[Inquiry] Re: Theme One Program

[Jon Awbrey]

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TOP.  Expository Note 14

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3.3.  Logical Cacti (cont.)

For the time being, the main things to take away from Tables 13 and 14 are
the ideas that the compositional structure of cactus graphs and expressions
can be articulated in terms of two different kinds of connective operations,
and that there are two distinct ways of mapping this compositional structure
into the compositional structure of propositional sentences, say, in English:

1.  The "node connective" joins a number of
    component cacti C_1, ..., C_k at a node:

    C_1 ... C_k
         @

2.  The "lobe connective" joins a number of
    component cacti C_1, ..., C_k to a lobe:

    C_1 C_2   C_k
     o---o-...-o
      \       /
       \     /
        \   /
         \ /
          @

Table 15 summarizes the existential and entitative
interpretations of the primitive cactus structures,
in effect, the graphical constants and connectives.

Table 15.  Existential & Entitative Interpretations of Cactus Structures
o-----------------o-----------------o-----------------o-----------------o
|  Cactus Graph   |  Cactus String  |  Existential    |   Entitative    |
|                 |                 | Interpretation  | Interpretation  |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        @        |       " "       |      true       |      false      |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        o        |                 |                 |                 |
|        |        |                 |                 |                 |
|        @        |       ( )       |      false      |      true       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|   C_1 ... C_k   |                 |                 |                 |
|        @        |   C_1 ... C_k   | C_1 & ... & C_k | C_1 v ... v C_k |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|  C_1 C_2   C_k  |                 |  Just one       |  Not just one   |
|   o---o-...-o   |                 |                 |                 |
|    \       /    |                 |  of the C_j,    |  of the C_j,    |
|     \     /     |                 |                 |                 |
|      \   /      |                 |  j = 1 to k,    |  j = 1 to k,    |
|       \ /       |                 |                 |                 |
|        @        | (C_1, ..., C_k) |  is not true.   |  is true.       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o

It is possible to specify "abstract rules of equivalence" (AROE's)
between cacti, rules for transforming one cactus into another that
are "formal" in the sense of being indifferent to the above choices
for logical or semantic interpretations, and that partition the set
of cacti into formal equivalence classes.

A "reduction" is an equivalence transformation
that is applied in the direction of decreasing
graphical complexity.

A "basic reduction" is a reduction that applies
to one of the two families of basic connectives.

Table 16 schematizes the two types of basic reductions
in a purely formal, interpretation-independent fashion.

Table 16.  Basic Reductions
o---------------------------------------o
|                                       |
|    C_1 ... C_k                        |
|         @         =         @         |
|                                       |
|    if and only if                     |
|                                       |
|    C_j = @ for all j = 1 to k         |
|                                       |
o---------------------------------------o
|                                       |
|   C_1 C_2   C_k                       |
|    o---o-...-o                        |
|     \       /                         |
|      \     /                          |
|       \   /                           |
|        \ /                            |
|         @         =         @         |
|                                       |
|   if and only if                      |
|                                       |
|         o                             |
|         |                             |
|   C_j = @ for exactly one j in [1, k] |
|                                       |
o---------------------------------------o

The careful reader will have noticed that we have begun to use
graphical paints like "a", "b", "c" and schematic proxies like
"C_1", "C_j", "C_k" in a variety of novel and unjustified ways.

The careful writer would have already introduced a whole bevy of
technical concepts and proved a whole crew of formal theorems to
justify their use before contemplating this stage of development,
but I have been hurrying to proceed with the informal exposition,
and this expedition must leave steps to the reader's imagination.

Of course I mean the "active imagination".
So let me assist the prospective exercise
with a few hints of what it would take to
guarantee that these practices make sense.

Jon Awbrey

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