ONT Re: Reductions Among Relations
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RAR. Note 12
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Compositional Analysis of Relations (cont.)
Let us then push on, in a retrograde way, returning to the orbit
of those very first relations that got us into the midst of this
quandary in the first place, to wit, the relations in medias res,
the relations of betwixt and between and all of their sundry kin.
But let us this time place the paltry special relations on which
we fixed the first time around back within the setting of a much
broader and a much more systematically examined context, that is,
an extended family of related relations or variations on a theme.
I hope that you will be able to recall that cousin of
the Between relation that we took up here once before,
and that you will be able to recognize its characters,
even if I now disguise it under a new name and partly
dissemble them under a new manner of parameterization.
Where you might take the name "IO" to mean "in order",
it is the relation defined on three real numbers thus:
Let the relation named "IO(213)"
such that IO(213) c R^3 = RxRxR,
be defined in the following way:
| IO(213)<x, a, b> iff [a < x < b],
|
| equivalently,
|
| IO(213)<x, a, b> iff [a < x] and [x < b].
Corresponding to the 3-adic relation IO(213) c R^3 = RxRxR,
there is a "proposition", a function io(213) : R^3 -> B,
that I will describe, until a better name comes along,
as the "relation map" that is "dual to" the relation.
It is also known as the "indicator" of that relation.
Consider the boolean analogue or the logical variant of IO, with
real domains of type R now replaced by boolean domains of type B.
The boolean analogue of the ordering "<" is the implication "=>",
so the logical variant of the relation IO(213) is given this way:
Let the relation named "IO(213)"
such that IO(213) c B^3 = BxBxB,
be defined in the following way:
| IO(213)<x, a, b>
|
| iff
|
| [a => x] and [x => b]
When it does not risk any confusion,
one can express this also like this:
| IO(213)<x, a, b>
|
| iff
|
| a => x => b
Corresponding to the 3-adic relation IO(213) c B^3 = BxBxB,
there is a "proposition", a function io(213) : B^3 -> B,
that I will describe, until a better name comes along,
as the "relation map" that is "dual to" the relation.
It is also known as the "indicator" of that relation.
At this point I want to try and get away with a bit
of additional flexibility in the syntax that I use,
reusing some of the same names for what are distinct
but closely related types of mathematical objects.
In particular, I would like to have the license
to speak a bit more loosely about these objects,
to ignore the distinction between "relations" of
the form Q c X_1 x ... x X_k and "relation maps"
of the form q : X_1 x ... x X_k -> B, and perhaps
on sundry informal occasions to use the very same
names for them -- The Horror! -- hoping to let the
context determine the appropriate type of object,
except where it may be necessary to maintain this
distinction in order to avoid risking confusion.
In order to keep track of all of the players -- not to mention all of the refs! --
it may help to re-introduce a diagram that I have used many times before, as a
kind of a play-book or programme, to sort out the burgeoning teams of objects
and the cryptic arrays of signs that we need to follow throughout the course
of this rather extended run-into-overtime game:
o-----------------------------o-----------------------------o
| Objective Framework (OF) | Interpretive Framework (IF) |
o-----------------------------o-----------------------------o
| Formal Objects | Formal Signs & Texts |
o-----------------------------o-----------------------------o
| | |
| Propositions | Expressions |
| (Logical) | (Logical) |
| o | o |
| | | | |
| | | | |
| o | o |
| / \ | / \ |
| / \ | / \ |
| o o | o o |
| Sets Maps | Set Names Map Names |
| (Geometric) (Functional) | (Geometric) (Functional) |
| | |
o-----------------------------o-----------------------------o
| | |
| B^k B^k -> B | "IO(213)" "io(213)" |
| R^k R^k -> B | "IO(213)" "io(213)" |
| X^k X^k -> B | "Q" "q" |
| | |
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Jon Awbrey
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