SUO: Re: Reductions Among Relations
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RARer & RARer Interest Group:
We have pursued the "projective analysis" of 3-adic relations,
tracing the pursuit via a ready quantity of concrete examples,
just far enough to arrive at this clearly demonstrable result:
| Some 3-adic relations are and
| some 3-adic relations are not
| uniquely reconstructible from,
| or informatively reducible to,
| their 2-adic projection data.
We now take up the "compositive analysis" of 3-adic relations,
coining a term to prevent confusion, like there's a chance in
the world of that, but still making use of nothing other than
that "standardly uptaken operation" of relational composition,
the one that constitutes the customary generalization of what
just about every formal, logical, mathematical community that
is known to the writer, anyway, dubs "functional composition".
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Note 6
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Compositive Reduction of Relations
The first order of business under this heading is straightforward enough:
to define what is standardly described as the "composition of relations".
[ Notes on the Ancestry, the Application, and
| the Anticipated Broadening of these Concepts:
|
| This is basically the same operation that C.S. Peirce described as
| "relative multiplication", except for the technical distinction that
| he worked primarily with so-called "relative terms", like "lover of",
| "sign of the object ~~~ to", and "warrantor of the fact that ~~~ to",
| rather than with the kinds of extensional and intensional relations
| to which the majority of us are probably more accustomed to use.
|
| It is with regard to this special notion of "composition", and it alone,
| that I plan to discuss the inverse notion of "decomposition". I try to
| respect other people's "reserved words" as much as I can, even if I can
| only go so far as to take them at their words and their own definitions
| of them in forming my interpretation of what they are apparently saying.
| Therefore, if I want to speak about other ways of building up relations
| from other relations and different ways of breaking down relations into
| other relations, then I will try to think up other names for these ways,
| or revert to a generic usage of terms like "analysis" and "combination".
|
| When a generalized definition of "relational composition" has been given,
| and its specialization to 2-adic relations is duly noted, then one will
| be able to notice that it is simply an aspect of this definition that
| the composition of two 2-adic relations yields nothing more than yet
| another 2-adic relation. This will, I hope, in more than one sense
| of the word, bring "closure" to this issue, of what can be reduced
| to compositions of 2-adic relations, to wit, just 2-adic relations.
]
A notion of relational composition is to be defined that generalizes the
usual notion of functional composition. The "composition of functions"
is that by which -- composing functions "on the right", as they say --
f : X -> Y and g : Y -> Z yield the "composite function" fg : X -> Z.
Accordingly, the "composition" of dyadic relations is that by which --
composing them here by convention in the same left to right fashion --
P sub XxY and Q sub YxZ yield the "composite relation" PQ sub XxZ.
There is a neat way of defining relational composition, one that
not only manifests its relationship to the projection operations
that go with any cartesian product space, but also suggests some
natural directions for generalizing relational composition beyond
the limits of the 2-adic case, and even beyond relations that have
any fixed arity, that is, to the general case of formal languages.
I often call this definition Tarski's Trick, though it probably
goes back further than that. This is what I will take up next.
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Note 7
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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 found 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).
[ The Author Addresses The Exasperants Of His Foibles:
|
| Please do not concern yourselves too much, or be too irritated by
| my extravagant struggles to hash out memorable hash codes for the
| produce the formal farm and the offerings of the logical scullery.
| I will eventually sort it all out and settle on some few that fit.
]
I am not accustomed to putting much stock in my own proofs
until I can reflect on them for a suitable period of time
or until a few other people have been able to go over them,
but until that happens I will just have to go 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,
namely:
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.
Many Regards,
Jon Awbrey
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