SUO: RE: Detached Ideas On Virally Important Topics
Dear Jon,
I have read what follows, and understood (I think) about 2/3.
My problem is that you are taking a way of turning triadic relations
into dyadic relations that I would not take, showing that it is
not sensible (I agree) and then concluding that because this way
doesn't work, you can't do it at all. Where as in reality you have
only shown that this way doesn't work.
I feel rather as I have done on some previous occassions that I have
been gagged and am only allowed to utter the words that you put there
because that is all I am allowed to have meant.
Let me take soemthing from below as an example.
If we have "John gave the flue to Mary" you argue that this
is an irreducibly triadic relation of the form:
Gives <John, Flue, Mary>.
I on the other hand put my analysts hat on and say:
There is anobject missing. There is an activity going on which I
can recognise as another object, say g, which is a member of the
Gives class (if I were being more precise I would describe this
as a physical transfer, but gives will do here).
There are 3 objects that are each involved in this activity with
different roles to play:
Giver <g, John>
Given <g, Flue>
Receiver <g, Mary>
Now from this (for those who want things triadic) I can easily
generate the original relation. On the other hand, on the second
occasion that John gives the flue to Mary, you might have difficulty
in distinguishing between the two occassions. This rather leads me to
think that my analysis provides an improved result.
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
<snip>
>
> 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.
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~EPISODE~3~¤~~~~~~~~~¤~~~~~~~~~¤
>
> 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.
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~EPISODE~4~¤~~~~~~~~~¤~~~~~~~~~¤
>
> 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
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
Regards
Matthew
===============================================================
Matthew West http://www.matthew-west.org.uk/
Principal Consultant Shell Visiting Professor
Operations & Asset Management The Keyworth Institute
Shell Services International The University of Leeds
http://www.shellservices.com/ http://www.keyworth.leeds.ac.uk/
H3229, Shell Centre, London, SE1 7NA, UK.
Tel: +44 207 934 4490 Fax: 7929 Mobile: +44 7796 336538
===============================================================