# Re: The MEANING of fuzzy AND and OR.

Scott Ferson (scott@ramas.com)
Sun, 28 Jun 1998 23:03:14 +0200 (MET DST)

The logical operations have been the focus of considerable
discussion over the history of fuzzy logic. Originally, the
min and max functions were proposed to model logical
conjunction and disjunction. These functions obviously
generalize the traditional Boolean operators, but it was
immediately recognized that they are not the only possible
functions that do. In the last three decades, many different
operators (called t-norms and t-conorms) have been
suggested for fuzzy logic, although it could be argued that
each is arbitrary in its own way. Klir and Yuan's book
"Fuzzy Sets and Fuzzy Logic" contains a discussion of the
variety of suggestions that have been made. Today, it's
proper to call any of many theories that use a t-norm
and its dual t-conorm a "fuzzy logic". Therefore, it's not
true anymore that "fuzzy logic uses only min".

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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