In this paper, a fuzzy controller is designed for a mechanical system with fractional damping without a priori knowledge of the system dynamics. Because of the constitutive equation of the damping, equations of motion of the system consist of fractional order terms. In the process of developing the fuzzy controller, the fuzzy rules are selected based on the human brain functions. The controller is first implemented for the case of a single inverted pendulum with destabilizing fractional dampings mounted on a cart, i.e. a two degree of freedom (DOF) system, where the functions of human brain in balancing a stick on a fingertip are used to train the fuzzy rules. Then, by extending the linguistic rules, the controller is applied to a double inverted pendulum with destabilizing fractional dampings mounted on a cart, i.e. a three DOF system. Since the linguistic rules are based on qualitative motion of the pendulums, the controller is capable bringing the system to rest at the unstable equilibrium point despite the fractional destabilizing damping in the system. Finally, the numerical results of the both examples are discussed.