C (A \cap B) will lie in the range: max(A+B-1,0)<= C <= min(A(x),B(x))
What is the fuzzy theory rationale for A ZAND B = min(A(x),B(x)) ?
> >
>
> It is probably improper to use the min-max operator here. If we are using
> probabilities, then (if we want to preserve the laws of excluded middle and
> contradiction)the proper operator to use depends on the prior association (or
> correlation) between A and B. In your case, the probability of failing question
> A and of failing question B are almost certainly somewhat imperfectly
> correlated.
I tried to choose a very bland case, making no assumptions about
priors or correlations. Just two percentages given and no other knowledge.
In this case I chose to use test questions which was misleading, since
one can make some empirically derived subjective constraints based on
that senario.
But in general, given membership percentages for two sets, and no
other information, the min-max would seem fundementally appropriate for
defining this set intersection completely. We have no reason to conclude
that it will be a fixed probability, but must instead assume a
distribution. (this is in a sense why von Neuman said: "there are
no dispersion-free states")
The violation of the law of contradiction, is in fact
quite an interesting aspect of fuzzy theory, because we see something
like this taking place in quantum mechanics. We see that when a particle
can travel between two possible paths, the degree of our knowledge of
which path it took determines the degree to which that particle obeys
classical probability theory.
Let A, and (not A) represent the two possible paths respectively.
Then the degree of knowledge of whether the particle took the path
A is given as a probability 0-1. If we do not know anything about
which path it took, then A=.5 and (not A)=.5 and in fuzzy theory
the overlap (A zand not A) > 0, which is actually OK since we observe
an interference pattern in this instance which cannot be explained
under a strict particle theory without this idea of (orthogonality)
contradiction violation. So for quantum physics at least, the violation
fits the bill nicely. I'm not sure, but I wouldn't be surprised if
"vacuum fluctuations" (the supposed something from nothing violations
that are theoretically allowed to happen) can be described similarly
> Let me review three of the inifinity of possible operators.
<really nice review snipped>
--http://www.bestweb.net/~ca314159/
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