SUO: Re: Anxiety Of Influence & Forms Of Communicability
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Jon Awbrey wrote:
>
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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.
>
> 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.
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John S (sans flu I trust), Matthew W, & All,
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:
John J exhales three viral particles numbered 1, 3, 5.
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.
That should be enough for today -- if not this week! --
but I pushed it just a bit too late for detail work,
so I will have to check it over tomorrow for errors.
Best Wishes,
Jon Awbrey
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>
> > 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 invove 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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