Using success probability as a measure of logical stochastic resonance, we mainly studied the phenomenon of logical stochastic resonance (LSR) in a one-dimensional bistable system induced by non-Gaussian noise in this paper. And we explained the cause of this phenomenon via the method of the mean first-passage time. Results show that the resonance peak of the success probability can be achieved only when in an appropriate range of noise intensity or correlation time. Through an optimization of system parameters, we have improved the reliability of realizing logic operation in the system.