SUO: Re: Brouillon Projet, Les Yeux Des Argues, La Laine Des Cartes
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Jean-Marc,
This thread touches on many motifs that I was just getting ready
to broacade upon the SUO (suo, suere, suturus sum) swatch, & sew
I take the liberty of cross-stitching through to that contextile.
Jean-Marc Orliaguet wrote:
JM: Isn't the ground of the nature of a "form"
or a relational structure? What else could
it be like?
JA: I am tempted to agree, and I probably would if I could use the
words "form" and "relational structure" in the ways that I am
already used to, but I cannot be sure yet of the way that you
may intend them, so I must hesitate until I know your meaning.
JM: my meaning would be, a collection of points and relations
between these points so that no point is left alone.
| "... the phaneron is made up entirely of qualities of
| feeling as truly as Space is entirely made up of points. ...
| no collection of points ... without the idea of the objects
| being brought together can in itself constitute space."
JM: What is yours?
Form. From Latin "forma" = "beauty".
There's more to say, of course, but
that is all you really need to know.
Relational Structure. Any relation
viewed with an eye to its form, q.v.
Relation. Here I see two cases:
1. Relation in Extension = a set of tuples.
Tuple = finite sequence of elements from
a predesignated set or collection of sets.
If the tuples all have the same cardinality k,
then they are called k-tuples and the relation
is said to have "arity", "adicity", "valence" k.
2. Relation in Intension = a property ("intension")
that is common to all of the elements in a set.
Nota bene: Saying that a property is shared by
all of the elements in a set is different from
saying that the property is a property of a set.
The elements of a relation in intension are known
as "elementary relations". These are the analogues,
in intension, of the tuples in extension.
For the past many years, all against my first inclinations,
I have been working to develop the extensional side of the
theory of sign relations, simply because this area is less
crowded, because far less work has been done on this face
of the mountain, and because this is the side of things
that makes a connection with empirical efforts, say,
in databases, ethology, and qualitative research.
JA: In the 1st category we find the relations of O to O, S to S, I to I.
In the 2nd category we find the relations of O to S, O to I, S to I.
In the 3rd category we find the relations of O, S, I, in 3-foldness.
JM: These would be the degenerate categories of thirdness.
I believe that it is better to build the categories so
that they are hierachized but still be independent of
each other. How do you express the fact that genuine
secondness is independent of genuine thirdness?
I have the feeling that "independent" may be another one of
those words that we use in different ways from one another.
But I may need to repeat that I am not trying to define
the Categories of 1-ness, 2-ness, 3-ness, as I consider
them to be primeval, primitive, undefined terms, and so,
in a peculiar sense, already independent "in terms of"
each other. Here, I am merely seeking to illustrate
how I understand their application to sign-theoretic
subject matter. It may help if I quote Chomsky again:
| In linguistic theory, we face the problem of constructing
| this system of levels in an abstract manner, in such a way
| that a simple grammar will result when this complex of abstract
| structures is given an interpretation in actual linguistic material.
|
| Since higher levels are not literally constructed out of lower ones,
| in this view, we are quite free to construct levels of a high degree
| interdependence, i.e., with heavy conditions of compatibility between
| them, without the fear of circularity that has been so widely stressed
| in recent theoretical work in lingustics. (Chomsky, LSOLT, page 100).
|
| Noam Chomsky, 'The Logical Structure of Linguistic Theory',
| Based on a widely circulated manuscript dated 1955.
| University of Chicago Press, Chicago, IL, 1975.
JA: Let me try to give a more straightforward answer,
as it strikes me on second reading that this way
of responding may appear evasive or even flippant.
JA: It did not occur to me that anyone would take what I said
as a strict definition of anything, since it was intended
more as a way of building relations among constructs that
are either primitive or else already sufficiently defined.
First of all, we already have a good enough definition of
the sign relation -- I personally consider the one in L75
to be the most clear, detailed, explicit, and formalized
of them all -- and this defines all of the roles O, S, I
simultaneously in relation to each other. Moreover, the
definition of the cartesian product, that comes into the
game as soon as we start to develop the theory of signs
along extensional lines, and which is almost inevitable
if we want to use sign relations as models of empirical
activities and natural forms of conduct, already brings
us the utilities of the various dimensional projections.
So my purpose here was more to elucidate or rationalize
the Categories as aspects or facets of 3-adic relations
than it was to define them on any particular foundation.
JM: [Quotes L75:]
| [I define a sign as] something, A, which brings something, B,
| its interpretant sign determined or created by it, into the
| same sort of correspondence with something, C, its object,
| as that in which itself stands to C. [Peirce, NEM 4, L75].
JA: More fully:
| On the Definition of Logic [Version 1]
|
| Logic will here be defined as 'formal semiotic'.
| A definition of a sign will be given which no more
| refers to human thought than does the definition
| of a line as the place which a particle occupies,
| part by part, during a lapse of time. Namely,
| a sign is something, 'A', which brings something,
| 'B', its 'interpretant' sign determined or created
| by it, into the same sort of correspondence with
| something, 'C', its 'object', as that in which it
| itself stands to 'C'. It is from this definition,
| together with a definition of "formal", that I
| deduce mathematically the principles of logic.
| I also make a historical review of all the
| definitions and conceptions of logic, and show,
| not merely that my definition is no novelty, but
| that my non-psychological conception of logic has
| 'virtually' been quite generally held, though not
| generally recognized. (CSP, NEM 4, 20-21).
|
| On the Definition of Logic [Version 2]
|
| Logic is 'formal semiotic'. A sign is something,
| 'A', which brings something, 'B', its 'interpretant'
| sign, determined or created by it, into the same
| sort of correspondence (or a lower implied sort)
| with something, 'C', its 'object', as that in
| which itself stands to 'C'. This definition no
| more involves any reference to human thought than
| does the definition of a line as the place within
| which a particle lies during a lapse of time.
| It is from this definition that I deduce the
| principles of logic by mathematical reasoning,
| and by mathematical reasoning that, I aver, will
| support criticism of Weierstrassian severity, and
| that is perfectly evident. The word "formal" in
| the definition is also defined. (CSP, NEM 4, 54).
|
| Charles Sanders Peirce,
|'The New Elements of Mathematics', Volume 4,
| Edited by Carolyn Eisele, Mouton, The Hague, 1976.
|
| Available at the Arisbe website:
|
| http://www.door.net/arisbe/menu/library/bycsp/L75/L75.htm
JM: The problem is that the definition only says
that the sign determines the interpretant.
It says nothing about the relation between
the object and the sign, i.e., that
the object determines the sign.
Are you under the impression that objects determine signs?
I will have to think about that. As you know, the proper
reading of the definition, if ever we arrive at it, will
depend on using the author's meanings for "correspondence"
and for "determination", which CSP gives in full, and at
length, needless to say, in many other prominent places.
But I still read this definition as defining a relation
among three roles of players or domains of components,
and so defining all of them in relation to each other.
JM: Now you say that the sign relation is a cartesian product <O,S,I>?
No, I say that a sign relation L is a subset of a cartesian product OxSxI.
At least, that is what I say on extensional days, which is most days of late.
JM: so you have three sets: O, S, and I and the cartesian product
is O x S x I = {(o, s, i) | o is in O, s is in S, i is in I}, i.e.
all possible combinations of elements from each set, corresponding
to "points in space" with coordinates (o,s,i) or ordered triplets,
which you project on lines, planes --?
Yes, that is a good description of the full product space OxSxI.
A sign relation L, then, is a subset L c OxSxI.
JM: But I believe that it is only begging the question:
what are S, O, and I? what are they sets of?
and why should it matter at all?
I do not understand. It is a form of description, no more.
It is not meant to tell you why you should care about this
or that sign relation. That is a matter for you to choose.
JM: Why not simply say as Peirce that when you have a triplet
you have three pairs, and when you have a pair you have
two units, no matter what the triplet is made of?
Why does the relation have to be a sign relation?
and how do you translate into the cartesian product
that idea that O determines S, S determines I,
and O determines I?
Again, this is just a form of description. As it happens,
and this is a very common tactic in mathematical practice,
it is very useful to begin by weakening it, and simply to
incorporate all subsets of such a space under a "nominal"
title of sign relations, only coming back at the second
or third pass to note that some of them qualify only in
a "trivial" way. The properties that they have are the
properties that they have. It is our job but to notice,
to describe, and to articulate them, species by species,
genus by genus, an so on. It is all very straightforward,
well, in principle, at least.
Jon Awbrey
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