[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D42

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The Radius Operator:  $e$

| And the tangible fact at the root of all our thought-distinctions,
| however subtle, is that there is no one of them so fine as to
| consist in anything but a possible difference of practice.
|
| William James, 'Pragmatism', [Jam, 46]

The operator identified as $d$^0 in the analytic diagram (Figure 33) has the
sole purpose of creating a proxy for F in the appropriately extended context.
Construed in terms of its broadest components, $d$^0 is equivalent to the
doubly tacit extension operator <!e!, !e!>, in recognition of which let
us redub it as "$e$".  Pursuing a geometric analogy, we may refer to
$e$ = <!e!, !e!> = $d$^0 as the "radius operator".  The operation
that is intended by all of these forms is defined by the equation:

   $e$F  =  <!e!, !e!> F

         =  <!e!F, !e!F>

         =  <!e!F_1, ..., !e!F_k,  !e!F_1, ..., !e!F_k>,

which is tantamount to the system of equations given below.

o--------------------------------------------------------------------------------o
|                                                                                |
|  x_1   =   !e!F_1 <u_1, ..., u_n,  du_1, ..., du_n>   =   F_1 <u_1, ..., u_n>  |
|                                                                                |
|  ...                                                                           |
|                                                                                |
|  x_k   =   !e!F_k <u_1, ..., u_n,  du_1, ..., du_n>   =   F_k <u_1, ..., u_n>  |
|                                                                                |
|                                                                                |
| dx_1   =   !e!F_1 <u_1, ..., u_n,  du_1, ..., du_n>   =   F_1 <u_1, ..., u_n>  |
|                                                                                |
|  ...                                                                           |
|                                                                                |
| dx_k   =   !e!F_k <u_1, ..., u_n,  du_1, ..., du_n>   =   F_k <u_1, ..., u_n>  |
|                                                                                |
o--------------------------------------------------------------------------------o

Jon Awbrey

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