Helmut, List,
I think it would be a good idea to continue reviewing basic concepts
and get better acquainted with the relational context that is needed
to ground all the higher order functions, properties, and structures
we might wish to think about. Once we understand what relations are
then we can narrow down to triadic relations and then sign relations
will fall more easily within our grasp.
I've written up intros to these topics many times before, and you
can find my latest editions, if still very much works in progress,
on the InterSciWiki site, though in this case it may be preferable
to take them up in order from special to general:
Sign Relations
http://intersci.ss.uci.edu/wiki/index.php/Sign_relation
Triadic Relations
http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation
Relation Theory
http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
I think most of the material you mentioned on Relational Reducibility,
Compositional and Projective, is summarized in the following article:
Relation Reduction
http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction
Regards,
Jon
On 4/14/2017 8:59 AM, Helmut Raulien wrote:
Jon, List,
Thank you for having found the thread from long ago! I think,
what often is confusing, that we, when we read "relation", think
of a connection merely, like a spoke of the triad. But in the case
of sign relations, I think, "relation" means: Kind of connection. So
"legisign" is a kind of connection of the representamen/sign with itself.
Perhaps this may also be said like: A kind of representamen/sign.
"Symbol" is a kind of connection between the representamen/sign
and the object, usually called "object relation of the sign".
Best,
Helmut
14. April 2017 um 05:48 Uhr
"Jon Awbrey" <jawb...@att.net> wrote:
Helmut, List ...
Given that sign relations are special cases of triadic relations,
we can get significant insight into the structures of both cases
by examining a few simple examples of triadic relations, without
getting distracted by all the extra features that come into play
with sign relations.
When I'm talking about a k-place relation L I will always be
thinking about a set of k-tuples. Each k-tuple has the form:
(x_1, x_2, ..., x_(k-1), x_k),
or, as Peirce often wrote them:
x_1 : x_2 : ... : x_(k-1) : x_k.
Of course, L could be a set of 1 k-tuple,
but that would be counted a trivial case.
That sums up the extensional view of k-place relations.
Using a single letter like “L” to refer to that set of
k-tuples is already the genesis of an intensional view,
since we now think of the elements of L as having some
property in common, even if it's only their membership
in L. When we turn to devising some sort of formalism
for working with relations in general, whether it's an
algebra, logical calculus, or graph-theoretic notation,
it's in the nature of the task to “unify the manifold”,
to represent a many as a one, to express a set of many
tuples by means of a single sign. That can be a great
convenience, producing formalisms of significant power,
but failing to discern the many in the one can lead to
no end of confusion.
It's late ...
more later ...
Jon
On 4/13/2017 8:08 PM, Jon Awbrey wrote:
Helmut, List,
Yes, I think that something in that vicinity might be
what's causing people so much trouble with this topic.
Let me just review a few things ...
One thing I always say at these junctures is that people really ought
to take Peirce's advice and study his logic of relative terms and its
relation to what most math and computer sci folks these days would call
the mathematical theory of relations. Personally I find his 1870 Logic
of Relatives very instructive, partly because he gives such concrete and
simple examples of every abstract abstrusity and (2) because he maintains
a healthy balance between the extensional and intensional views of things,
drawing on both our empiricist and rationalist ways of thinking. Thereby
hangs another problem people often have with understanding Peirce's logic
and semiotics. We have what might be called diverse “cognitive styles” or
“intellectual inclinations” that range or swing between the above two poles.
I doubt if there's anything like pure types in the human arena, but thinkers
do tend to lean in one direction or the other, at least, at any given moment.
As a rule, though, we are almost always operating at two different levels of
abstraction, whether we are aware of it or not, and our task is to get better
at doing that, through increased awareness of how thought works. There is the
level of intension, or rational concepts, and there is the level of extension,
or empirical cases.
Well, the striking of the grandfather clock tells me
it's time for Big Bang Theory, so I'll have to break ...
Regards,
Jon
On 4/13/2017 3:45 PM, Helmut Raulien wrote:
Jon [A. Schmidt], List,
You wrote:
“To be honest, given that the Sign relation
is genuinely /triadic/, I have never fully
understood why Peirce initially classified
Signs on the basis of one correlate and two
/dyadic/ relations. Perhaps others on the
List can shed some light on that.”
I have a guess about that: I remember from a thread
with Jon Awbrey about relation reduction something
like the following:
A triadic relation is called irreducible, because
it cannot compositionally be reduced to three dyadic
relations. Compositional reduction is the real kind
of reduction. But there is another kind of reduction,
called projective (or projectional?) reduction, which
is a kind of consolation prize for people, who want to
reduce. It is possible for some triadic relations.
Now a triadic relation, say, (S,O,I) might be
reduced projectionally to (S,O), (O,I), (I,S).
My guess is now, that Peirce uses another kind
of projectional reduction: (S,S), (S,O), (S,I).
It is only a guess, because I am not a mathematician.
But at least I would say, that mathematically a relation
with itself is possible, so the representamen relation
can be called relation too, instead of correlate.
Best,
Helmut
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