Re: Fuzzy Sets and Triadic Relations

JA = Jon Awbrey
SB = Stefan Berwing

JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/9075
cf: http://inquiryintoinquiry.com/2012/11/27/fuzzy-sets-and-triadic-relations/
SB: http://permalink.gmane.org/gmane.science.philosophy.peirce/9077

JA: 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.

JA: 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.

JA: 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.

SB: This sounds bayesian to me, because being in a set or being not in a set
     depends on the on the "belief" of the judges.  Take for example a country
     that is neither a democracy nor a authoritarian regime.  It belongs to both
     classes because it has features of both a democracy and a authoritarian regime
     and not because judges believe it to be a democracy or a authoritarian regime.

SB: I think this should be taken into account from the frequentist (peircean) point of view.

Stefan & All,

Yes, that is probably one way to think about it.

Those two triadic relations:

• x \in^r S = “x is in S to the degree r”

• x \in_j S = “x is in S to the judge j”

along with their intensional counterparts:

• x \is^r P = “x is P to the degree r”

• x \is_j P = “x is P to the judge j”

are all defined abstractly enough that you could probably
fit them out with almost any reasonable statistical model.

One adds concreteness by specifying how the triadic relations
combine under operations analogous to the usual set-theoretic
and logical operations.  It's been a while since I studied it,
but I think most of the fuzzy literature works with rules for
combining degrees that cannot be interpreted as probabilities
but as what they like to call “possibilities”.  But I'm a bit
fuzzy on that ...

How one combines interpreters is a really good question.
This may be another one of those places where we have to
turn Peirce's trick of replacing them with interpretants.

Regards,

Jon

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