This paper presents a new method for the probabilistic logic satisfiability problem, based on the variable neighborhood search metaheuristic. The solution space consists of 0-1 variables, while the associated probabilities are found by our fast approximate variable neighborhood descent procedure combined with the Nelder-Mead nonlinear optimization method. Computational experience shows that, with our approach, problem instances with up to 200 propositional letters can be solved successfully. They are, to the best of our knowledge, the largest solved instances of the PSAT problem appeared in the literature.