ONT Re: Differential Logic -- Series A

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DLOG.  Note A8

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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 form of boundary relationship is exhibited for any cell
of origin that one might elect to indicate, say, by means of the
conjunction of positive and negative basis features u_1 ... u_k,
where 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 the cell u_1 ... u_k.

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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