[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 11

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We have been contemplating functions of the type f : X -> 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 a pair of basic connectives in the cactus language for ZOL.

Consider the proposition f of concrete type f : !P! x !Q! x !R! -> B
and abstract type f : B^3 -> B that is written as "(p, q, r)" in the
cactus syntax.  Taken as an assertion in what C.S. Peirce called the
"existential interpretation", the so-called boundary form "(p, q, r)"
asserts that one and only one of the propositions p, q, r is false.
It is instructive to consider this assertion in relation to the
conjunction "p q r" of the same propositions.  A venn diagram
for the boundary form (p, q, r) is shown in Figure 11.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o-------------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` P ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o--o----------o ` o----------o--o ` ` ` ` ` ` |
| ` ` ` ` ` `/` ` \%%%%%%%%%%\ /%%%%%%%%%%/ ` `\` ` ` ` ` ` |
| ` ` ` ` ` / ` ` `\%%%%%%%%%%o%%%%%%%%%%/` ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` ` ` ` \%%%%%%%%/ \%%%%%%%%/ ` ` ` `\` ` ` ` ` |
| ` ` ` ` / ` ` ` ` `\%%%%%%/ ` \%%%%%%/` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` \%%%%/` ` `\%%%%/ ` ` ` ` ` `\` ` ` ` |
| ` ` ` o ` ` ` ` ` ` `o--o-------o--o` ` ` ` ` ` ` o ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` `Q` ` ` ` |%%%%%%%| ` ` ` `R` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` o ` ` ` ` ` ` ` ` o%%%%%%%o ` ` ` ` ` ` ` ` o ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` `\%%%%%/` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` ` ` \%%%/ ` ` ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` `\%/` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` `/ \` ` ` ` ` ` ` `/` ` ` ` ` ` |
| ` ` ` ` ` ` o-------------o ` o-------------o ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 11.  Boundary Form (p, q, r)

In relation to the center cell indicated by the conjunction pqr
the region indicated by (p, q, r) 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, pqr.

More generally speaking, in a k-dimensional universe of discourse
that is based on the "alphabet" of features !X! = {x_1, ..., x_k},
the same form of boundary relationship is manifested for any cell
of origin that one might choose 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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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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