ONT Re: Differential Logic
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DLOG. Note D38
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Operators on Propositions and Transformations
The next step is naturally inclined toward objects of the next higher order,
namely, with operators that take in argument lists of logical transformations
and that give back specified types of logical transformations as their results.
For our present aims, we do not need to consider the most general class of such
operators, nor any one of them for its own sake. Rather, we are interested in
the special sorts of operators that arise in the study and analysis of logical
transformations. Figuratively speaking, these operators serve as instruments
for the live tomography (and hopefully not the vivisection) of the forms of
change under view. Beyond that, they open up ways to implement the changes
of view that we need to grasp all the variations on a transformational theme,
or to appreciate enough of its significant features to "get the drift" of the
change occurring, to form a passing acquaintance or a synthetic comprehension
of its general character and disposition.
The simplest type of operator is one that takes a single transformation
as an argument and returns a single transformation as a result, and most
of the operators that I will explicitly consider here are of this kind.
Figure 31 illustrates the typical situation.
o---------------------------------------o
| |
| |
| U% F X% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| !W! | | !W! |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| !W!U% !W!F !W!X% |
| |
| |
o---------------------------------------o
Figure 31. Operator Diagram (1)
In this Figure, "!W!" serves as a generic name for an operator, in this case
one that takes a logical transformation F of type (U% -> X%) into a logical
transformation !W!F of the type (!W!U% -> !W!X%). Thus, the operator !W!
must be viewed as making assignments for both families of objects that we
have previously considered, both for universes of discourse like U% and X%
and for logical transformations like F.
NB. Strictly speaking, an operator like !W! works between two whole categories
of universes and transformations, which we call the "source" and the "target"
categories of !W!. Given this setting, !W! specifies for each universe U%
in its source category a definite universe !W!U% in its target category,
and to each transformation F in its source category it assigns a unique
transformation !W!F in its target category. Naturally, this only works
if !W! takes the source U% and the target X% of the map F over to the
source !W!U% and the target !W!X% of the map !W!F. With luck or care
enough, we can avoid ever having to put anything like that in words
again, letting diagrams do the work. In the situations of present
concern we are usually focused on a single transformation F, and
thus we can take it for granted that the assignment of universes
under !W! is defined appropriately at the source and the target
ends of F. It is not always the case, though, that we need to
use the particular names (like "!W!U%" and "!W!X%") that !W!
assigns by default to its operative image universes. In most
contexts we will usually have a prior acquaintance with these
universes under other names, and it is only necessary that we
can tell from the information associated with an operator !W!
what universes they are.
In Figure 31 the maps F and !W!F are displayed horizontally, the way that one
normally orients functional arrows in a written text, and !W! rolls the map F
downward into the images that are associated with !W!F. In Figure 32 the same
information is redrawn so that the maps F and !W!F flow down the page, and !W!
unfurls the map F rightward into domains that are the eminent purview of !W!F.
o---------------------------------------o
| |
| |
| U% !W! !W!U% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| F | | !W!F |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| X% !W! !W!X% |
| |
| |
o---------------------------------------o
Figure 32. Operator Diagram (2)
The latter arrangement, as it appears in Figure 32, is more congruent with the
thinking about operators that we shall be doing in the rest of this discussion,
since all logical transformations from here on out will be pictured vertically,
after the fashion of Figure 30.
Jon Awbrey
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