By Geological Survey (U.S.); United States. National Aeronautics and Space Administration

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**Example text**

Note that while llB(A) is symmetrical in both arguments since it expresses to what extent A II B (defined pointwisely as min(A(·) ,B(·» ) is not empty, NB(A) is not so, since it expresses a degree of inclusion of B into A. The set-functions llB and NB are still max and min-decomposable respectively on fuzzy events. However, max(llB(A), I - NB(A) is no longer equal to 1 when A is fuzzy, since we may have llB(A) < I and llB(AC) < 1. Fortunately, llB(A) ~ NB(A) is always satisfied for A and B normalized.

Some authors have considered interval-valued operations. This comes from noticing that normal conjunctive forms and normal disjunctive forms do not coincide in multiple-valued logic. Turksen (1986) uses these forms to compute I(a, b) as an interval. The problem is to make sure that the interval is sufficiently narrow to be informative and sufficiently wide to encompass many implications. Results by Turksen and Yao (1984), Turksen (1986, 1989) indicate that this is the case to some extent. In the case of implication, the disjunctive (DNF) and conjunctive (CNF) normal forms are as follows CNF(p ~ q) = ""p v q DNF(p ~ q) =(p /\ q) v (""p /\ q) v (""p /\ ""q) FUZZY SETS IN APPROXIMATE AND PLAUSmLE REASONING 31 that lead to the following many-valued counterparts for the implication: an S-implication for CNF(a -t b) DNF(a -t b) =S(T(a, b), T(n(a), b), T(n(a), n(b))).

Dubois and Prade (l984a) have shown that S-implications and R-implications could be merged into a single family, provided that the class of triangular norms is enlarged to non-commutative conjunction operators. For instance, Kleene-Dienes implication I(a, b) = max(l - a, b) can be obtained by residuation from the non-commutative conjunction T*(a b) = {b if a + ~ > 1 , 0 otherWIse. This non-commutative conjunction can be obtained from GOdel's iinplication and conversely as T*(a, b) = n(a ~ neb)) and a ~ b = n(T*(a, neb)) respectively.