Fuzzy Sets and Triadic Relations

Re: Lotfi Zadeh
At: http://cs.nyu.edu/pipermail/fom/2012-November/016780.html

Lotfi,

Perhaps I might ask you about a more general issue.

Way back during my first foundational crisis (1967-1972),
I had been willing to consider almost any alternatives to
the usual set theories, so I can remember looking at early
accounts of fuzzy set theory.  There was in addition a link
to certain issues that came up in my studies of C.S. Peirce,
especially the idea that many of the dyadic relations we use
in logic, mathematics, and semantics -- typically functions
that assign meanings and values to symbols and expressions --
are better understood if taken in the context of triadic
relations that serve to complete and generalize them.

My line of thought went a bit like this:

Consider a fuzzy set as a triadic relation of the form x \in^r S
among an element x, a degree of membership r, and a set S.

Ask yourself:  Where do these assigned degrees of membership come from?
Imagine that they come from averaging the results of many judges making
binary {0, 1} = {out, in} decisions.

Now consider the more fundamental triadic relation from which this
data is derived, the relation of the form x in_j S that exists among
an element x, an interpreter (judge, observer, user) j, and a set S.

That formulates fuzzy sets in a way that links up with many Peircean themes.

Regards,

Jon Awbrey

-- 

academia: http://independent.academia.edu/JonAwbrey
my word press blog: http://inquiryintoinquiry.com/
inquiry list: http://stderr.org/pipermail/inquiry/
mwb: http://www.mywikibiz.com/Directory:Jon_Awbrey
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache