ONT Re: Differential Logic
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DLOG. Note D65
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Terminological Interlude (cont.)
At the beginning of this Division I recast the natural form of a proposition
J : U -> B into the thematic role of a transformation J : U% -> [x], where x
was a variable recruited to express the newly independent ¢(J). However, in
my computations and representations of operator actions I immediately lapsed
back to viewing the results as native elements of the extended universe EU%,
in other words, as propositions WJ : EU -> B, where W ranged over the set
{!e!, E, D, d, r}. That is as it should be. In fact, I have worked hard
to devise a language that gives us all of these competing advantages, the
flexibility to exchange terms and types that bear equal information value,
and the capacity to reflect as quickly and as wittingly as a controlled
reflex on the fibers of our propositions, independently of whether they
express amusements, beliefs, or conjectures.
As we take on target spaces of increasing dimension, however, these
types of confusions (and confusions of types) become less and less
permissible. For this reason, Tables 54 and 55 present a rather
detailed summary of the notation and the terminology that I am
using here, as applied to the case of J = uv. The rationale
of these Tables is not so much to train more elephant guns
on this poor drosophila of an example, but to establish
the general paradigm with enough solidity to bear the
weight of abstraction that is coming on down the road.
Table 54 provides basic notation and descriptive information for the
objects and operators that are used used in this Example, giving the
generic type (or broadest defined type) for each entity. Here, the
sans serif operators, $W$ in {$e$, $E$, $D$, $d$, $r$}, and their
components W in {!e!, !h!, E, D, d, r} both have the same broad
type $W$, W : (U% -> X%) -> (EU% -> EX%), as would be expected
of operators that map transformations J : U% -> X% to extended
transformations $W$J, WJ : EU% -> EX%.
Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators
o------o-------------------------o------------------o----------------------------o
| Item | Notation | Description | Type |
o------o-------------------------o------------------o----------------------------o
| | | | |
| U% | = [u, v] | Source Universe | [B^2] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| X% | = [x] | Target Universe | [B^1] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| EU% | = [u, v, du, dv] | Extended | [B^2 x D^2] |
| | | Source Universe | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| EX% | = [x, dx] | Extended | [B^1 x D^1] |
| | | Target Universe | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| J | J : U -> B | Proposition | (B^2 -> B) c [B^2] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| J | J : U% -> X% | Transformation, | [B^2] -> [B^1] |
| | | or Mapping | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| W | W : | Operator | |
| | U% -> EU%, | | [B^2] -> [B^2 x D^2], |
| | X% -> EX%, | | [B^1] -> [B^1 x D^1], |
| | (U%->X%)->(EU%->EX%), | | ([B^2] -> [B^1]) |
| | for each W among: | | -> |
| | e!, !h!, E, D, d | | ([B^2 x D^2]->[B^1 x D^1]) |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | |
| !e! | | Tacit Extension Operator !e! |
| !h! | | Trope Extension Operator !h! |
| E | | Enlargement Operator E |
| D | | Difference Operator D |
| d | | Differential Operator d |
| | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| $W$ | $W$ : | Operator | |
| | U% -> $T$U% = EU%, | | [B^2] -> [B^2 x D^2], |
| | X% -> $T$X% = EX%, | | [B^1] -> [B^1 x D^1], |
| | (U%->X%)->($T$U%->$T$X%)| | ([B^2] -> [B^1]) |
| | for each $W$ among: | | -> |
| | $e$, $E$, $D$, $T$ | | ([B^2 x D^2]->[B^1 x D^1]) |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | |
| $e$ | | Radius Operator $e$ = <!e!, !h!> |
| $E$ | | Secant Operator $E$ = <!e!, E > |
| $D$ | | Chord Operator $D$ = <!e!, D > |
| $T$ | | Tangent Functor $T$ = <!e!, d > |
| | | |
o------o-------------------------o-----------------------------------------------o
Jon Awbrey
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