[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D15

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The Extended Universe of Discourse

| At the moment of speaking, I would like to have perceived a nameless voice,
| long preceding me, leaving me merely to enmesh myself in it, taking up its
| cadence, and to lodge myself, when no one was looking, in its interstices
| as if it had paused an instant, in suspense, to beckon to me.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]

Next, we define the so-called "extended alphabet" or "bundled alphabet" E!A! as:

   E!A!  =  !A! |_| d!A!  =  {a_1, ..., a_n,  da_1, ..., da_n}.

This supplies enough material to construct the "differential extension" EA,
or the "tangent bundle" over the initial space A, in the following fashion:

   EA  =  A x dA  =  <|E!A!|>  =  <|!A! |_| d!A!|>

       =  <|a_1, ..., a_n,  da_1, ..., da_n|>,

thus giving EA the type B^n x D^n.

Finally, the tangent universe EA% = [E!A!] is constituted from the totality
of points and maps, or interpretations and propositions, which are based on
the extended set of features E!A!:

   EA%  =  [E!A!]  =  [a_1, ..., a_n,  da_1, ..., da_n],

thus giving the tangent universe E!A! the type
(B^n x D^n +-> B) = (B^n x D^n, (B^n x D^n -> B).

A proposition in the tangent universe [E!A!] is
called a "differential proposition" and forms the
analogue of a system of differential equations,
constraints, or relations in ordinary calculus.

With these constructions, to be specific, the differential extension EA
and the differential proposition h : EA -> B, we have arrived, in concept
at least, at one of the major subgoals of this study.  At this juncture,
I pause by way of summary to set another Table with the current crop of
mathematical produce (Table 8).

Table 8.  Notation for the Differential Extension of Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol  | Notation          | Description       | Type              |
o---------o-------------------o-------------------o-------------------o
| d!A!    | {da_1, ..., da_n} | Alphabet of       | [n]  =  #n#       |
|         |                   | differential      |                   |
|         |                   | features          |                   |
o---------o-------------------o-------------------o-------------------o
| dA_i    | {(da_i), da_i}    | Differential      |  D                |
|         |                   | dimension i       |                   |
o---------o-------------------o-------------------o-------------------o
| dA      | <|d!A!|>          | Tangent space     |  D^n              |
|         | <|da_i,...,da_n|> | at a point:       |                   |
|         | {<da_i,...,da_n>} | Set of changes,   |                   |
|         | dA_1 x ... x dA_n | motions, steps,   |                   |
|         | Prod_i dA_i       | tangent vectors   |                   |
|         |                   | at a point        |                   |
o---------o-------------------o-------------------o-------------------o
| dA*     | (hom : dA -> B)   | Linear functions  | (D^n)*  ~=~  D^n  |
|         |                   | on dA             |                   |
o---------o-------------------o-------------------o-------------------o
| dA^     | (dA -> B)         | Boolean functions |  D^n -> B         |
|         |                   | on dA             |                   |
o---------o-------------------o-------------------o-------------------o
| dA%     | [d!A!]            | Tangent universe  | (D^n, (D^n -> B)) |
|         | (dA, dA^)         | at a point of A%, | (D^n +-> B)       |
|         | (dA +-> B)        | based on the      | [D^n]             |
|         | (dA, (dA -> B))   | tangent features  |                   |
|         | [da_1, ..., da_n] | {da_1, ..., da_n} |                   |
o---------o-------------------o-------------------o-------------------o

The adjectives "differential" or "tangent" are systematically attached to
every construct based on the differential alphabet d!A!, taken by itself.
Strictly speaking, we probably ought to call d!A! the set of "cotangent"
features derived from !A!, but the only time this distinction really
seems to matter is when we need to distinguish the tangent vectors
as maps of type (B^n -> B) -> B from cotangent vectors as elements
of type D^n.  In like fashion, having defined E!A! = !A! |_| d!A!,
we can systematically attach the adjective "extended" or the
substantive "bundle" to all the constructs associated with
this full complement of 2n features.

Eventually we may want to extend our basic alphabet even further,
to allow for discussion of higher order differential expressions.
For those who want to run ahead, and would like to play through,
I submit the following gamut of notation (Table 9).

Table 9.  Higher Order Differential Features
o----------------------------------------o----------------------------------------o
|                                        |                                        |
| !A!   = d^0.!A! = {a_1, ..., a_n}      | E^0.!A!  = d^0.!A!                     |
|                                        |                                        |
| d!A!  = d^1.!A! = {da_1, ..., da_n}    | E^1.!A!  = d^0.!A! |_| d^1.!A!         |
|                                        |                                        |
|         d^k.!A! = {d^k.a_1,...,d^k.a_n}| E^k.!A!  = d^0.!A! |_| ... |_| d^k.!A! |
|                                        |                                        |
| d*!A! = {d^0.!A!, ..., d^k.!A!, ...}   | E^oo.!A! = d*!A!                       |
|                                        |                                        |
o----------------------------------------o----------------------------------------o

Jon Awbrey

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