Taking the fuzz out of fuzzy logic

This paper gives an algorithm which, given any boolean function F whose n arguments are known probabilities, will deduce the tightest possible upper and lower bounds (assuming nothing is known about the correlations among the probabilities) on the value of F. This may be accomplished by solving a 2 n-dimensional linear program. (It is not suprising that the linear program has exponential dimensionality, since if subexponential dimensionality were possible, then SAT would be soluble in subexponential time.) If something is known about the correlations this knowledge may also be incorporated into the linear program. With little additional trouble one may also nd the max-entropy estimate of F. \Fuzzy logic" attempts to generalize logic to concern \fuzzy truth values" { real numbers in the interval 0; 1]. We now can have simultaneously: (1) a probabilistic interpretation of fuzzy truth values, and (2) total rigor. This comes at the cost of an exponential amount of computation, but: (1) we gain more understanding of the (polytopal) structure of fuzz-land, and (2) Arguably: only small values of n arise in practice, or else the boolean functions encountered in practice are \modularized" with each \module" having a small complexity, hence this exponential complexity is acceptable. Fuzzy Wuzzy was a bear Fuzzy Wuzzy lost his hair Fuzzy Wuzzy wasn't fuzzy, was he? No, he was a bare bear! { Unattributable.