Re: Algebraic sum

N.G. du Bois (n.dubois@wxs.nl)
Sun, 14 Jun 1998 22:26:56 +0200 (MET DST)

ca314159 wrote:
>
> WSiler wrote:
> >
> > CA13159 wrote:
> >
> > > If A = percentage of students failing question A and,
> > > B = percentage of students failing question B,
> > > then if
> > > C = percentage of students failing both questions A and B,
>
> 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/

I like the discussion and the answers, but I still have questions
I also encountered the term "bounded sum". Is'nt that a better term for
sums like A + B - AB; or A + B + C - AB - AC - BC; etcetera.
And should the term algebraic sum not only be used for sums like A + B;
or A + B + C and so on?

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