Thread:
HP:http://permalink.gmane.org/gmane.science.philosophy.peirce/15155
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15160
HP:http://permalink.gmane.org/gmane.science.philosophy.peirce/15161
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15162
HP:http://permalink.gmane.org/gmane.science.philosophy.peirce/15163
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15164
ET:http://permalink.gmane.org/gmane.science.philosophy.peirce/15166
HP:http://permalink.gmane.org/gmane.science.philosophy.peirce/15172
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15183
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15184
SJ:http://permalink.gmane.org/gmane.science.philosophy.peirce/15185
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15205
RM:http://permalink.gmane.org/gmane.science.philosophy.peirce/15206
SJ:http://permalink.gmane.org/gmane.science.philosophy.peirce/15208
ET:http://permalink.gmane.org/gmane.science.philosophy.peirce/15209
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15211
JBD:http://permalink.gmane.org/gmane.science.philosophy.peirce/15514
Jeff, List,
I think it goes without saying that the extensional and intensional approaches
to concepts and data are both incomplete in themselves, and this is precisely
why Peirce's information-theoretic approach to inquiry, logic, and semiotics
is so revealing of new insights into their complementarity and relativity to
the total information.
The outlines of Peirce's information-theoretic integration of comprehension and
extension are already sketched to an appreciable degree in his Harvard and
Lowell
lectures of 1865-1866. The level of concrete detail and supply of examples in
these
lectures makes them the best place to begin getting a sense of his overall
vision of
information. But they do tend to meander down a long and winding road that
puts some
readers off the trail. So this time around I tried taking up a reading of
selections
from the 1880 "Algebra of Logic", which still preserves traces of an attempt to
handle
extensional aggregates of "individuals" and comprehensive compositions of
"simples" in
a relatively even-handed way.
Regards,
Jon
On 1/27/2015 2:08 PM, Jeffrey Brian Downard wrote:
Hi Jon, Lists,
I've been thinking about the way you are characterizing triadic relations in
terms of ordered triples. For a while
now, I've been wondering if there are limits to such an approach that might
make it difficult to explain what is
special about a genuinely triadic relation. Here are a few quick comments that
I wanted to register--with the hope
that we might, over time, compare Peirce's more algebraic and geometric
approaches for setting up formal systems. In
particular, I'm wondering if there might be limitations that come with the more
algebraic approach when we turn from
formal logic to semiotics and try to put those systems to work for the sake of
philosophical analysis and
explanation.
So, here are two passages I wanted to put on the table:
First, Peirce says the following in "The Logic of Mathematics, An Attempt to Develop
My Categories from Within":
"What is a triad? It is a three, But a three what? If we say it is three
subjects, we take at the outset an
incomplete view of it." (CP, 1.471)
Second, he says the following in his discussion of the improvement of the gamma
graphs: "For although I have always
recognized that a possibility may be real, ... I have invariably recognized, as
one great class of relations, the
class of references, as I have called them, where one correlate is an existent,
and another is a mere possibility;
yet whenever I have undertaken to develop the logic of relations, I have always
left these references out of account,
notwithstanding their manifest importance, simply because the algebras or other
forms of diagrammatization which I
employed did not seem to afford me any means of representing them." (CP, 4.579)
I don't yet have a clear way of stating many of the questions that have been
puzzling me. Having said that, let me
try to get one question out so that we might start a conversation about the
limitations that might be involved in
trying to analyze the phenomena associated with reasoning and then build
philosophical explanations by relying too
heavily on the idea of ordered triples and algebraic expressions of the
relations between such triples.
Here is the starting question: Doesn't the notion of an ordered triple require
that we already have things sorted
out in such a way that we are able to ascribe quantitative values to each
subject that is a correlate of the triadic
relation?
Here are a few comments to put some flesh to the bare bones of what I'm trying
to ask. It seems to me that one
advantage of a more topological approach in setting up logical systems is that
we can iconically represent relations
between things that do not yet have such quantitative values. After all, there
are many kinds of relations (e.g.,
those involved in relations of similarity between feelings) that may not have a
determinate scale for the attribution
of a quantity to it. That is one reason Peirce is so keen to point out in his
discussion of quantity that the
conception of a quanta is not grounded arithmetically. Rather, it is
fundamentally a geometric notion. My hunch is
that Peirce is trying to set things up so that he can explain what is necessary
for establishing such ordered
relations between things serving as signs, objects and interpretants in the
growth of understanding. The second
quote listed above from the essay on the improvement of the gamma graphs seems
to suggest that Peirce saw limitations
to more algebraic ways of trying to lay out the formal relations, and he
couldn't find any way around the problem
until he moved to more geometric (i.e., iconic) approaches in the existential
graphs.
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Jon Awbrey [[email protected]]
Sent: Wednesday, December 17, 2014 9:12 PM
To: Howard Pattee
Cc: [email protected]; 'Peirce-L'
Subject: [PEIRCE-L] Re: Triadic Relations
Howard,
At this point we can can distinguish two forms
of decomposability or reducibility -- along with
their corresponding negations, indecomposability
or irreducibility -- that commonly arise.
1. Reducibility under relational composition P o Q.
All triadic relations are irreducible in this sense.
This is because relational compositions of monadic
and dyadic relations can produce only more monadic
and dyadic relations.
2. Reducibility under projections. For that we need some definitions:
Every triadic relation, say L contained as a subset of the cartesian product
X x Y x Z, determines three dyadic relations, namely, the projections of L on
the "planes" X x Y, X x Y, and Y x Z.
In particular:
Every sign relation, say Q contained as a subset of the cartesian product
O x S x I, those being the sets of objects, signs, and interpretant signs
respectively under discussion, determines three dyadic relations, which we
may notate as follows:
* proj_{OS}(Q), the projection of Q on the O x S plane;
* proj_{OI}(Q), the projection of Q on the O x I plane;
* proj_{SI}(Q), the projection of Q on the S x I plane.
To visualize the situation for sign relations, see the following paper:
☞ http://www.iupui.edu/~arisbe/menu/library/aboutcsp/awbrey/integrat.htm
And contemplate the following figure:
☞ http://www.iupui.edu/~arisbe/menu/library/aboutcsp/awbrey/FIG3.gif
Here is the critical point. The triadic relation always determines
the three dyadic projections but the three dyadic projections may or
may not determine the triadic relation. Thus we have two cases:
1. If the dyadic projections determine the triadic relation, that is,
there is only one triadic relation that has those three projections,
then the triadic relation is said to be "projectively reducible" to
those three dyadic relations.
2. If the dyadic projections do not determine the triadic relation,
that is, there is more than one triadic relation that has those
same three projections, then the triadic relation is said to be
"projectively irreducible".
Regards,
Jon
Howard Pattee wrote:
At 12:12 AM 12/17/2014, Jon Awbrey wrote:
What do I see in a picture like this?
```````s``
``````/```
o---<R````
``````\```
```````i``
The "R" brings to mind a triadic relation R, which collateral
knowledge tells me is a set of 3-tuples. What sort of 3-tuples? The
picture sets a place for them by means the place-names "o", "s", "i",
in no particular order. Without loss of generality I can take them up
in the ordered triple (o, s, i). All of this is just mnemonic
machination meant to say that a typical element is (o, s, i) in R.
It's up to me to remember that R is a subset of O x S x I, with o in
O, s in S, and i in I. The diagram is just a mnemonic catalyst. You
have to know the codebook to decode it.
I can see that with the help of your words. But I also see dyadic
relations, which I agree are not to be confused as corresponding to
"parts of signs." I see no harm in recognizing their formal necessity,
as long as they are not misinterpreted.
Howard
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