# [FOM] Logical Correctness -- Fuzzyness & Triadic Relations

Jon Awbrey jawbrey at att.net
Fri Nov 23 00:24:18 EST 2012

Lotfi,

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
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
http://inquiryintoinquiry.com