LOR Comment 11.12
JA:http://inquiryintoinquiry.com/2014/05/12/peirces-1870-logic-of-relatives-%E2%80%A2-comment-11-12/
JA:http://web.archive.org/web/20140512190003/http://permalink.gmane.org/gmane.science.philosophy.peirce/12829
SJ:http://web.archive.org/web/20140516220032/http://permalink.gmane.org/gmane.science.philosophy.peirce/12862
JA:http://web.archive.org/web/20140516221002/http://permalink.gmane.org/gmane.science.philosophy.peirce/12864
JLRC:http://web.archive.org/web/20140517172001/http://permalink.gmane.org/gmane.science.philosophy.peirce/12868
SJ:http://web.archive.org/web/20140605215048/http://permalink.gmane.org/gmane.science.philosophy.peirce/13081
JA:http://web.archive.org/web/20140605221000/http://permalink.gmane.org/gmane.science.philosophy.peirce/13083
SJ:http://web.archive.org/web/20140606023002/http://permalink.gmane.org/gmane.science.philosophy.peirce/13084
Sung,
I belong to the school of thought that uses "sign" as the general term,
leaving "represent-amen" to another chapter and verse of subtilization.
That was good enough for Peirce in his best definitions of sign relations,
and so it's good enough for me. Thus, I view a sign relation in extension
as a subset L of a cartesian product O x S x I, where O is the set of objects,
S is the set of signs, and I is the set of interpretant signs being considered
in a given context of discussion. In symbols this is written as L ⊆ O × S × I.
This are course further conditions on L before it qualifies as a sign relation,
but O × S × I is just the "uncarved block" from which L must be sculpted to form
the generals and particulars of its being.
There is a general way of expressing an arbitrary sign relation L ⊆ O × S × I in functional terms,
namely, by using the mathematical device of a "characteristic function" or an "indicator function".
The indicator function f_L for L ⊆ O × S × I is defined as f_L : O × S × I → B = {0, 1} such that
f_L (o, s, i) = 1 for (o, s, i) in L and f_L (o, s, i) = 0 for (o, s, i) not in L.
I think that's about the most one can say at this eagle's eye level of
generality.
Regards,
Jon
Sungchul Ji wrote:
(For an undistorted figure, see the attached.)
Jon,
Thanks for the beautiful 3-D figures depicting the sign relation.
The following thoughts occurred to me:
(1) Can S (sign) in the figure be replaced by R (i.e., representamen), so
that the Peircean triadic sign can be viewed as a point (or a set of
points) in the O-R-I space, namely, S = f(O, R, I), where f is the
function or relation? As it stands in your figure, the algebraic
representation of the sign would be S = f(O, S, I), so that S appears
twice in the equation as both dependent and independent variables. This
awkwardness could be avoided if we can replace the S-axis with the R-axis
and assign the locus of S within the volume of the O-R-I space proper
rather than on its periphery, i.e., on one of the three axes.
(2) The Peircean sign can be alternatively represented as a system of
Borromean rings, O, R, and I, so that removing any one of them will lead
to the reduction of the three-ring system to two-ring systems, i.e., R-O
(denotation), R-I (connotation), or O-I systems (grounding or embodiment
?), the last of which your figure excludes without any explanation.
(3) If we can represent the Peircean sign algebraically as S = f(R, O, I),
we can justify representing it geometrically as a 4-node network as I have
long been advocating (despite strong oppositions from Edwina and John):
R
|
|
|
S <====> S = f(R, O, I)
/ \
/ \
/ \
O I
Figure 1. The geometric and algebraic representations of the Pericean
sign (S). R = representamen, O = object, and I = interpretant.
With all the best.
Sung
__________________________________________________
Sungchul Ji, Ph.D.
Associate Professor of Pharmacology and Toxicology
Department of Pharmacology and Toxicology
Ernest Mario School of Pharmacy
Rutgers University
Piscataway, N.J. 08855
732-445-4701
www.conformon.net
Sung,
The most generic picture we can form by way of visualizing an arbitrary 3-adic
relation in spatial
terms would be as a "body", an arbitrary subset, residing in a 3-dimensional
space. In the case of
a 3-adic sign relation, the 3 axes would be formed by the object, sign, and
interpretant domains and
the sign relation L would be a subset of the cartesian product space O x S x I.
There is a picture of this in a paper that Susan Awbrey and I wrote:
Sign Relation Figure
http://cspeirce.com/menu/library/aboutcsp/awbrey/fig3.gif
Conference Paper
http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm
Published Paper
http://org.sagepub.com/content/8/2/269.abstract
Regards,
Jon
--
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