A convergence analysis with accompanying error estimate is provided for the convex-splitting scheme of the Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. In particular, a higher order consistency analysis, accomplished by carefully defined supplementary functions, is performed to obtain the second truncation error. In turn, the rough error estimate leads to the separation property of numerical solution and the separation property further leads to the refined error estimate of the logarithmic term. The rough and refined error estimates help us to derive an optimal rate convergence.