r1          |
 +
|    q4_r4          |
 +
|    q5_r2        | <A>
 +
o-------------------o
 +
 +
The number at least checks with all of the best authorities,
 +
so I can breathe a sigh of relief on that account, at least.
 +
I am sure that there just has to be a more clever way to do
 +
this, that is to say, within the bounds of ZOT reason alone,
 +
but the above is the best that I could figure out with the
 +
time that I had at the time.
 +
 +
References:  [BaC, 166], [VaH, 122], [Wir, 143].
 +
 +
[BaC]  Ball, W.W. Rouse, & Coxeter, H.S.M.,
 +
      'Mathematical Recreations and Essays',
 +
      13th ed., Dover, New York, NY, 1987.
 +
 +
[VaH]  Van Hentenryck, Pascal,
 +
      'Constraint Satisfaction in Logic Programming,
 +
      MIT Press, Cambridge, MA, 1989.
 +
 +
[Wir]  Wirth, Niklaus,
 +
      'Algorithms + Data Structures = Programs',
 +
      Prentice-Hall, Englewood Cliffs, NJ, 1976.
 +
 +
http://mathworld.wolfram.com/QueensProblem.html
 +
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000170
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
I turn now to another golden oldie of a constraint satisfaction problem
 +
that I would like to give here a slightly new spin, but not so much for
 +
the sake of these trifling novelties as from a sense of old time's ache
 +
and a duty to -- well, what's the opposite of novelty?
 +
 +
Phobic Apollo
 +
 +
| Suppose Peter, Paul, and Jane are musicians.  One of them plays
 +
| saxophone, another plays guitar, and the third plays drums.  As
 +
| it happens, one of them is afraid of things associated with the
 +
| number 13, another of them is afraid of cats, and the third is
 +
| afraid of heights.  You also know that Peter and the guitarist
 +
| skydive, that Paul and the saxophone player enjoy cats, and
 +
| that the drummer lives in apartment 13 on the 13th floor.
 +
|
 +
| Soon we will want to use these facts to reason
 +
| about whether or not certain identity relations
 +
| hold or are excluded.  Assume X(Peter, Guitarist)
 +
| means "the person who is Peter is not the person who
 +
| plays the guitar".  In this notation, the facts become:
 +
|
 +
| 1.  X(Peter, Guitarist)
 +
| 2.  X(Peter, Fears Heights)
 +
| 3.  X(Guitarist, Fears Heights)
 +
| 4.  X(Paul, Fears Cats)
 +
| 5.  X(Paul, Saxophonist)
 +
| 6.  X(Saxophonist, Fears Cats)
 +
| 7.  X(Drummer, Fears 13)
 +
| 8.  X(Drummer, Fears Heights)
 +
|
 +
| Exercise attributed to Kenneth D. Forbus, pages 449-450 in:
 +
| Patrick Henry Winston, 'Artificial Intelligence', 2nd ed.,
 +
| Addison-Wesley, Reading, MA, 1984.
 +
 +
Here is one way to represent these facts in the form of a ZOT
 +
and use it as a logical program to draw a succinct conclusion:
 +
 +
Logical Input File:  ConSat.Log
 +
o-----------------------------------------------------------------------o
 +
|                                                                      |
 +
|  (( pete_plays_guitar ),( pete_plays_sax ),( pete_plays_drums ))      |
 +
|  (( paul_plays_guitar ),( paul_plays_sax ),( paul_plays_drums ))      |
 +
|  (( jane_plays_guitar ),( jane_plays_sax ),( jane_plays_drums ))      |
 +
|                                                                      |
 +
|  (( pete_plays_guitar ),( paul_plays_guitar ),( jane_plays_guitar ))  |
 +
|  (( pete_plays_sax    ),( paul_plays_sax    ),( jane_plays_sax    ))  |
 +
|  (( pete_plays_drums  ),( paul_plays_drums  ),( jane_plays_drums  ))  |
 +
|                                                                      |
 +
|  (( pete_fears_13 ),( pete_fears_cats ),( pete_fears_height ))        |
 +
|  (( paul_fears_13 ),( paul_fears_cats ),( paul_fears_height ))        |
 +
|  (( jane_fears_13 ),( jane_fears_cats ),( jane_fears_height ))        |
 +
|                                                                      |
 +
|  (( pete_fears_13    ),( paul_fears_13    ),( jane_fears_13    ))  |
 +
|  (( pete_fears_cats  ),( paul_fears_cats  ),( jane_fears_cats  ))  |
 +
|  (( pete_fears_height ),( paul_fears_height ),( jane_fears_height ))  |
 +
|                                                                      |
 +
|  ((                                                                  |
 +
|                                                                      |
 +
|  ( pete_plays_guitar )                                                |
 +
|  ( pete_fears_height )                                                |
 +
|                                                                      |
 +
|  ( pete_plays_guitar  pete_fears_height )                            |
 +
|  ( paul_plays_guitar  paul_fears_height )                            |
 +
|  ( jane_plays_guitar  jane_fears_height )                            |
 +
|                                                                      |
 +
|  ( paul_fears_cats )                                                  |
 +
|  ( paul_plays_sax  )                                                  |
 +
|                                                                      |
 +
|  ( pete_plays_sax  pete_fears_cats )                                  |
 +
|  ( paul_plays_sax  paul_fears_cats )                                  |
 +
|  ( jane_plays_sax  jane_fears_cats )                                  |
 +
|                                                                      |
 +
|  ( pete_plays_drums  pete_fears_13 )                                  |
 +
|  ( paul_plays_drums  paul_fears_13 )                                  |
 +
|  ( jane_plays_drums  jane_fears_13 )                                  |
 +
|                                                                      |
 +
|  ( pete_plays_drums  pete_fears_height )                              |
 +
|  ( paul_plays_drums  paul_fears_height )                              |
 +
|  ( jane_plays_drums  jane_fears_height )                              |
 +
|                                                                      |
 +
|  ))                                                                  |
 +
|                                                                      |
 +
o-----------------------------------------------------------------------o
 +
 +
Sense Outline:  ConSat.Sen
 +
o-----------------------------o
 +
| pete_plays_drums            |
 +
|  paul_plays_guitar          |
 +
|  jane_plays_sax            |
 +
|    pete_fears_cats          |
 +
|    paul_fears_13          |
 +
|      jane_fears_height      |
 +
o-----------------------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Phobic Apollo (cont.)
 +
 +
It might be instructive to review various aspects
 +
of how the Theme One Study function actually went
 +
about arriving at its answer to that last problem.
 +
Just to prove that my program and I really did do
 +
our homework on that Phobic Apollo ConSat problem,
 +
and didn't just provoke some Oracle or other data
 +
base server to give it away, here is the middling
 +
output of the Model function as run on ConSat.Log:
 +
 +
Model Outline:  ConSat.Mod
 +
o-------------------------------------------------o
 +
| pete_plays_guitar -                            |
 +
| (pete_plays_guitar )                            |
 +
|  pete_plays_sax                                |
 +
|  pete_plays_drums -                            |
 +
|  (pete_plays_drums )                          |
 +
|    paul_plays_sax -                            |
 +
|    (paul_plays_sax )                            |
 +
|    jane_plays_sax -                            |
 +
|    (jane_plays_sax )                          |
 +
|      paul_plays_guitar                          |
 +
|      paul_plays_drums -                        |
 +
|      (paul_plays_drums )                      |
 +
|        jane_plays_guitar -                      |
 +
|        (jane_plays_guitar )                    |
 +
|        jane_plays_drums                        |
 +
|          pete_fears_13                          |
 +
|          pete_fears_cats -                    |
 +
|          (pete_fears_cats )                    |
 +
|            pete_fears_height -                  |
 +
|            (pete_fears_height )                |
 +
|            paul_fears_13 -                    |
 +
|            (paul_fears_13 )                    |
 +
|              jane_fears_13 -                    |
 +
|              (jane_fears_13 )                  |
 +
|              paul_fears_cats -                |
 +
|              (paul_fears_cats )                |
 +
|                paul_fears_height -              |
 +
|                (paul_fears_height ) -          |
 +
|          (pete_fears_13 )                      |
 +
|          pete_fears_cats -                    |
 +
|          (pete_fears_cats )                    |
 +
|            pete_fears_height -                  |
 +
|            (pete_fears_height ) -              |
 +
|        (jane_plays_drums ) -                  |
 +
|      (paul_plays_guitar )                      |
 +
|      paul_plays_drums                          |
 +
|        jane_plays_drums -                      |
 +
|        (jane_plays_drums )                      |
 +
|        jane_plays_guitar                      |
 +
|          pete_fears_13                          |
 +
|          pete_fears_cats -                    |
 +
|          (pete_fears_cats )                    |
 +
|            pete_fears_height -                  |
 +
|            (pete_fears_height )                |
 +
|            paul_fears_13 -                    |
 +
|            (paul_fears_13 )                    |
 +
|              jane_fears_13 -                    |
 +
|              (jane_fears_13 )                  |
 +
|              paul_fears_cats -                |
 +
|              (paul_fears_cats )                |
 +
|                paul_fears_height -              |
 +
|                (paul_fears_height ) -          |
 +
|          (pete_fears_13 )                      |
 +
|          pete_fears_cats -                    |
 +
|          (pete_fears_cats )                    |
 +
|            pete_fears_height -                  |
 +
|            (pete_fears_height ) -              |
 +
|        (jane_plays_guitar ) -                  |
 +
|      (paul_plays_drums ) -                    |
 +
|  (pete_plays_sax )                              |
 +
|  pete_plays_drums                              |
 +
|    paul_plays_drums -                          |
 +
|    (paul_plays_drums )                          |
 +
|    jane_plays_drums -                          |
 +
|    (jane_plays_drums )                        |
 +
|      paul_plays_guitar                          |
 +
|      paul_plays_sax -                          |
 +
|      (paul_plays_sax )                        |
 +
|        jane_plays_guitar -                      |
 +
|        (jane_plays_guitar )                    |
 +
|        jane_plays_sax                          |
 +
|          pete_fears_13 -                        |
 +
|          (pete_fears_13 )                      |
 +
|          pete_fears_cats                      |
 +
|            pete_fears_height -                  |
 +
|            (pete_fears_height )                |
 +
|            paul_fears_cats -                  |
 +
|            (paul_fears_cats )                  |
 +
|              jane_fears_cats -                  |
 +
|              (jane_fears_cats )                |
 +
|              paul_fears_13                    |
 +
|                paul_fears_height -              |
 +
|                (paul_fears_height )            |
 +
|                jane_fears_13 -                |
 +
|                (jane_fears_13 )                |
 +
|                  jane_fears_height *            |
 +
|                  (jane_fears_height ) -        |
 +
|              (paul_fears_13 )                  |
 +
|                paul_fears_height -              |
 +
|                (paul_fears_height ) -          |
 +
|          (pete_fears_cats )                    |
 +
|            pete_fears_height -                  |
 +
|            (pete_fears_height ) -              |
 +
|        (jane_plays_sax ) -                    |
 +
|      (paul_plays_guitar )                      |
 +
|      paul_plays_sax -                          |
 +
|      (paul_plays_sax ) -                      |
 +
|  (pete_plays_drums ) -                        |
 +
o-------------------------------------------------o
 +
 +
This is just the traverse of the "arboreal boolean expansion" (ABE) tree
 +
that Model function germinates from the propositional expression that we
 +
planted in the file Consat.Log, which works to describe the facts of the
 +
situation in question.  Since there are 18 logical feature names in this
 +
propositional expression, we are literally talking about a function that
 +
enjoys the abstract type f : %B%^18 -> %B%.  If I had wanted to evaluate
 +
this function by expressly writing out its truth table, then it would've
 +
required 2^18 = 262144 rows.  Now I didn't bother to count, but I'm sure
 +
that the above output does not have anywhere near that many lines, so it
 +
must be that my program, and maybe even its author, has done a couple of
 +
things along the way that are moderately intelligent.  At least, we hope.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
AK = Antti Karttunen
 +
JA = Jon Awbrey
 +
 +
AK: Am I (and other SeqFanaticians) missing something from this thread?
 +
 +
AK: Your previous message on seqfan (headers below) is a bit of the same topic,
 +
    but does it belong to the same thread?  Where I could obtain the other
 +
    messages belonging to those two threads?  (I'm just now starting to
 +
    study "mathematical logic", and its relations to combinatorics are
 +
    very interesting.)  Is this "cactus" language documented anywhere?
 +
 +
here i was just following a courtesy of copying people
 +
when i reference their works, in this case neil's site:
 +
 +
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000170
 +
 +
but then i thought that the seqfantasians might be amused, too.
 +
 +
the bit on higher order propositions, in particular,
 +
those of type h : (B^2 -> B) -> B, i sent because
 +
of the significance that 2^2^2^2 = 65536 took on
 +
for us around that time.  & the ho, ho, ho joke.
 +
 +
"zeroth order logic" (zol) is just another name for
 +
the propositional calculus or the sentential logic
 +
that comes before "first order logic" (fol), aka
 +
first intens/tional logic, quantificational logic,
 +
or predicate calculus, depending on who you talk to.
 +
 +
the line of work that i have been doing derives from
 +
the ideas of c.s. peirce (1839-1914), who developed
 +
a couple of systems of "logical graphs", actually,
 +
two variant interpretations of the same abstract
 +
structures, called "entitative" and "existential"
 +
graphs.  he organized his system into "alpha",
 +
"beta", and "gamma" layers, roughly equivalent
 +
to our propositional, quantificational, and
 +
modal levels of logic today.
 +
 +
on the more contemporary scene, peirce's entitative interpretation
 +
of logical graphs was revived and extended by george spencer brown
 +
in his book 'laws of form', while the existential interpretation
 +
has flourished in the development of "conceptual graphs" by
 +
john f sowa and a community of growing multitudes.
 +
 +
a passel of links:
 +
 +
http://members.door.net/arisbe/
 +
http://www.enolagaia.com/GSB.html
 +
http://www.cs.uah.edu/~delugach/CG/
 +
http://www.jfsowa.com/
 +
http://www.jfsowa.com/cg/
 +
http://www.jfsowa.com/peirce/ms514w.htm
 +
http://users.bestweb.net/~sowa/
 +
http://users.bestweb.net/~sowa/peirce/ms514.htm
 +
 +
i have mostly focused on "alpha" (prop calc or zol) --
 +
though the "func conception of quant logic" thread was
 +
a beginning try at saying how the same line of thought
 +
might be extended to 1st, 2nd, & higher order logics --
 +
and i devised a particular graph & string syntax that
 +
is based on a species of cacti, officially described as
 +
the "reflective extension of logical graphs" (ref log),
 +
but more lately just referred to as "cactus language".
 +
 +
it turns out that one can do many interesting things
 +
with prop calc if one has an efficient enough syntax
 +
and a powerful enough interpreter for it, even using
 +
it as a very minimal sort of declarative programming
 +
language, hence, the current thread was directed to
 +
applying "zeroth order theories" (zot's) as brands
 +
of "zeroth order programs" (zop's) to a set of old
 +
constraint satisfaction and knowledge rep examples.
 +
 +
more recent expositions of the cactus language have been directed
 +
toward what some people call "ontology engineering" -- it sounds
 +
so much cooler than "taxonomy" -- and so these are found in the
 +
ieee standard upper ontology working group discussion archives.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Let's now pause and reflect on the mix of abstract and concrete material
 +
that we have cobbled together in spectacle of this "World Of Zero" (WOZ),
 +
since I believe that we may have seen enough, if we look at it right, to
 +
illustrate a few of the more salient phenomena that would normally begin
 +
to weigh in as a major force only on a much larger scale.  Now, it's not
 +
exactly like this impoverished sample, all by itself, could determine us
 +
to draw just the right generalizations, or force us to see the shape and
 +
flow of its immanent law -- it is much too sparse a scattering of points
 +
to tease out the lines of its up and coming generations quite so clearly --
 +
but it can be seen to exemplify many of the more significant themes that
 +
we know evolve in more substantial environments, that is, On Beyond Zero,
 +
since we have already seen them, "tho' obscur'd", in these higher realms.
 +
 +
One the the themes that I want to to keep an eye on as this discussion
 +
develops is the subject that might be called "computation as semiosis".
 +
 +
In this light, any calculus worth its salt must be capable of helping
 +
us do two things, calculation, of course, but also analysis.  This is
 +
probably one of the reasons why the ordinary sort of differential and
 +
integral calculus over quantitative domains is frequently referred to
 +
as "real analysis", or even just "analysis".  It seems quite clear to
 +
me that any adequate logical calculus, in many ways expected to serve
 +
as a qualitative analogue of analytic geometry in the way that it can
 +
be used to describe configurations in logically circumscribed domains,
 +
ought to qualify in both dimensions, namely, analysis and computation.
 +
 +
With all of these various features of the situation in mind, then, we come
 +
to the point of viewing analysis and computation as just so many different
 +
kinds of "sign transformations in respect of pragmata" (STIROP's).  Taking
 +
this insight to heart, let us next work to assemble a comprehension of our
 +
concrete examples, set in the medium of the abstract calculi that allow us
 +
to express their qualitative patterns, that may hope to be an increment or
 +
two less inchoate than we have seen so far, and that may even permit us to
 +
catch the action of these fading fleeting sign transformations on the wing.
 +
 +
Here is how I picture our latest round of examples
 +
as filling out the framework of this investigation:
 +
 +
o-----------------------------o-----------------------------o
 +
|    Objective Framework    |  Interpretive Framework    |
 +
o-----------------------------o-----------------------------o
 +
|                                                          |
 +
|                              s_1 = Logue(o)      |      |
 +
|                              /                    |      |
 +
|                            /                      |      |
 +
|                            @                      |      |
 +
|                          ·  \                      |      |
 +
|                        ·    \                    |      |
 +
|                      ·        i_1 = Model(o)      v      |
 +
|                    ·          s_2 = Model(o)      |      |
 +
|                  ·          /                    |      |
 +
|                ·            /                      |      |
 +
|    Object = o · · · · · · @                      |      |
 +
|                ·            \                      |      |
 +
|                  ·          \                    |      |
 +
|                    ·          i_2 = Tenor(o)      v      |
 +
|                      ·        s_3 = Tenor(o)      |      |
 +
|                        ·    /                    |      |
 +
|                          ·  /                      |      |
 +
|                            @                      |      |
 +
|                            \                      |      |
 +
|                              \                    |      |
 +
|                              i_3 = Sense(o)      v      |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure.  Computation As Semiotic Transformation
 +
 +
The Figure shows three distinct sign triples of the form <o, s, i>, where
 +
o = ostensible objective = the observed, indicated, or intended situation.
 +
 +
| A.  <o, Logue(o), Model(o)>
 +
|
 +
| B.  <o, Model(o), Tenor(o)>
 +
|
 +
| C.  <o, Tenor(o), Sense(o)>
 +
 +
Let us bring these several signs together in one place,
 +
to compare and contrast their common and their diverse
 +
characters, and to think about why we make such a fuss
 +
about passing from one to the other in the first place.
 +
 +
1.  Logue(o)  =  Consat.Log
 +
o-----------------------------------------------------------------------o
 +
|                                                                      |
 +
|  (( pete_plays_guitar ),( pete_plays_sax ),( pete_plays_drums ))      |
 +
|  (( paul_plays_guitar ),( paul_plays_sax ),( paul_plays_drums ))      |
 +
|  (( jane_plays_guitar ),( jane_plays_sax ),( jane_plays_drums ))      |
 +
|                                                                      |
 +
|  (( pete_plays_guitar ),( paul_plays_guitar ),( jane_plays_guitar ))  |
 +
|  (( pete_plays_sax    ),( paul_plays_sax    ),( jane_plays_sax    ))  |
 +
|  (( pete_plays_drums  ),( paul_plays_drums  ),( jane_plays_drums  ))  |
 +
|                                                                      |
 +
|  (( pete_fears_13 ),( pete_fears_cats ),( pete_fears_height ))        |
 +
|  (( paul_fears_13 ),( paul_fears_cats ),( paul_fears_height ))        |
 +
|  (( jane_fears_13 ),( jane_fears_cats ),( jane_fears_height ))        |
 +
|                                                                      |
 +
|  (( pete_fears_13    ),( paul_fears_13    ),( jane_fears_13    ))  |
 +
|  (( pete_fears_cats  ),( paul_fears_cats  ),( jane_fears_cats  ))  |
 +
|  (( pete_fears_height ),( paul_fears_height ),( jane_fears_height ))  |
 +
|                                                                      |
 +
|  ((                                                                  |
 +
|                                                                      |
 +
|  ( pete_plays_guitar )                                                |
 +
|  ( pete_fears_height )                                                |
 +
|                                                                      |
 +
|  ( pete_plays_guitar  pete_fears_height )                            |
 +
|  ( paul_plays_guitar  paul_fears_height )                            |
 +
|  ( jane_plays_guitar  jane_fears_height )                            |
 +
|                                                                      |
 +
|  ( paul_fears_cats )                                                  |
 +
|  ( paul_plays_sax  )                                                  |
 +
|                                                                      |
 +
|  ( pete_plays_sax  pete_fears_cats )                                  |
 +
|  ( paul_plays_sax  paul_fears_cats )                                  |
 +
|  ( jane_plays_sax  jane_fears_cats )                                  |
 +
|                                                                      |
 +
|  ( pete_plays_drums  pete_fears_13 )                                  |
 +
|  ( paul_plays_drums  paul_fears_13 )                                  |
 +
|  ( jane_plays_drums  jane_fears_13 )                                  |
 +
|                                                                      |
 +
|  ( pete_plays_drums  pete_fears_height )                              |
 +
|  ( paul_plays_drums  paul_fears_height )                              |
 +
|  ( jane_plays_drums  jane_fears_height )                              |
 +
|                                                                      |
 +
|  ))                                                                  |
 +
|                                                                      |
 +
o-----------------------------------------------------------------------o
 +
 +
2.  Model(o)  =  Consat.Mod  ><>  http://suo.ieee.org/ontology/msg03718.html
 +
 +
3.  Tenor(o)  =  Consat.Ten (Just The Gist Of It)
 +
o-------------------------------------------------o
 +
| (pete_plays_guitar )                            | <01> -
 +
|  (pete_plays_sax )                              | <02> -
 +
|  pete_plays_drums                              | <03> +
 +
|    (paul_plays_drums )                          | <04> -
 +
|    (jane_plays_drums )                        | <05> -
 +
|      paul_plays_guitar                          | <06> +
 +
|      (paul_plays_sax )                        | <07> -
 +
|        (jane_plays_guitar )                    | <08> -
 +
|        jane_plays_sax                          | <09> +
 +
|          (pete_fears_13 )                      | <10> -
 +
|          pete_fears_cats                      | <11> +
 +
|            (pete_fears_height )                | <12> -
 +
|            (paul_fears_cats )                  | <13> -
 +
|              (jane_fears_cats )                | <14> -
 +
|              paul_fears_13                    | <15> +
 +
|                (paul_fears_height )            | <16> -
 +
|                (jane_fears_13 )                | <17> -
 +
|                  jane_fears_height *            | <18> +
 +
o-------------------------------------------------o
 +
 +
4.  Sense(o)  =  Consat.Sen
 +
o-------------------------------------------------o
 +
| pete_plays_drums                                | <03>
 +
|  paul_plays_guitar                              | <06>
 +
|  jane_plays_sax                                | <09>
 +
|    pete_fears_cats                              | <11>
 +
|    paul_fears_13                              | <15>
 +
|      jane_fears_height                          | <18>
 +
o-------------------------------------------------o
 +
 +
As one proceeds through the subsessions of the Theme One Study session,
 +
the computation transforms its larger "signs", in this case text files,
 +
from one to the next, in the sequence:  Logue, Model, Tenor, and Sense.
 +
 +
Let us see if we can pin down, on sign-theoretic grounds,
 +
why this very sort of exercise is so routinely necessary.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We were in the middle of pursuing several questions about
 +
sign relational transformations in general, in particular,
 +
the following Example of a sign transformation that arose
 +
in the process of setting up and solving a classical sort
 +
of constraint satisfaction problem.
 +
 +
o-----------------------------o-----------------------------o
 +
|    Objective Framework    |  Interpretive Framework    |
 +
o-----------------------------o-----------------------------o
 +
|                                                          |
 +
|                              s_1 = Logue(o)      |      |
 +
|                              /                    |      |
 +
|                            /                      |      |
 +
|                            @                      |      |
 +
|                          ·  \                      |      |
 +
|                        ·    \                    |      |
 +
|                      ·        i_1 = Model(o)      v      |
 +
|                    ·          s_2 = Model(o)      |      |
 +
|                  ·          /                    |      |
 +
|                ·            /                      |      |
 +
|    Object = o · · · · · · @                      |      |
 +
|                ·            \                      |      |
 +
|                  ·          \                    |      |
 +
|                    ·          i_2 = Tenor(o)      v      |
 +
|                      ·        s_3 = Tenor(o)      |      |
 +
|                        ·    /                    |      |
 +
|                          ·  /                      |      |
 +
|                            @                      |      |
 +
|                            \                      |      |
 +
|                              \                    |      |
 +
|                              i_3 = Sense(o)      v      |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure.  Computation As Semiotic Transformation
 +
 +
1.  Logue(o)  =  Consat.Log
 +
o-----------------------------------------------------------------------o
 +
|                                                                      |
 +
|  (( pete_plays_guitar ),( pete_plays_sax ),( pete_plays_drums ))      |
 +
|  (( paul_plays_guitar ),( paul_plays_sax ),( paul_plays_drums ))      |
 +
|  (( jane_plays_guitar ),( jane_plays_sax ),( jane_plays_drums ))      |
 +
|                                                                      |
 +
|  (( pete_plays_guitar ),( paul_plays_guitar ),( jane_plays_guitar ))  |
 +
|  (( pete_plays_sax    ),( paul_plays_sax    ),( jane_plays_sax    ))  |
 +
|  (( pete_plays_drums  ),( paul_plays_drums  ),( jane_plays_drums  ))  |
 +
|                                                                      |
 +
|  (( pete_fears_13 ),( pete_fears_cats ),( pete_fears_height ))        |
 +
|  (( paul_fears_13 ),( paul_fears_cats ),( paul_fears_height ))        |
 +
|  (( jane_fears_13 ),( jane_fears_cats ),( jane_fears_height ))        |
 +
|                                                                      |
 +
|  (( pete_fears_13    ),( paul_fears_13    ),( jane_fears_13    ))  |
 +
|  (( pete_fears_cats  ),( paul_fears_cats  ),( jane_fears_cats  ))  |
 +
|  (( pete_fears_height ),( paul_fears_height ),( jane_fears_height ))  |
 +
|                                                                      |
 +
|  ((                                                                  |
 +
|                                                                      |
 +
|  ( pete_plays_guitar )                                                |
 +
|  ( pete_fears_height )                                                |
 +
|                                                                      |
 +
|  ( pete_plays_guitar  pete_fears_height )                            |
 +
|  ( paul_plays_guitar  paul_fears_height )                            |
 +
|  ( jane_plays_guitar  jane_fears_height )                            |
 +
|                                                                      |
 +
|  ( paul_fears_cats )                                                  |
 +
|  ( paul_plays_sax  )                                                  |
 +
|                                                                      |
 +
|  ( pete_plays_sax  pete_fears_cats )                                  |
 +
|  ( paul_plays_sax  paul_fears_cats )                                  |
 +
|  ( jane_plays_sax  jane_fears_cats )                                  |
 +
|                                                                      |
 +
|  ( pete_plays_drums  pete_fears_13 )                                  |
 +
|  ( paul_plays_drums  paul_fears_13 )                                  |
 +
|  ( jane_plays_drums  jane_fears_13 )                                  |
 +
|                                                                      |
 +
|  ( pete_plays_drums  pete_fears_height )                              |
 +
|  ( paul_plays_drums  paul_fears_height )                              |
 +
|  ( jane_plays_drums  jane_fears_height )                              |
 +
|                                                                      |
 +
|  ))                                                                  |
 +
|                                                                      |
 +
o-----------------------------------------------------------------------o
 +
 +
2.  Model(o)  =  Consat.Mod  ><>  http://suo.ieee.org/ontology/msg03718.html
 +
 +
3.  Tenor(o)  =  Consat.Ten (Just The Gist Of It)
 +
o-------------------------------------------------o
 +
| (pete_plays_guitar )                            | <01> -
 +
|  (pete_plays_sax )                              | <02> -
 +
|  pete_plays_drums                              | <03> +
 +
|    (paul_plays_drums )                          | <04> -
 +
|    (jane_plays_drums )                        | <05> -
 +
|      paul_plays_guitar                          | <06> +
 +
|      (paul_plays_sax )                        | <07> -
 +
|        (jane_plays_guitar )                    | <08> -
 +
|        jane_plays_sax                          | <09> +
 +
|          (pete_fears_13 )                      | <10> -
 +
|          pete_fears_cats                      | <11> +
 +
|            (pete_fears_height )                | <12> -
 +
|            (paul_fears_cats )                  | <13> -
 +
|              (jane_fears_cats )                | <14> -
 +
|              paul_fears_13                    | <15> +
 +
|                (paul_fears_height )            | <16> -
 +
|                (jane_fears_13 )                | <17> -
 +
|                  jane_fears_height *            | <18> +
 +
o-------------------------------------------------o
 +
 +
4.  Sense(o)  =  Consat.Sen
 +
o-------------------------------------------------o
 +
| pete_plays_drums                                | <03>
 +
|  paul_plays_guitar                              | <06>
 +
|  jane_plays_sax                                | <09>
 +
|    pete_fears_cats                              | <11>
 +
|    paul_fears_13                              | <15>
 +
|      jane_fears_height                          | <18>
 +
o-------------------------------------------------o
 +
 +
We can worry later about the proper use of quotation marks
 +
in discussing such a case, where the file name "Yada.Yak"
 +
denotes a piece of text that expresses a proposition that
 +
describes an objective situation or an intentional object,
 +
but whatever the case it is clear that we are knee & neck
 +
deep in a sign relational situation of a modest complexity.
 +
 +
I think that the right sort of analogy might help us
 +
to sort it out, or at least to tell what's important
 +
from the things that are less so.  The paradigm that
 +
comes to mind for me is the type of context in maths
 +
where we talk about the "locus" or the "solution set"
 +
of an equation, and here we think of the equation as
 +
denoting its solution set or describing a locus, say,
 +
a point or a curve or a surface or so on up the scale.
 +
 +
In this figure of speech, we might say for instance:
 +
 +
| o is
 +
| what "x^3 - 3x^2 + 3x - 1 = 0" denotes is
 +
| what "(x-1)^3 = 0" denotes is
 +
| what "1" denotes
 +
| is 1.
 +
 +
Making explicit the assumptive interpretations
 +
that the context probably enfolds in this case,
 +
we assume this description of the solution set:
 +
 +
{x in the Reals : x^3 - 3x^2 + 3x -1 = 0} = {1}.
 +
 +
In sign relational terms, we have the 3-tuples:
 +
 +
| <o, "x^3 - 3x^2 + 3x - 1 = 0", "(x-1)^3 = 0">
 +
|
 +
| <o, "(x-1)^3 = 0", "1">
 +
|
 +
| <o, "1", "1">
 +
 +
As it turns out we discover that the
 +
object o was really just 1 all along.
 +
 +
But why do we put ourselves through the rigors of these
 +
transformations at all?  If 1 is what we mean, why not
 +
just say "1" in the first place and be done with it?
 +
A person who asks a question like that has forgetten
 +
how we keep getting ourselves into these quandaries,
 +
and who it is that assigns the problems, for it is
 +
Nature herself who is the taskmistress here and the
 +
problems are set in the manner that she determines,
 +
not in the style to which we would like to become
 +
accustomed.  The best that we can demand of our
 +
various and sundry calculi is that they afford
 +
us with the nets and the snares more readily
 +
to catch the shape of the problematic game
 +
as it flies up before us on its own wings,
 +
and only then to tame it to the amenable
 +
demeanors that we find to our liking.
 +
 +
In sum, the first place is not ours to take.
 +
We are but poor second players in this game.
 +
 +
That understood, I can now lay out our present Example
 +
along the lines of this familiar mathematical exercise.
 +
 +
| o is
 +
| what Consat.Log denotes is
 +
| what Consat.Mod denotes is
 +
| what Consat.Ten denotes is
 +
| what Consat.Sen denotes.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
It will be good to keep this picture before us a while longer:
 +
 +
o-----------------------------o-----------------------------o
 +
|    Objective Framework    |  Interpretive Framework    |
 +
o-----------------------------o-----------------------------o
 +
|                                                          |
 +
|                              s_1  =  Logue(o)    |      |
 +
|                              /                    |      |
 +
|                            /                      |      |
 +
|                            @                      |      |
 +
|                          ·  \                      |      |
 +
|                        ·    \                    |      |
 +
|                      ·        i_1  =  Model(o)    v      |
 +
|                    ·          s_2  =  Model(o)    |      |
 +
|                  ·          /                    |      |
 +
|                ·            /                      |      |
 +
|  Object  =  o · · · · · · @                      |      |
 +
|                ·            \                      |      |
 +
|                  ·          \                    |      |
 +
|                    ·          i_2  =  Tenor(o)    v      |
 +
|                      ·        s_3  =  Tenor(o)    |      |
 +
|                        ·    /                    |      |
 +
|                          ·  /                      |      |
 +
|                            @                      |      |
 +
|                            \                      |      |
 +
|                              \                    |      |
 +
|                              i_3  =  Sense(o)    v      |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure.  Computation As Semiotic Transformation
 +
 +
The labels that decorate the syntactic plane and indicate
 +
the semiotic transitions in the interpretive panel of the
 +
framework point us to text files whose contents rest here:
 +
 +
http://suo.ieee.org/ontology/msg03722.html
 +
 +
The reason that I am troubling myself -- and no doubt you --
 +
with the details of this Example is because it highlights
 +
a number of the thistles that we will have to grasp if we
 +
ever want to escape from the traps of YARNBOL and YARWARS
 +
in which so many of our fairweather fiends are seeking to
 +
ensnare us, and not just us -- the whole web of the world.
 +
 +
YARNBOL  =  Yet Another Roman Numeral Based Ontology Language.
 +
YARWARS  =  Yet Another Representation Without A Reasoning System.
 +
 +
In order to avoid this, or to reverse the trend once it gets started,
 +
we just have to remember what a dynamic living process a computation
 +
really is, precisely because it is meant to serve as an iconic image
 +
of dynamic, deliberate, purposeful transformations that we are bound
 +
to go through and to carry out in a hopeful pursuit of the solutions
 +
to the many real live problems that life and society place before us.
 +
So I take it rather seriously.
 +
 +
Okay, back to the grindstone.
 +
 +
The question is:  "Why are these trips necessary?"
 +
 +
How come we don't just have one proper expression
 +
for each situation under the sun, or all possible
 +
suns, I guess, for some, and just use that on any
 +
appearance, instance, occasion of that situation?
 +
 +
Why is it ever necessary to begin with an obscure description
 +
of a situation? -- for that is exactly what the propositional
 +
expression caled "Logue(o)", for Example, the Consat.Log file,
 +
really is.
 +
 +
Maybe I need to explain that first.
 +
 +
The first three items of syntax -- Logue(o), Model(o), Tenor(o) --
 +
are all just so many different propositional expressions that
 +
denote one and the same logical-valued function p : X -> %B%,
 +
and one whose abstract image we may well enough describe as
 +
a boolean function of the abstract type q : %B%^k -> %B%,
 +
where k happens to be 18 in the present Consat Example.
 +
 +
If we were to write out the truth table for q : %B%^18 -> %B%
 +
it would take 2^18 = 262144 rows.  Using the bold letter #x#
 +
for a coordinate tuple, writing #x# = <x_1, ..., x_18>, each
 +
row of the table would have the form <x_1, ..., x_18, q(#x#)>.
 +
And the function q is such that all rows evalue to %0% save 1.
 +
 +
Each of the four different formats expresses this fact about q
 +
in its own way.  The first three are logically equivalent, and
 +
the last one is the maximally determinate positive implication
 +
of what the others all say.
 +
 +
From this point of view, the logical computation that we went through,
 +
in the sequence Logue, Model, Tenor, Sense, was a process of changing
 +
from an obscure sign of the objective proposition to a more organized
 +
arrangement of its satisfying or unsatisfying interpretations, to the
 +
most succinct possible expression of the same meaning, to an adequate
 +
positive projection of it that is useful enough in the proper context.
 +
 +
This is the sort of mill -- it's called "computation" -- that we have
 +
to be able to put our representations through on a recurrent, regular,
 +
routine basis, that is, if we expect them to have any utility at all.
 +
And it is only when we have started to do that in genuinely effective
 +
and efficient ways, that we can even begin to think about facilitating
 +
any bit of qualitative conceptual analysis through computational means.
 +
 +
And as far as the qualitative side of logical computation
 +
and conceptual analysis goes, we have barely even started.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We are contemplating the sequence of initial and normal forms
 +
for the Consat problem and we have noted the following system
 +
of logical relations, taking the enchained expressions of the
 +
objective situation o in a pairwise associated way, of course:
 +
 +
Logue(o)  <=>  Model(o)  <=>  Tenor(o)  =>  Sense(o).
 +
 +
The specifics of the propositional expressions are cited here:
 +
 +
http://suo.ieee.org/ontology/msg03722.html
 +
 +
If we continue to pursue the analogy that we made with the form
 +
of mathematical activity commonly known as "solving equations",
 +
then there are many salient features of this type of logical
 +
problem solving endeavor that suddenly leap into the light.
 +
 +
First of all, we notice the importance of "equational reasoning"
 +
in mathematics, by which I mean, not just the quantitative type
 +
of equation that forms the matter of the process, but also the
 +
qualitative type of equation, or the "logical equivalence",
 +
that connects each expression along the way, right up to
 +
the penultimate stage, when we are satisfied in a given
 +
context to take a projective implication of the total
 +
knowledge of the situation that we have been taking
 +
some pains to preserve at every intermediate stage
 +
of the game.
 +
 +
This general pattern or strategy of inference, working its way through
 +
phases of "equational" or "total information preserving" inference and
 +
phases of "implicational" or "selective information losing" inference,
 +
is actually very common throughout mathematics, and I have in mind to
 +
examine its character in greater detail and in a more general setting.
 +
 +
Just as the barest hint of things to come along these lines, you might
 +
consider the question of what would constitute the equational analogue
 +
of modus ponens, in other words the scheme of inference that goes from
 +
x and x=>y to y.  Well the answer is a scheme of inference that passes
 +
from x and x=>y to x&y, and then being reversible, back again.  I will
 +
explore the rationale and the utility of this gambit in future reports.
 +
 +
One observation that we can make already at this point,
 +
however, is that these schemes of equational reasoning,
 +
or reversible inference, remain poorly developed among
 +
our currently prevailing styles of inference in logic,
 +
their potentials for applied logical software hardly
 +
being broached in our presently available systems.
 +
 +
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 +
 +
Extra Examples
 +
 +
1.  Propositional logic example.
 +
Files:  Alpha.lex + Prop.log
 +
Ref:    [Cha, 20, Example 2.12]
 +
 +
2.  Chemical synthesis problem.
 +
Files:  Chem.*
 +
Ref:    [Cha, 21, Example 2.13]
 +
 +
3.  N Queens problem.
 +
Files:  Queen*.*,  Q8.*,  Q5.*
 +
Refs:  [BaC, 166], [VaH, 122], [Wir, 143].
 +
Notes:  Only the 5 Queens example will run in 640K memory.
 +
        Use the "Queen.lex" file to load the "Q5.eg*" log files.
 +
 +
4.  Five Houses puzzle.
 +
Files:  House.*
 +
Ref:    [VaH, 132].
 +
Notes:  Will not run in 640K memory.
 +
 +
5.  Graph coloring example.
 +
Files:  Color.*
 +
Ref:    [Wil, 196].
 +
 +
6.  Examples of Cook's Theorem in computational complexity,
 +
    that propositional satisfiability is NP-complete.
 +
 +
Files:  StiltN.* = "Space and Time Limited Turing Machine",
 +
        with N units of space and N units of time.
 +
        StuntN.* = "Space and Time Limited Turing Machine",
 +
        for computing the parity of a bit string,
 +
        with Number of Tape cells of input equal to N.
 +
Ref:    [Wil, 188-201].
 +
Notes:  Can only run Turing machine example for input of size 2.
 +
        Since the last tape cell is used for an end-of-file marker,
 +
        this amounts to only one significant digit of computation.
 +
        Use the "Stilt3.lex" file  to load the "Stunt2.egN" files.
 +
        Their Sense file outputs appear on the "Stunt2.seN" files.
 +
 +
7.  Fabric knowledge base.
 +
Files:  Fabric.*, Fab.*
 +
Ref:    [MaW, 8-16].
 +
 +
8.  Constraint Satisfaction example.
 +
Files:  Consat1.*, Consat2.*
 +
Ref:    [Win, 449, Exercise 3-9].
 +
Notes:  Attributed to Kenneth D. Forbus.
 +
 +
References
 +
 +
| Angluin, Dana,
 +
|"Learning with Hints", in
 +
|'Proceedings of the 1988 Workshop on Computational Learning Theory',
 +
| edited by D. Haussler & L. Pitt, Morgan Kaufmann, San Mateo, CA, 1989.
 +
 +
| Ball, W.W. Rouse, & Coxeter, H.S.M.,
 +
|'Mathematical Recreations and Essays', 13th ed.,
 +
| Dover, New York, NY, 1987.
 +
 +
| Chang, Chin-Liang & Lee, Richard Char-Tung,
 +
|'Symbolic Logic and Mechanical Theorem Proving',
 +
| Academic Press, New York, NY, 1973.
 +
 +
| Denning, Peter J., Dennis, Jack B., and Qualitz, Joseph E.,
 +
|'Machines, Languages, and Computation',
 +
| Prentice-Hall, Englewood Cliffs, NJ, 1978.
 +
 +
| Edelman, Gerald M.,
 +
|'Topobiology:  An Introduction to Molecular Embryology',
 +
| Basic Books, New York, NY, 1988.
 +
 +
| Lloyd, J.W.,
 +
|'Foundations of Logic Programming',
 +
| Springer-Verlag, Berlin, 1984.
 +
 +
| Maier, David & Warren, David S.,
 +
|'Computing with Logic:  Logic Programming with Prolog',
 +
| Benjamin/Cummings, Menlo Park, CA, 1988.
 +
 +
| McClelland, James L. and Rumelhart, David E.,
 +
|'Explorations in Parallel Distributed Processing:
 +
| A Handbook of Models, Programs, and Exercises',
 +
| MIT Press, Cambridge, MA, 1988.
 +
 +
| Peirce, Charles Sanders,
 +
|'Collected Papers of Charles Sanders Peirce',
 +
| edited by Charles Hartshorne, Paul Weiss, & Arthur W. Burks,
 +
| Harvard University Press, Cambridge, MA, 1931-1960.
 +
 +
| Peirce, Charles Sanders,
 +
|'The New Elements of Mathematics',
 +
| edited by Carolyn Eisele, Mouton, The Hague, 1976.
 +
 +
|'Charles S. Peirce:  Selected Writings;  Values in a Universe of Chance',
 +
| edited by Philip P. Wiener, Dover, New York, NY, 1966.
 +
 +
| Spencer Brown, George,
 +
|'Laws of Form',
 +
| George Allen & Unwin, London, UK, 1969.
 +
 +
| Van Hentenryck, Pascal,
 +
|'Constraint Satisfaction in Logic Programming',
 +
| MIT Press, Cambridge, MA, 1989.
 +
 +
| Wilf, Herbert S.,
 +
|'Algorithms and Complexity',
 +
| Prentice-Hall, Englewood Cliffs, NJ, 1986.
 +
 +
| Winston, Patrick Henry,
 +
|'Artificial Intelligence, 2nd ed.,
 +
| Addison-Wesley, Reading, MA, 1984.
 +
 +
| Wirth, Niklaus,
 +
|'Algorithms + Data Structures = Programs',
 +
| Prentice-Hall, Englewood Cliffs, NJ, 1976.
 +
 +
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 +
 +
Cactus Town Cartoons
 +
 +
01.  http://suo.ieee.org/ontology/msg03567.html
 +
02.  http://suo.ieee.org/ontology/msg03571.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Differential Analytic Turing Automata (DATA)
 +
 +
01.  http://suo.ieee.org/ontology/msg00596.html
 +
02.  http://suo.ieee.org/ontology/msg00618.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Differential Logic
 +
 +
01.  http://suo.ieee.org/ontology/msg04040.html
 +
02.  http://suo.ieee.org/ontology/msg04041.html
 +
03.  http://suo.ieee.org/ontology/msg04045.html
 +
04.  http://suo.ieee.org/ontology/msg04046.html
 +
05.  http://suo.ieee.org/ontology/msg04047.html
 +
06.  http://suo.ieee.org/ontology/msg04048.html
 +
07.  http://suo.ieee.org/ontology/msg04052.html
 +
08.  http://suo.ieee.org/ontology/msg04054.html
 +
09.  http://suo.ieee.org/ontology/msg04055.html
 +
10.  http://suo.ieee.org/ontology/msg04067.html
 +
11.  http://suo.ieee.org/ontology/msg04068.html
 +
12.  http://suo.ieee.org/ontology/msg04069.html
 +
13.  http://suo.ieee.org/ontology/msg04070.html
 +
14.  http://suo.ieee.org/ontology/msg04072.html
 +
15.  http://suo.ieee.org/ontology/msg04073.html
 +
16.  http://suo.ieee.org/ontology/msg04074.html
 +
17.  http://suo.ieee.org/ontology/msg04077.html
 +
18.  http://suo.ieee.org/ontology/msg04079.html
 +
19.  http://suo.ieee.org/ontology/msg04080.html
 +
20.
 +
 +
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 +
 +
Extensions Of Logical Graphs
 +
 +
01.  http://www.virtual-earth.de/CG/cg-list/old/msg03351.html
 +
02.  http://www.virtual-earth.de/CG/cg-list/old/msg03352.html
 +
03.  http://www.virtual-earth.de/CG/cg-list/old/msg03353.html
 +
04.  http://www.virtual-earth.de/CG/cg-list/old/msg03354.html
 +
05.  http://www.virtual-earth.de/CG/cg-list/old/msg03376.html
 +
06.  http://www.virtual-earth.de/CG/cg-list/old/msg03379.html
 +
07.  http://www.virtual-earth.de/CG/cg-list/old/msg03381.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Functional Conception Of Quantificational Logic
 +
 +
01.  http://suo.ieee.org/ontology/msg03562.html
 +
02.  http://suo.ieee.org/ontology/msg03563.html
 +
03.  http://suo.ieee.org/ontology/msg03577.html
 +
04.  http://suo.ieee.org/ontology/msg03578.html
 +
05.  http://suo.ieee.org/ontology/msg03579.html
 +
06.  http://suo.ieee.org/ontology/msg03580.html
 +
07.  http://suo.ieee.org/ontology/msg03581.html
 +
08.  http://suo.ieee.org/ontology/msg03582.html
 +
09.  http://suo.ieee.org/ontology/msg03583.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Propositional Equation Reasoning Systems (PERS)
 +
 +
01.  http://suo.ieee.org/email/msg04187.html
 +
02.  http://suo.ieee.org/email/msg04305.html
 +
03.  http://suo.ieee.org/email/msg04413.html
 +
04.  http://suo.ieee.org/email/msg04419.html
 +
05.  http://suo.ieee.org/email/msg04422.html
 +
06.  http://suo.ieee.org/email/msg04423.html
 +
07.  http://suo.ieee.org/email/msg04432.html
 +
08.  http://suo.ieee.org/email/msg04454.html
 +
09.  http://suo.ieee.org/email/msg04455.html
 +
10.  http://suo.ieee.org/email/msg04476.html
 +
11.  http://suo.ieee.org/email/msg04510.html
 +
12.  http://suo.ieee.org/email/msg04517.html
 +
13.  http://suo.ieee.org/email/msg04525.html
 +
14.  http://suo.ieee.org/email/msg04533.html
 +
15.  http://suo.ieee.org/email/msg04536.html
 +
16.  http://suo.ieee.org/email/msg04542.html
 +
17.  http://suo.ieee.org/email/msg04546.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Reflective Extension Of Logical Graphs (RefLog)
 +
 +
01.  http://suo.ieee.org/email/msg05694.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Sequential Interactions Generating Hypotheses
 +
 +
01.  http://suo.ieee.org/email/msg02607.html
 +
02.  http://suo.ieee.org/email/msg02608.html
 +
03.  http://suo.ieee.org/email/msg03183.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Sowa's Top Level Categories
 +
 +
01.  http://suo.ieee.org/email/msg01949.html
 +
02.  http://suo.ieee.org/email/msg01956.html
 +
03.  http://suo.ieee.org/email/msg01966.html
 +
 +
04.  http://suo.ieee.org/ontology/msg00048.html
 +
05.  http://suo.ieee.org/ontology/msg00051.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Zeroth Order Logic (ZOL)
 +
 +
01.  http://suo.ieee.org/email/msg01246.html
 +
02.  http://suo.ieee.org/email/msg01406.html
 +
03.  http://suo.ieee.org/email/msg01546.html
 +
04.  http://suo.ieee.org/email/msg01561.html
 +
05.  http://suo.ieee.org/email/msg01670.html
 +
06.  http://suo.ieee.org/email/msg01739.html
 +
07.  http://suo.ieee.org/email/msg01966.html
 +
08.  http://suo.ieee.org/email/msg01985.html
 +
09.  http://suo.ieee.org/email/msg01988.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Zeroth Order Theories (ZOT's)
 +
 +
01.  http://suo.ieee.org/ontology/msg03680.html
 +
02.  http://suo.ieee.org/ontology/msg03681.html
 +
03.  http://suo.ieee.org/ontology/msg03682.html
 +
04.  http://suo.ieee.org/ontology/msg03683.html
 +
05.  http://suo.ieee.org/ontology/msg03685.html
 +
06.  http://suo.ieee.org/ontology/msg03687.html
 +
07.  http://suo.ieee.org/ontology/msg03689.html
 +
08.  http://suo.ieee.org/ontology/msg03691.html
 +
09.  http://suo.ieee.org/ontology/msg03693.html
 +
10.  http://suo.ieee.org/ontology/msg03694.html
 +
11.  http://suo.ieee.org/ontology/msg03695.html
 +
12.  http://suo.ieee.org/ontology/msg03696.html
 +
13.  http://suo.ieee.org/ontology/msg03700.html
 +
14.  http://suo.ieee.org/ontology/msg03701.html
 +
15.  http://suo.ieee.org/ontology/msg03702.html
 +
16.  http://suo.ieee.org/ontology/msg03703.html
 +
17.  http://suo.ieee.org/ontology/msg03705.html
 +
18.  http://suo.ieee.org/ontology/msg03706.html
 +
19.  http://suo.ieee.org/ontology/msg03707.html
 +
20.  http://suo.ieee.org/ontology/msg03708.html
 +
21.  http://suo.ieee.org/ontology/msg03709.html
 +
22.  http://suo.ieee.org/ontology/msg03711.html
 +
23.  http://suo.ieee.org/ontology/msg03712.html
 +
24.  http://suo.ieee.org/ontology/msg03715.html
 +
25.  http://suo.ieee.org/ontology/msg03716.html
 +
26.  http://suo.ieee.org/ontology/msg03717.html
 +
27.  http://suo.ieee.org/ontology/msg03718.html
 +
28.  http://suo.ieee.org/ontology/msg03720.html
 +
29.  http://suo.ieee.org/ontology/msg03721.html
 +
30.  http://suo.ieee.org/ontology/msg03722.html
 +
31.  http://suo.ieee.org/ontology/msg03723.html
 +
32.  http://suo.ieee.org/ontology/msg03724.html
    
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 
</pre>
 
</pre>
Jon Awbrey
12,080

edits

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