[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D39

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Differential Analysis of Propositions and Transformations

| The resultant metaphysical problem now is this:
|
| 'Does the man go round the squirrel or not?'
|
| William James, 'Pragmatism', [Jam, 43]

The approach to the differential analysis of logical propositions and
transformations of discourse that will be pursued here is carried out
in terms of particular operators !W! that act on propositions F or on
transformations F to yield the corresponding operator maps !W!F.  The
operator results then become the subject of a series of further stages
of analysis, which take them apart into their propositional components,
rendering them as a set of purely logical constituents.  After this is
done, all the parts are then re-integrated to reconstruct the original
object in the light of a more complete understanding, at least in ways
that enable one to appreciate certain aspects of it with fresh insight.

NB.  Remark on Strategy.  At this point I run into a set of conceptual
difficulties that force me to make a strategic choice in how I proceed.
Part of the problem can be remedied by extending my discussion of tacit
extensions to the transformational context.  But the troubles that remain
are much more obstinate and lead me to try two different types of solution.
The approach that I develop first makes use of a variant type of extension
operator, the "trope extension", to be defined below.  This method is more
conservative and requires less preparation, but has features which make it
seem unsatisfactory in the long run.  A more radical approach, but one with
a better hope of long term success, makes use of the notion of "contingency
spaces".  These are an even more generous type of extended universe than the
kind I currently use, but are defined subject to certain internal constraints.
The extra work needed to set up this method forces me to put it off to a later
stage.  However, as a compromise, and to prepare the ground for the next pass,
I call attention to the various conceptual difficulties as they arise along
the way and try to give an honest estimate of how well my first approach
deals with them.

I now describe in general terms the particular operators that are
instrumental to this form of analysis.  The main series of operators
all have the form !W! : (U% -> X%) -> (EU% -> EX%).  If we assume that
the source universe U% and the target universe X% have finite dimensions
n and k, respectively, then each operator !W! is encompassed by the same
abstract type:

   !W!  :  ([B^n] -> [B^k])  ->  ([B^n x D^n] -> [B^k x D^k]).

Since the range features of the operator result !W!F : [B^n x D^n] -> [B^k x D^k]
can be sorted out by their ordinary versus their differential qualities and the
component maps can be examined independently, the complete operator !W! can be
separated accordingly into two components, in the form !W! = <!e!, W>.  Given
a fixed context of source and target universes of discourse, !e! is always
the same type of operator, a multiple component elaboration of the tacit
extension operators that were articulated earlier.  In this context !e!
has the shape:

   Concrete type.  !e!  :  ( U%   ->  X%  )  ->  (    EU%     ->  X%  )

   Abstract type.  !e!  :  ([B^n] -> [B^k])  ->  ([B^n x D^n] -> [B^k])

On the other hand, the operator W is specific to
each !W!.  In this context W always has the form:

   Concrete type.   W   :  ( U%   ->  X%  )  ->  (    EU%     ->  dX% )

   Abstract type.   W   :  ([B^n] -> [B^k])  ->  ([B^n x D^n] -> [D^k])

In the types just assigned to !e! and W, and implicitly to their results
!e!F and WF, I have listed the most restrictive ranges defined for them,
rather than the more expansive target spaces that subsume these ranges.
When there is need to recognize both, we may use type indications like
the following:

   !e!F  :  (EU% ->  X% c EX%)  ~=~  ([B^n x D^n] -> [B^k] c [B^k x D^k])

     WF  :  (EU% -> dX% c EX%)  ~=~  ([B^n x D^n] -> [D^k] c [B^k x D^k])

Hopefully, though, a general appreciation of these subsumptions will prevent
us from having to make such declarations more often than absolutely necessary.

In giving names to these operators I am attempting to preserve as much of the
traditional nomenclature and as many of the classical associations as possible.
The chief difficulty in doing this is occasioned by the distinction between the
"sans serif" operators !W! and their "serified" components W, which forces me
to find two distinct but parallel sets of terminology.  Here is the plan that
I have settled on.  First, the component operators W are named by analogy
with the corresponding operators in the classical difference calculus.
Next, the complete operators !W! = <!e!, W> are assigned their titles
according to their roles in a geometric or trigonometric allegory,
if only to ensure that the tangent functor, that belongs to this
family and whose exposition I am still working toward, comes out
fit with its customary name.  Finally, the operator results !W!F
and WF can be fixed in this frame of reference by tethering the
operative adjective for !W! or W to the anchoring epithet "map",
in conformity with an already standard practice.

Jon Awbrey

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