This paper revisits the classical question of the stability of the nonlinear Wonham filter. The novel contributions of this paper are two-fold: (i) definition of the stabilizability for the (control-theoretic) dual to the nonlinear filter; and (ii) the use of this definition to obtain conclusions on the stability of the Wonham filter. Specifically, it is shown that the stabilizability of the dual system is necessary for filter stability and conversely stabilizability implies that the filter asymptotically detects the correct ergodic class. The formulation and the proofs are based upon a recently discovered duality result whereby the nonlinear filtering problem is cast as a stochastic optimal control problem for a backward stochastic differential equation (BSDE). The control-theoretic proof techniques and results may be viewed as a generalization of the classical work on the stability of the Kalman filter.