SUO: Detached Ideas On Virally Important Topics
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Sufferers of Influenza,
Because of my need in this discussion to be constantly referring
to things that were written at an earlier stage, not to mention
the need to cut-and-paste complex arrays of data and diagrams,
I will continue to keep the whole case history in one file.
Matthew has been raising the issue of measurement,
and there are a few themes here that bear on that
question, and so I will make the attempt to begin
addressing measurement in the present installment.
Measurement, of course, is a very important topic,
and the need to encompass what all it involves is
at the very hub or nub of the efforts that I have
been making over the last ten years or so, rather
more deliberately than I ever have before, trying
to arrange better coordinations among the methods
and the resources, as supported by the conceptual
and the computational frameworks, of course, that
we currently enjoy with regard to two broad areas
of data analysis and knowledge systems, typically:
1. Qualitative: Categorical, Linguistic, Logical.
2. Quantitative: Numerical, Rational, Statistical.
[ Here, of course, I mean "rational" in the sense
| of rational number or real number types of data. ]
And on this very score I need to begin paying off some IOU's
with regard to some assertions that I made a while back about
how these classification schemes, "ontologies" and "taxonomies",
as diverse people differentially prefer to classify their arrays,
and after which we are so hot in pursuit, are not too absurdly
analogous to the coordinate systems, the "reference frames",
as they are often referred to, that serve to coordinate the
measurements of particular observers abroad in the cosmos.
But I must confess, I really did believe at the time that I made them
that my statements about "frames of reference" (FOR's) and the brands
of "trans-FOR-mations" that it would be necessary to consider between
them were "yet another trivial truism expressed redundantly" (YATTER)
of the sort for which I am commonly acknowledged to have such a knack,
and mayhap there is still a silent majority of folks who still regard
them as no more, no less, but the off-list remarks that I have gotten
since that time have clued me in to the suspicion that this is not as
trite as I had hoped to style it, but rather novel, or wrong, or both.
Now, all of this is the very topic that I have tried to raise
by means of my several other threads, the "manifold" lines of
inquiry, just to mention one half-ravelled-half-tangled skein,
and so at some point I may cast off the rest of this knitwork
to one of those other more suited textures of text or context.
The story of "A Case of Influenza"
continues at "Episode 5".
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John F. Sowa wrote:
>
> Jay Halcomb sent me an interesting counterexample
> to my claim that giving involves intentionality:
>
> John gave Mary the flu.
>
> The verb "give" may be used in a triadic form when
> there is no intention involved. But in such a case,
> it is possible to break the triad into dyads:
>
> John sneezed out the flu virus.
> Mary caught the virus.
>
> Peirce called examples of this kind "degenerate triads"
> because they have the appearance of taking 3 arguments,
> but they can be decomposed into two independent sentences
> that only involve two participants at a time.
>
> Following is another example, which uses the verb "throw",
> which frequently (but not always) has an intended goal:
>
> The quarterback threw the ball to the wide receiver.
>
> According to the rules of football, the quarterback would
> normally intend to throw the ball to a player on his own
> team. If a defensive player caught the ball, that would
> thwart the intention. Therefore, the triad can be broken
> down into two sentences, each of which has two participants:
>
> The quarterback threw the ball.
> One of the defenders intercepted it.
>
> If the quarterback is particularly inept, a sports commentator
> might make a statement like the following with a tone of sarcasm:
>
> The quarterback threw the ball to the defenders.
>
> This statement suggests that the quarterback
> deliberately made a losing play.
>
> Bottom line: Intentionality requires three arguments, and
> if a sentence with verbs like "giving" and "throwing" can
> be split in two parts, each of which involves only two
> of the participants, the intentionality is missing.
>
> John Sowa
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A Case of Influenza
There is this 3-adic transaction among
three relational domains. Let us say:
"Transmitters", "Vectors", "Receivers",
and let us symbolize: C c T x V x R.
In order to prove the proposition, for instance, as in a court of law,
that "John J gave-the-flu-to Mary M", it is necessary but by no means
enough to convince an arbiter that an infectious colony or a virulent
sample of particular micro-organisms of the genus known as "influenza"
was transported from John J (SSN 1 TBN) to Mary M (SSN 2 TBN) on some
well-specified occasion in question.
In other words, the "evidence" for the 2-adic relation that bears
the form and the description F c T x R : "-- gave-the-flu-to --",
is found solely within the 3-adic relation of "communication" C.
Let us assume that this long chain of causal and physical "influences"
can be more conveniently summarized, for our present purposes, in the
form of a 3-adic relation that connects a transmitter t, a "vector" v,
and a receiver r. Thus a bona fide incident or a genuine instance of
the "communication relation" C c TxVxR will be "minimally adequately",
as they say in epidemiology, charted in a datum of the form <t, v, r>.
What is the character of the relationship between
the 3-adic relation of "communication" C c TxVxR
and the 2-adic relation "-- gave-the-flu-to --"?
This particular relation among relations --
you may be about to read my mention, but
will not if I can help it find me to use
the term "meta-relation" for this notion --
is broadly nomenclated as a "projection",
with type here being Proj : TxVxR -> TxR.
Our use of it in this presenting case is
an example of how we transit from caring
about the "detail of the evidence" (DOTE)
to desiring only a brief sum of the fact.
For now, let us stipulate that we have the following
sample of data about the 3-adic relation C c TxVxR :
{..., <John J, Agent A, Mary M>, ...}.
In other words, we are fixing on a single element:
<John J, Agent A, Mary M> in C c TxVxR.
Let us now contemplate the generalization of ordinary functional composition
to 2-adic relations, called, not too surprisingly, "relational composition",
and roughly information-equivalent to Peirce's "relative multiplication".
I will employ the data of our present case to illustrate two different
styles of picture that we can use to help us reason out the operation
of this particular form of relational composition.
First I show one of my favorite genres of pictures for 2-adic relations,
availing itself of the species of graphs known as "bipartite graphs",
or "bigraphs", for short.
Let an instance of the 2-adic relation E c TxV
informally defined by {<t, v> : t exhales v},
be expressed in the form "t exhales v".
Let an instance of the 2-adic relation I c VxR
informally defined by {<v, r> : v infects r},
be expressed in the form "v infects r".
Just for concreteness in the example, let us imagine that:
1. John J exhales three viral particles numbered 1, 3, 5.
2. Mary M inhales three viral particles numbered 3, 5, 7,
each of which infects her with influenza.
The 2-adic relation E that exists in this situation is
imaged by the bigraph on the T and the V columns below.
The 2-adic relation I that exists in this situation is
imaged by the bigraph on the V and the R columns below.
E I
T---->V---->R
o 1 o
/
/
o / 2 o
/
/
J o-----3 o
\ \
\ \
o \ 4 \ o
\ \
\ \
o 5-----o M
/
/
o 6 / o
/
/
o 7 o
Let us now use this picture to illustrate for ourselves,
by way of concrete examples, many of the distinct types
of set-theoretic constructs that would arise in general
when contemplating any similar relational configuration.
First of all, there is in fact a particular 3-adic relation Q
that is determined by the data of these two 2-adic relations.
It cannot be what we are calling the "relational composition"
or the "relative product", of course, since that is defined --
forgive me if I must for this moment be emphatic -- DEFINED
to be yet another 2-adic relation. Just about every writer
that I have read who has discovered this construction has
appeared to come up with a different name for it, and I
have already forgotten the one that I was using last,
so let me just define it and we will name it later:
What we want is easy enough to see in visible form,
as far as the present case goes, if we look at the
composite sketch already given. There the mystery
3-adic relation has exactly the 3-tuples <t, v, r>
that are found on the marked paths of this diagram.
That much insight should provide enough of a hint
to find a duly officious set-theoretic definition:
Q = {<t, v, r> : <t, v> in E and <v, r> in I}.
There is yet another, very convenient, way to define this,
the recipe of which construction proceeds by these stages:
1. For 2-adic relation G c TxV, define GxR,
named the "extension" of G to TxVxR, as:
{<t, v, r> in TxVxR : <t, v> in G}.
2. For 2-adic relation H c VxR, define TxH,
named the "extension" of H to TxVxR, as:
{<t, v, r> in TxVxR : <v, r> in H}.
In effect, these extensions just keep the constraint
of the 2-adic relation "in its places" while letting
the other elements roam freely.
Given the ingredients of these two extensions,
at the elemental level enjoying the two types:
TxV -> TxVxR and VxR -> TxVxR, respectively,
we can define the 3-adic Q as an intersection:
Q(G, H) = GxR |^| TxH
One way to comprehend what this construction means
is to recognize that it is the largest relation on
TxVxR that is congruent with having its projection
on TxV be G and its projection on VxR be H.
Thus, the particular Q in our present example is:
Q(E, I) = ExR |^| TxI
This is the relation on TxVxR, to us, embodying an assumption
about the "evidence" that underlies the case, which restricts
itself to the information given, imposing no extra constraint.
And finally -- though it does amount to something like the "scenic tour",
it will turn out to be useful that we did things in this roundabout way --
we define the relational composition of the 2-adic relations G and H as:
G o H = Proj<T, R> Q(G, H) = Proj<T, R> (GxR |^| TxH)
[ Reference:
|
| Although it no doubt goes way back, I am used to thinking
| of this formula as "Tarski's Trick", because I first took
| notice of it in a book by Ulam, who made this attribution.
|
| Ulam & Bednarek,
| "On the Theory of Relational Structures
| and Schemata for Parallel Computation",
| Original report dated May 1977, in:
| Ulam, 'Analogies Between Analogies',
| University of California Press, Berkely, CA, 1990.
]
Applying this general formula to our immediate situation:
E o I = Proj<T, R> Q(E, I) = Proj<T, R> (ExR |^| TxI)
We arrive at this picture of the composition E o I c TxR:
EoI
T---->R
o o
o o
J o o
\
\
o \ o
\
\
o o M
o o
o o
In summation, E o I = {<John J, Mary M>}.
By the way, you may have noticed that I am using here
what strikes me as a more natural order for composing
2-adic relations, but the opposite of what is usually
employed for functions. In the present ordering, one
can read the appearances of the relational domains in
what seems like a much more straightforward way, just
as they are invoked by the series of relation symbols.
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What has gone so far adequately covers the case of
composing 2-adic relations, at least, in the usual,
so-called "generic" or "non-degenerate" fashion of
Peirce's times and ours. One could easily develop
this limited subject matter along a line analogous
to the mathematical "theory of categories" (TOC) --
and a few people have, so far as I know, in rather
notable detail by Peter Freyd and Andre Scedrov in
their book 'Categories, Allegories', North-Holland,
Amsterdam, 1990. One can further develop the area,
as was done by Peirce, his students, Ladd-Franklin
and Mitchell, especially, along with Schroeder and
a host of others, to discover a genuine cornucopia
of different sorts of operations and products, the
staples of the drygoods store and sundries counter
of Nineteenth Century Logic, that, like the Theory
of Invariants or Substitutional Analysis, you just
cannot get at a supermarket anymore. Yes, I know --
but we now have FAST, and Google, and Vivisimo!
So let us push on, and do not pass "go".
Of this late lamented lamia of the "triadic irreducible relation" (TIR),
let me just say on this occasion, one more time, and I hope still more
clearly than I have ever done before, that it is ONLY with respect to
the form of operation that Peirce described as the "relative product",
and that we today more or less associate with "relational composition",
that he ever expended any large degree of logical clamor, or exhausted
any significant quantity of breath to explore, or otherwise lit up or
into any other brand of hue and cry about. And when it comes to the
substance of this claim about 3-adic relations under relative product,
the facts are just about as controversial as our post*modern anxieties
about the number of primes less than or equal to 3. All 3-adics are
irreducible to 2-adics, at least, in this sense, since the collection
of 2-adic relations is "closed" under this operation of composition --
the operation is defined precisely in order to enjoy the fruits of
this, analytically speaking, hugely beneficial "closure" property.
Furthermore, it is only with regard to relational composition that
the analogy or the morphism that Peirce proposed between the pair
<Relations, Arities> and the pair <Nodes, Degrees> makes any sense.
And so, if there is something that we are worrying about under the
heading of these TIR's, then that "residue of worry" (ROW) really
belongs under another heading, and we would do well to seek it
there instead, whether it be ill-founded, well-founded, or
wholly groundless. I hope that relieves these TIR's.
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Let us declare a "logical basis" -- and leave it
as an exercise for the reader to improvise a fit
definition of what is, and what ought to be that! --
but anyway a collection of elements of this form:
Basic Entia K = T |_| V |_| R
Transmissia T = {t1, t2, t3, t4, t5, t6, t7}
Viral Entia V = {v1, v2, v3, v4, v5, v6, v7}
Receptacula R = {r1, r2, r3, r4, r5, r6, r7}
Just by way of orientation to the way that we speak "way out here",
t3 = John.
r5 = Mary.
So far, so good, but here we have come to one of those junctures
where personal tastes are noticed to be notoriously divergent in
matters of notation, and so at this point I will simply describe
a few of the most popular options:
1. One may lump all of these elements together and work
with a cubic array that has dimensions 21 x 21 x 21,
taking its projections into square matrices 21 x 21.
2. One may consider the very like possibilty, here only,
that the T's and the R's are abstractly the same set,
and reduce the representation in a corresponding way.
3. One may treat the relational domains T, V, R as three
distinct sets, start with a 3-adic relation Q c TxVxR
represented as a cubic array of dimensions 7 x 7 x 7,
taking its projections into square matrices of 7 x 7.
Option 3 seems easier for us here,
just as a way of conserving space.
The extensions of the 2-adic relations E and I
are the following collections of ordered pairs:
E = {<t3, v1>, <t3, v3>, <t3, v5>}
I = {<v3, r5>, <v5, r5>, <v7, r5>}
Peirce represented 2-adic relations in this form:
E = t3:v1 + t3:v3 + t3:v5
I = v3:r5 + v5:r5 + v7:r5
It is very often convenient, though by no means obligatory,
to arrange these quasi-algebraic terms in forms like these:
T x V =
[
| t1:v1, t1:v2, t1:v3, t1:v4, t1:v5, t1:v6, t1:v7,
| t2:v1, t2:v2, t2:v3, t2:v4, t2:v5, t2:v6, t2:v7,
| t3:v1, t3:v2, t3:v3, t3:v4, t3:v5, t3:v6, t3:v7,
| t4:v1, t4:v2, t4:v3, t4:v4, t4:v5, t4:v6, t4:v7,
| t5:v1, t5:v2, t5:v3, t5:v4, t5:v5, t5:v6, t5:v7,
| t6:v1, t6:v2, t6:v3, t6:v4, t6:v5, t6:v6, t6:v7,
| t7:v1, t7:v2, t7:v3, t7:v4, t7:v5, t7:v6, t7:v7,
]
V x R =
[
| v1:r1, v1:r2, v1:r3, v1:r4, v1:r5, v1:r6, v1:r7,
| v2:r1, v2:r2, v2:r3, v2:r4, v2:r5, v2:r6, v2:r7,
| v3:r1, v3:r2, v3:r3, v3:r4, v3:r5, v3:r6, v3:r7,
| v4:r1, v4:r2, v4:r3, v4:r4, v4:r5, v4:r6, v4:r7,
| v5:r1, v5:r2, v5:r3, v5:r4, v5:r5, v5:r6, v5:r7,
| v6:r1, v6:r2, v6:r3, v6:r4, v6:r5, v6:r6, v6:r7,
| v7:r1, v7:r2, v7:r3, v7:r4, v7:r5, v7:r6, v7:r7,
]
Now, taking these generic motifs as scenic -- or, at least, schematic --
backdrops, one can permit the particular characters of one's favorite
2-adic relations to represent themselves and to play out their action
on this stage, by attaching affirming or nullifying "coefficients" to
the appropriate places of the thus-arrayed company of possible actors.
E =
[
| 0 t1:v1, 0 t1:v2, 0 t1:v3, 0 t1:v4, 0 t1:v5, 0 t1:v6, 0 t1:v7,
| 0 t2:v1, 0 t2:v2, 0 t2:v3, 0 t2:v4, 0 t2:v5, 0 t2:v6, 0 t2:v7,
| 1 t3:v1, 0 t3:v2, 1 t3:v3, 0 t3:v4, 1 t3:v5, 0 t3:v6, 0 t3:v7,
| 0 t4:v1, 0 t4:v2, 0 t4:v3, 0 t4:v4, 0 t4:v5, 0 t4:v6, 0 t4:v7,
| 0 t5:v1, 0 t5:v2, 0 t5:v3, 0 t5:v4, 0 t5:v5, 0 t5:v6, 0 t5:v7,
| 0 t6:v1, 0 t6:v2, 0 t6:v3, 0 t6:v4, 0 t6:v5, 0 t6:v6, 0 t6:v7,
| 0 t7:v1, 0 t7:v2, 0 t7:v3, 0 t7:v4, 0 t7:v5, 0 t7:v6, 0 t7:v7,
]
I =
[
| 0 v1:r1, 0 v1:r2, 0 v1:r3, 0 v1:r4, 0 v1:r5, 0 v1:r6, 0 v1:r7,
| 0 v2:r1, 0 v2:r2, 0 v2:r3, 0 v2:r4, 0 v2:r5, 0 v2:r6, 0 v2:r7,
| 0 v3:r1, 0 v3:r2, 0 v3:r3, 0 v3:r4, 1 v3:r5, 0 v3:r6, 0 v3:r7,
| 0 v4:r1, 0 v4:r2, 0 v4:r3, 0 v4:r4, 0 v4:r5, 0 v4:r6, 0 v4:r7,
| 0 v5:r1, 0 v5:r2, 0 v5:r3, 0 v5:r4, 1 v5:r5, 0 v5:r6, 0 v5:r7,
| 0 v6:r1, 0 v6:r2, 0 v6:r3, 0 v6:r4, 0 v6:r5, 0 v6:r6, 0 v6:r7,
| 0 v7:r1, 0 v7:r2, 0 v7:r3, 0 v7:r4, 1 v7:r5, 0 v7:r6, 0 v7:r7,
]
And then there are times when it is not so convenient!
At any rate, it is then conceivable to push the level
of abstraction in our so-arrayed representations even
one step further, and so long as we keep in mind what
the now-suppressed row-indices and column-indices are
supposed to signify, logically speaking, in the first
place, then we can push them even deeper into the dim
and tacit background of the overriding interpretation.
E =
[
| 0, 0, 0, 0, 0, 0, 0,
| 0, 0, 0, 0, 0, 0, 0,
| 1, 0, 1, 0, 1, 0, 0,
| 0, 0, 0, 0, 0, 0, 0,
| 0, 0, 0, 0, 0, 0, 0,
| 0, 0, 0, 0, 0, 0, 0,
| 0, 0, 0, 0, 0, 0, 0,
]
I =
[
| 0, 0, 0, 0, 0, 0, 0,
| 0, 0, 0, 0, 0, 0, 0,
| 0, 0, 0, 0, 1, 0, 0,
| 0, 0, 0, 0, 0, 0, 0,
| 0, 0, 0, 0, 1, 0, 0,
| 0, 0, 0, 0, 0, 0, 0,
| 0, 0, 0, 0, 1, 0, 0,
]
When all of this is said and done, that is to say,
when all of this is said and done the fitting way,
then one can represent relative multiplication or
relational composition in terms of an appropriate
quasi-algebraic "multiplication" operation on the
rectangular matrices that represent the relations.
The logical operation of the relative product has
to be qualified as "quasi-algebraic" just to help
us keep in mind the fact that it is not precisely
the one that algebraically-minded folks would put
on the same brands of {0, 1}-coefficient matrices.
¤~~~~~~~~~¤~~~~~~~~~¤~EPISODE~5~¤~~~~~~~~~¤~~~~~~~~~¤
There are so many things that I could do at this point,
all of which I ought to do sooner or later, that it is
rather difficult to figure out what is best to do next.
1. I could pick up right where I left off last time,
and define the relative product as it appears in
the matrix formalism.
2. I could develop the generalization of category theory
that covers 2-adic relations, and abstracts from them.
3. I could launch into wide open ocean of k-adic relations
and all the many sights that there are to be seen there.
4. I could address the bearing of the "logic of relatives" (LOR)
on the more timely issues of manifolds and measurements (MAM).
I think that I will choose to compromise, even to temporize,
and say just a little about each of these options, save for
the waves of k-adic relations that I need to save for later.
Let me begin with an incidental remark on the character of measurement
that is fresh on my mind because it came up in this triple interaction
that occurred quite recently among John, Lee, and Matthew:
[John Sowa stated:]
| As Lee [Auspitz] has said, this act of semiosis
| can be represented as a triangle, which relates
| other signs (such as the marks on a thermometer)
| to a sign that serves as a standard unit to a
| number, which serves as a sign of the count
| of units that represent the result.
[Matthew West replied:]
| Oh I thought that a unit of measure was
| a particular mapping between a type of
| characteristic (say temperature) and
| a number space. I don't see how signs
| get into it, except as representations
| of the mapping.
As I personally understand the topic, a measurement
is "classically" -- with that word I mean to ignore
issues of "relativity and quantum mecahnics" (RAQM),
at least, for the moment, still, not to exclude all
chance of taking them up on some future occasion --
represented by a function f : X -> R from the space
of interest, the object cosmos or the source domain,
to what "without loss of generality" (WOLOG) may be
taken to be the target domain of the real numbers R.
So, it may appear to be a matter of a 2-adic relation,
as all functions indeed are, but a person who fancies
this has never had the experience of being stuck with
a "pile of numbers" (PON), duly recorded and archived
away as an "aftermath" of some exorbitantly expensive
experiment, only to find to his inconsolable dismay a
few years or decades or centuries later that some GDI
apprentice or clerk or scribe has omitted to preserve
what the dolt in question regarded as the "irrelevant"
details of the logical labels on the rows and columns
and files of this once precious, now meaningless data.
Time For Lynch!
Jon Awbrey
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