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:
"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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-----------------------------
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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
>
--
inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey
isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
facebook page: https://www.facebook.com/JonnyCache
-----------------------------
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