ONT Re: Reductions Among Relations
RAR. Note 8
Compositional Analysis of Relations (cont.)
There are one or two confusions that demand to
be cleared up before I can proceed any further.
We had been enjoying our long-anticipated breakthrough on the
allegedly "easy case" of projective reduction, having detected
hidden within that story of our old friends and usual suspects
A and B two examples of 3-adic relations, L(A) and L(B), that
are indeed amenable, not only to being distinguished, one from
the other, between the two of them, but also to being uniquely
determined amongst all of their kin by the information that is
contained in their 2-dimensional projections. So far, so good.
Had I been thinking fast enough, I would have assigned these the
nomen "triadics reducible in projections over dyadics" (TRIPOD's).
Other good names: "triadics reducible over projections" (TROP's),
or perhaps "triadics reconstructible out of projections" (TROOP's).
Then we fell upon two examples of triadic relations, L_0 and L_1,
that I described as "projectively irreducible triadics" (PIT's),
because they collapse into an indistinct mass of non-descript
flatness on having their dyadic pictures taken. That acronym
does not always work for me, so I will give them the alias of
"triadics irreducible by projections over dyadics" (TIBPOD's),
or perhaps "triadics irreducible over projections" (TIOP's).
I'm not accustomed to putting much stock in my own proofs
until I can reflect on them for a suitable period of time,
or until some other people have been able to go over them,
but until that day comes I will just have to move forward
with these results as I presently see them.
In reply to my notes on these topics, Matthew West
has contributed the following pair of commentaries:
1. Regarding L(A) and L(B)
| Whilst I appreciate the academic support for showing
| that any triadic relation can be represented by some
| number of dyadic relations, the real point is to use
| this fact to seek for an improved analysis based on
| more fundamental concepts. It is not the objective
| to do something mechanical.
2. Regarding L_0 and L_1
| I don't think you have shown very much except that reducing
| triadic relations to dyadic relations using the mechanical
| process you have defined can loose information. I am not
| surprised by this. My experience of doing this with real,
| rather than abstract examples, is that there are often
| extra things to do.
So I need to clarify that what I think that I showed was
that "some" triadic relations are "reducible" in a given
informational sense to the information that is contained
in their dyadic projections, e.g., as L(A) and L(B) were,
but that others are not reducible in this particular way,
e.g., as L_0 and L_1 were not.
Now, aside from this, I think that Matthew is raising
a very important issue here, which I personally grasp
in terms of two different ways of losing information,
1. The information that we lose in forming a trial model,
in effect, in going from the unformalized "real world"
over to the formal context or the syntactic medium in
which models are constrained to live out their lives.
2. The information that we lose in "turning the crank"
on the model, that is, in drawing inferences from
the admittedly reductive and "off'n'wrong" model
in a way that loses even the initial information
that it captured about the real-world situation.
To do it justice, though, I will need to return
to this issue in a less frazzled frame of mind.
This will complete the revision of this RARified thread from last Autumn.
I will wind it up, as far as this part of it goes, by recapitulating the
development of the "Rise" relation, from a couple of days ago, this time
working through its analysis and its synthesis as fully as I know how at
the present state of my knowledge. The good of this exercise, of course,
the reason for doing all of this necessary work, is not because the Rise
relation is so terribly interesting in itself, but rather to demonstrate
the utility of the functional framework and its sundry attached tools in
their application to a nigh unto minimal and thus least obstructive case.