Аннотация:We study the generation of rational probabilities by Boolean functions. A probability a is generated by a set H of probabilities if a is the probability of f(x 1, ... , x n )=1 for some Boolean function f provided that for any i the probability of x i =1 belongs to H and all the values of x 1, ... , x n are independent. The closure of the set H is the set of all numbers generated by H. A set of probabilities is called closed if it coincides with its closure. We give an explicit characterization of closures for all sets of rational probabilities. Using this result, we describe all closed and all finitely generated closed sets of rational probabilities. Moreover, we determine the structure of the lattice formed of these sets.