[FOM] Logical Correctness -- Fuzzyness & Triadic Relations

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


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.


Jon Awbrey

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