Fréchet derived the tightest possible bounds on probabilities
of logical combinations, irrespective of any assumption about
independence between the underlying events:
conjunction (AND) [ max(0, a+b–1), min(a, b) ],
disjunction (OR) [ max(a, b), min(1, a+b) ],
where a and b represent probabilities. Both the upper
and lower bounds are used by some fuzzy logic. What may
be surprising is that not all of the t-norms and t-conorms that
have been defined in fuzzy logic as models for conjunction
and disjunction are contained within the Fréchet intervals.
For instance, Klir and Yuan mention the t-conorm "drastic
intersection", defined as
a, when b=1
b, when a=1
0, otherwise,
as a possible AND operator. So, you see, your question
might well have been "why does fuzzy logic allow lower than
the lower bound allowed by probability theory?"
Scott Ferson <scott@ramas.com>
Applied Biomathematics, 516-751-4350, fax -3435
> Subject: Re: The MEANING of fuzzy AND and OR.
> Date: Sun, 21 Jun 1998 15:05:51 +0200 (MET DST)
> From: ca314159@bestweb.net
> To: nafips-l@sphinx.Gsu.EDU
<snip>
> degree of overlap, which can be in the range:
>
> max(A+B-1,0) <= (A AND B) <= min(A,B) [1]
>
> if A and B are not disjoint and A + B <=2.
>
> My initial question was why does fuzzy theory only
> use the upper bound of the expression [1] above ?
> Why does fuzzy theory ignore the lower bound
> max(A+B-1,0) which is significant when the membership
> grades are both > .5 which can happen in non-disjoint sets ?
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