Binary classification (rain or shine, disease or not, increase or decrease) is a fundamental problem in machine learning. We present an algorithm that can take any standard online binary classification algorithm and provably improve its performance under very weak assumptions, given the right to refuse to make predictions in certain cases. The extent of improvement will depend on the data size, stability of the algorithm, and room for improvement in the algorithms performance. Our experiments on standard machine learning data sets and standard algorithms (k-nearest neighbors and random forests) show the effectiveness of our approach, even beyond what is possible using previous work on conformal predictors upon which our approach is based. Though we focus on binary classification, our theory could be extended to multiway classification. Our code and data are available upon request.