## Triadic Relations, Intentions, Fuzzy Subsets : 1

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 dyadic relations we use in logic, mathematics, and semantics — as a rule being 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 ∈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 ∈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.

### 1 Response to Triadic Relations, Intentions, Fuzzy Subsets : 1

1. George Berger says:

Excellent. I was perplexed when I first read about fuzzy sets, several decades ago. I figured they covered up some more basic notions that could either replace fuzzy sets and reasoning with complex uses of ordinary set theory, or e.g. by combining some ordinary set theory with theorems resulting from Kolmogoroff’s axioms (say, measure theory). I guess something along these lines is possible, but why not give your approach a chance? Sellars uses a vaguely related technique in his essays on “Induction as Vindication,” that is, an explicit reference to persons and their judgemental acts.

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