a AND b = a * b
a OR b = a + b - a * b
where a AND b are truth values (grades of membership of a and b in a fuzzy set
of things known to be true), 0 <= a <= 1.
If a and b are known a priori to be statistically independent (uncorrelated),
and represent the probabilities that a and b are true, then the probabilities
that (a AND b) and (a OR b) are true are given by these formulae from simple
statistics.
Incidentally, if a and b are known a priori to be positively correlated to the
maximum possible extent, then the Zadeh max-min logic is correct; if a and b
are known a priori to be negatively correlated to the maximum possible extent,
then the Lukasiewicz logic is correct. The Lukasiewicz logic is:
a AND b = max(0,1 - (a+b))
a OR b = min(1, a + b)
In all these cases, NOT a = 1 - a.
Jim Buckley and I have a paper coming out in Fuzzy Sets and Systems which goes
into multivalued logics which obey the laws of excluded middle and
contradiction in considerable detail.
William Siler