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ONT Re: Differential Logic




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DLOG.  Note D66

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Terminological Interlude (concl.)

Table 55 supplies a more detailed outline of terminology for operators and
their results.  Here, I list the restrictive subtype (or narrowest defined
subtype) that applies to each entity, and I indicate across the span of the
Table the whole spectrum of alternative types that color the interpretation
of each symbol.  Accordingly, each of the component operator maps WJ, since
their ranges are 1-dimensional (of type B^1 or D^1), can be regarded either
as propositions WJ : EU -> B or as logical transformations WJ : EU% -> X%.
As a rule, the plan of the Table allows us to name each entry by detaching
the adjective at the left of its row and prefixing it to the generic noun
at the top of its column.  In one case, however, it is customary to depart
from this scheme.  Because the phrase "differential proposition", applied
to the result dJ : EU -> D, does not distinguish it from the general run
of differential propositions G : EU -> B, it is usual to single out dJ
as the "tangent proposition" of J.

Table 55.  Synopsis of Terminology:  Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
|              | Operator             | Proposition        | Map                  |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Tacit        | !e! :                | !e!J :             | !e!J :               |
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x]     |
|              | (U%->X%)->(EU%->X%)  | B^2 x D^2 -> B     | [B^2 x D^2 ->[B^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Trope        | !h! :                | !h!J :             | !h!J :               |
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2 ->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Enlargement  | E :                  | EJ :               | EJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2 ->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Difference   | D :                  | DJ :               | DJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2 ->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Differential | d :                  | dJ :               | dJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2 ->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Remainder    | r :                  | rJ :               | rJ :                 |
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
|              | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2 ->[D^1]   |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Radius       | $e$ = <!e!, !h!> :   |                    | $e$J :               |
| Operator     | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Secant       | $E$ = <!e!, E> :     |                    | $E$J :               |
| Operator     | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Chord        | $D$ = <!e!, D> :     |                    | $D$J :               |
| Operator     | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o
|              |                      |                    |                      |
| Tangent      | $T$ = <!e!, d> :     | dJ :               | $T$J :               |
| Functor      | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[x, dx] |
|              | (U%->X%)->(EU%->EX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[B x D] |
|              |                      |                    |                      |
o--------------o----------------------o--------------------o----------------------o

Jon Awbrey

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