SUO: Re: Expostulation
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Matthew,
I am going to put some more time in on the concrete examples,
and clear away everything that's been said a number of times,
leaving only the lines of inquiry that still await an answer.
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MW: So I'm afraid that is not an argument that will
convince me on its own. If you wish to do that
you will have to present a specific example of
where the information in some triadic relation
cannot be transformed into some set of binary
relations. If you provide the triadic relation,
I will provide the binary relations from which
the triadic relation can be constructed (or not
and admit you are right).
JA: Just to refresh your memory,
here are L^0 and L^1 once again:
---------------------------------------------------
|
| Relation L^0
|
| L^0 = {<x, y, z> in B^3 : x + y + z = 0}.
|
| L^0 has the following four triples
| of the form <x, y, z> in B^3:
|
| <0, 0, 0>
| <0, 1, 1>
| <1, 0, 1>
| <1, 1, 0>
|
---------------------------------------------------
|
| Relation L^1
|
| L^1 = {<x, y, z> in B^3 : x + y + z = 1}.
|
| L^1 has the following four triples
| of the form <x, y, z> in B^3:
|
| <0, 0, 1>
| <0, 1, 0>
| <1, 0, 0>
| <1, 1, 1>
|
---------------------------------------------------
|
| Most recent exposition of L^0 & L^1 at:
|
| http://suo.ieee.org/email/msg04575.html
|
---------------------------------------------------
Perhaps the difficulty in getting started on this task lies
in knowing exactly what it would mean that "the information
in some triadic relation [can] be transformed into some set
of binary relations", or even just what, in operational and
practical terms, may be incumbent on a person who is called
for to "provide the binary relations from which the triadic
relation can be constructed". My sense of it tells me that
the word "information" plays a critical role in the problem,
but I will need to think a bit more about how to tackle the
elicitation and the explication of its bearing on the issue.
JA: Let me tell you where I think that you and I stand at present.
I believe that we have agreed on the ground rules that nothing of
any rational sense will come of arguing about any puported form of
"analysis", "decomposition", "discombobulation", "reconstruction",
"reduction", "transliteration", "transmogrification", or whatever,
unless we have clearly defined the particular species of relation
that we are talking about in a case at issue. Now, for my own part,
I regard the argument as being effectively over right at this point,
since I never knew a "form of analysis" (FOA) that pretended to be
anything other than a relationship among one thing, the analysand,
and at least two other things, the analytic components. One plus
at least two implies at least three, and so the very idea of a FOA
has us neck deep already in a 3-adic relation.
MW: But as usual you skip from three things to necessary triadicity.
In fact the only thing that is required for 3 things (or any
number of things for that matter) to be related is that in some
network of relations there is a path that links them together.
JA: This is one way that we can say, if we speak loosely enough,
that "three things are related". People sometimes describe
this by saying that they are "pairwise related" in such and
such a way that chosen triples are connected in 2-adic pairs.
It is what is what we might call some kind of "discombobulatable
3-adic relation", say, "conjunctively reducible", which is the
same thing, if I remember correctly, or related to, at least,
the "projectively reducible" 3-adic relation. Anyway, there
are many such special cases of relations, but their existence
does not affect the general truth that many 3-adic relations
are not of this kind. And, once again, the logical forms of
these very reductions involve 3-adic relations at every step
of the corresponding analyses.
JA: For example, contemplate the 3-adic relation that is expressed by
the rheme "--- is an ordered pair consisting of --- and then ---",
of which one instantiation is <<Jack, Jill>, Jack, Jill>, and which
one single instance might be diagrammed as follows:
| Jack
| o
| /
| <Jack, Jill> o---<|
| \
| o
| Jill
JA: Let us symbolize this 3-adic relation by writing statements
of the form "LP12 (q, x, y)", of which we have the example
where LP12 (<Jack, Jill>, Jack, Jill) is true.
JA: The 3-adic relation that is meant to be exemplified here,
of which <<Jack, Jill>, Jack, Jill> is but one exemplar,
is just the sort of thing that can be expressed in terms
of certain pairwise connections, say, as we might try to
suggest by means of the following scheme:
| Jack
| [LP1]--o
| / \
| <Jack, Jill> o [L12]
| \ /
| [LP2]--o
| Jill
JA: In this picture, the 2-adic relations LP1, LP2, and L12
are defined to capture forms of pairwise relations like:
LP1 (q, x) <=> x is the 1st component
in the ordered pair q.
LP2 (q, y) <=> y is the 2nd component
in the ordered pair q.
L12 (x, y) <=> x is the 1st component &
y is the 2nd component
in some ordered pair.
JA: Now, to justify the claim that the original 3-adic relation can be
defined or expressed in terms of these particular 2-adic relations,
it is necessary to write out the requisite definition, for example:
LP12 (q, x, y) <=> LP1 (q, x) and LP2 (q, y).
JA: This, I think, does it. So here is an example of a 3-adic relation
that is "definable in terms of" (DITO) 2-adic relations. This is
all well and good to say, but it can only be well and good to say
if one chooses one's words very carefully, as I have in this case.
There is no mention of a 3-adic relation being "decomposable" to
2-adic relations, for that would invoke the notion of relational
composition, on analogy to functional composition, in minds that
have been exposed to certain early and formative life experiences.
JA: I would like to interject a note of reservation at this juncture,
since the definition of LP12 that I gave above is not yet in the
form that I would consider the tip-top shape for perfect clarity.
The main thing that it lacks in its current shape is any sort of
careful circumscriptions of the so-called "domains of definition"
for the assorted relations that serve defined and defining roles.
Just as a thing to think about, you might well see reason to ask:
"What is the domain of definition for the variable q?" But I do
not want to get bogged down in all of that now, so let's move on.
Still, it helps to name these domains: so "Q", "X", "Y" will do.
JA: Also, while I'm at it, allow me to convert my temporary terminology
into what is more like the "standard-usage-onyms" where I come from.
As a start, notice that the 2-adic relations LP1 and LP2 are actually
functions from the domain of q to the domains of x and y, respectively.
In recognition of this fact, it is usual to give them functional names,
commonly something like the following reformulations of the definitions:
1. Proj<1> : XxY -> X such that Proj<1> : <x, y> ~> x.
2. Proj<2> : XxY -> Y such that Proj<2> : <x, y> ~> y.
JA: And yet, once again, this entire skirmish to mop-up the residual bits
of 2-adic structure that may be squeegeed out of the original 3-adic
relation is being waged only after the general issue has been conceded,
for we had to use the 3-adic relation associated with the truth-functional
connective "and" just to express the definition of LP12.
JA: Still, I do not believe that the example that I gave of L^0 and L^1
is definable or reducible over any 2-adics in even this sort of way.
At least, I gave a proof that they cannot be projectively reducible,
and I am yet waiting for you to come up with the promised analysis
along whatever lines that you might be able to devise.
JA: What we have just now established, a little ways above,
is that an ordered pair q is determined by this "data":
a couple of 2-adic relations, in particular, functions,
whose names are not of any great importance, but which
I am now calling the "projections" Proj<1> amd Proj<2>,
together with the values of q's projections in X and Y.
JA: Here is a sort of picture that can be useful in helping us
to remember what is essential in this kind of a relational
situation, beneath the vast bewildering array of notations:
| Q
| =
| X x Y
| o
| o o
| o o
| o o
| o o
| o o
| o q o
| x7 o / \ o y7
| x6 o / * y6
| x5 o / o y5
| x4 * o y4
| x3 o o y3
| x2 o y2
| X x1 y1 Y
|
| This picture indicates that Q = X x Y,
| that X = {x1, x2, x3, x4, x5, x6, x7},
| that Y = {y1, y2, y3, y4, y5, y6, y7},
| that q = <x3, y5>, or, in other words,
| that Proj<1>(q) = x3, Proj<2>(q) = y5.
JA: At this juncture, I would like to demonstrate a few amusing
diversions that can be found in this genre of illustrations.
JA: We have just illustrated the circumstance -- coincidence? I think not! --
that a deux-tuple is determined by the details of its projective pieces --
still, according to a sage bit of wisdom, if a deux-tuple did not exist
that was determined by the data of its projective parts, then it should
be necessary to invent one that was -- which is parably what this bible
of pan-categorics tells us about the genesis of every cartesian product:
JA: So what if we add another character, yclept "p", let us say,
to this little drama of relations that we have set out here?
| Q
| =
| X x Y
| o
| o o
| o o
| o o
| o o
| o o
| o p q o
| x7 o / \ / \ o y7
| x6 * . * y6
| x5 o / \ o y5
| x4 * * y4
| x3 o o y3
| x2 o y2
| X x1 y1 Y
|
| This picture indicates that Q = X x Y,
| that X = {x1, x2, x3, x4, x5, x6, x7},
| that Y = {y1, y2, y3, y4, y5, y6, y7},
| that p = <x5, y3>, or, in other words,
| that Proj<1>(p) = x5, Proj<2>(p) = y3,
| that q = <x3, y5>, or, in other words,
| that Proj<1>(q) = x3, Proj<2>(q) = y5.
We are now looking at a non-trivial 2-adic relation,
that is, one that has more than one 2-tuple to its name.
Speaking of which, it has none yet, so let's give it the
moniker "M<35>", mnemonically, say, for "Move 3 <-> 5".
Now, one of the interesting things that we can do with
the "coordinate projections" Proj<1> and Proj<2> is to
extend their definitions from single points or tuples
to entire relations, comprising many points or tuples.
Definition. For the 2-adic relation L c X x Y, and Proj<j>, j = 1, 2,
the "projection of L on the j^th axis (or j^th domain)" is defined as:
Proj<j>(L) = {Proj<j>(e) : e in L}.
Thus, in the present example:
1. Proj<1>(M<35>) = {Proj<1>(p), Proj<1>(q)} = {x3, x5} c X.
2. Proj<2>(M<35>) = {Proj<2>(p), Proj<2>(q)} = {y3, y5} c Y.
Well, that was such fun that I'm tempted to try another one.
Let us then contemplate another non-trivial 2-adic relation,
oh, I don't know, just pulling a random one out of thin air:
| Q
| =
| X x Y
| o
| o o
| o o
| o o
| o r o
| o / \ o
| o / \ o
| x7 o / \ o y7
| x6 * s * y6
| x5 o / \ o y5
| x4 * * y4
| x3 o o y3
| x2 o y2
| X x1 y1 Y
|
| This picture indicates that Q = X x Y,
| that X = {x1, x2, x3, x4, x5, x6, x7},
| that Y = {y1, y2, y3, y4, y5, y6, y7},
| that r = <x5, y5>, or, in other words,
| that Proj<1>(r) = x5, Proj<2>(r) = y5,
| that s = <x3, y3>, or, in other words,
| that Proj<1>(s) = x3, Proj<2>(s) = y3.
Let us dub this one as "L<35>", say, on the basis
of a likely graphical analogy, for "Loop 3 and 5".
Computing the projections for this example:
1. Proj<1>(L<35>) = {Proj<1>(r), Proj<1>(s)} = {x3, x5} c X.
2. Proj<2>(L<35>) = {Proj<2>(r), Proj<2>(s)} = {y3, y5} c Y.
Thus this renders it beyond the shadow of a doubt,
if it was not already evident from their mugshots,
that L<35> & M<35> have the very same projections.
So, what have we learned from this state of affairs?
Just that, even though single tuples are determined
by their projection data, that relations themselves,
generally speaking, are not.
I think that this is something to think about.
Jon Awbrey
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The rest is epilogue.
MW: But to be honest, I don't think what you are really arguing
for is triadicity in relations as such, but for some things
that relations are used to represent. If this were the case,
then I would probably agree with you. So let me make a few
statement about how I see relations.
MW: To me relations are purely mathematical constructs.
They are sets of tuples. So first of all I see an
infinity of tuples consisting of all the possible
combinations of things, and then I see a further
infinity of sets of these (but let us restrict
ourselves to those where all the members of the
set have the same adacity).
MW: The good news is that we don't have to care about all of them,
but to a very small subset we choose (note the word choose)
to map our experiences of the world to in terms of connections
(broad sense) that we see between things.
MW: As I see it there is no "necessary" mapping between the world
and relations (i.e. there is only one choice how to do it).
I'm not sure you share that view.
MW: I would suggest that we talked more about what was being represented
rather than how it was being represented. I suspect that we would
then find more to agree about.
JA: I am afraid that we can only talk about things in so far as we represent them.
JA: As an object example, taken for the moment in the converse direction
of synthesis, joining an x and a y to form the set {x, y} is already
an element of a 3-adic relation, just as every other "binary operation"
is a 3-adic relation.
MW: Sorry, I don't agree. We've been here before. Binary operations
are of the form F : a -> b, and even though a and b might be quite
complex, the essential form is binary.
JA: I would say that the accidental form, even the superficial form is binary --
anything at all can be glossed over with a superficially simple appearance --
it is our job precisely to detect and to reveal the underlying complexities.
JA: Here, let me try to explain it to you this way.
The following table presents the 3-adic relation
that is otherwise known as the 2-ary operation of
conjunction, commonly symbolized by "and" or "&":
o-----o-----o---------o
| x | y | z = x&y |
o-----o-----o---------o
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
o-----o-----o---------o
JA: Now, let us suppose that you are a humble Data Scrivener,
as I once was, and your not-so-humble Boss comes to your
cubicle one day, no doubt pursuant to that data downsize
directive that I encouraged you to explore in a previous
scenario, and tells you that you need to get this simple
structure into a 2-column table at most, which being the
true believer in economy that you are you promptly do as
follows:
o-----------o---------o
|< x , y >| z = x&y |
o-----------o---------o
|< 0 , 0 >| 0 |
|< 0 , 1 >| 0 |
|< 1 , 0 >| 0 |
|< 1 , 1 >| 1 |
o-----------o---------o
JA: Whereupon you are promptly rewarded with a well-deserved
promotion at twice your former salary for the incredible
shrinking cost you have thereby beneficed your unit with!
JA: In yer dreams ...
MW: Personally I think we are discussing the wrong things.
We are talking about representation of things, rather
than what things are.
JA: The day that we ascend to this direct, unmediated knowledge of being,
is not a day that I think any of our sort will see this side of heaven.
The only way that we "fallible and mortal finite information creatures"
have to approach the One and the Universal is via the Many Appearances,
by taking up as many different styles and types of representations as
we can fathom, and then, by way of transformations on them, to distill
what invariants, half concealing and half disclosing, they may manifest.
MW: If we set the representation by relations of particular adacity
aside, I think we would find much more to agree about.
JA: Well, that is a heavenly notion.
JA: The only residual dross of the matter appears to turn on the
circumstance that some people cannot yet distinguish a relation
from an element of one.
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MW: OK well let me first try to state a method,
you will have to forgive some lack of formality,
but it should not be so unclear what I mean.
MW: Where I come from this method would be
informally known as reification.
JA: Gasp!
I do not know why I was so astonished by this usage
that I could not even comment on it for several days.
I have certainly seen enough "cargo cultism" here and
there in my time -- and web culture seems to be utterly
drenched in it -- to know that new rites for old spells
are constanly burbling up all the time, and it appears
that the furthest thought from the minds of the sort
of flock who are busily compillaging our brave new
wörterbücher would be to cast their whatevery eyes
upon the decks of those rapedly obsolescing brands.
And so it should not shock me too much that old words
are getting new twists, often to the point of torture,
day by day -- and who knows, it may be that I, too, have
inadvertently been responsible for a turn or two, myself,
but then I remind myself that the word that has been turned,
by degrees, to the point where it now bears a sense 180 about
from the sense that it onetime had, well, if things keep going
on this way, then one day that word may come to be revolved 360,
and thus have its leaning restored to its once and future meaning.
At any rate, when I was I child, we called this "codification",
but now that I am no longer a child I know it as "sublimation".
I am collating the various versions of your methodology below,
and I am also altering the notation slightly to avoid clashes.
MW: Purpose:
V1: To take any n-adic relation occurrence
and turn it into n binary relations
without loss of meaning.
V2: To take what is represented by an n-ary relation,
and represent it by n binary relations and
a set/class of objects.
I am taking the sense of your later explanations to be
that "n-adic relation occurrence" is just an "n-tuple",
otherwise known as an "n-adic relation instance". (?)
Of course, it would only be interesting if an entire n-adic relation,
not just a single n-tuple, was being converted into 2-adic relations,
right?
MW: Method:
V1: Create an object (say e) that stands for
the n-adic relation [instance] as a whole.
V1: e replaces the tuple rather than a relation.
So e replaces <element1, element2, ...>.
V1: This does not mean that e = <element1, element2, ...>
V1: there is a relationship, but it is one of
derivability rather than equality.
V1: Take each each element in the relation occurrence
V1: The relation occurrence is a tuple,
an element is the thing in a place in a tuple.
So <element1, element2, ...>
V1: and create a relation between it and x.
V2: For each tuple
<component1, component2, ...>
that is a member of the n-ary relation (L) create exactly
one object (e) that is a member of the set of objects (E).
JA: Just for the sake of concreteness as I attempt to follow along,
let me try this method out on one or two of the above examples.
Relation L(&) = {<x, y, z> in B^3 : x & y = z}.
o-----o-----o---------o
| x | y | z = x&y |
o-----o-----o---------o
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
o-----o-----o---------o
JA: So, for each tuple <x, y, z> in L(&)
create exactly one object e in set E.
E = {e<0>, e<1>, e<2>, e<3}.
JA: Why don't we get really clever and
use a binary index for the e's?
E = {e<00>, e<01>, e<10>, e<11>}.
MW: For each component of the tuple create a binary tuple
that is a member of a relation that indicates the place
in the original tuple that the component stands in,
e.g.
place1 <e, component1>
JA: I will revert to my custom here and employ the
standard-use-onyms of "coordinate projections".
Proj<1>(e) = x.
Proj<2>(e) = y.
Proj<3>(e) = z.
MW: e.g. if I have:
(gives1 John Mary Book1)
MW: If I apply the above method I might get:
(gives2 e)
(giver-in John e)
(receiver-in Mary e)
(given-in Book1 e)
JA: To be honest, I worry a little about this rendition,
but I shall render under unto KIFer what is KIFer's.
As it stands, I do not know what it's trying to say.
Is there supposed to be some sort of equivalence?
I do not see that "(gives2 e)" is at all simpler.
I have trouble understanding how the materials that
you have assembled here are supposed to do anything
at all but restate an application of the relation
in a different font, as it were.
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MW: Use the role that the element plays
in the original relation to name
this binary relation.
| Example:
|
| (<=> (gives John Mary Book1)
| (and (gives x)
| (giver John)
| (receiver Mary)
| (given book1)
| )
| )
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