Automated protein function prediction is one of the grand challenges in computational biology. Multi-label learning is widely used to predict functions of proteins. Most of multi-label learning methods make prediction for unlabeled proteins under the assumption that the labeled proteins are completely annotated, i.e., without any missing functions. However, in practice, we may have a subset of the ground-truth functions for a protein, and whether the protein has other functions is unknown. To predict protein functions with incomplete annotations, we propose a Protein Function Prediction method with Weak-label Learning (ProWL) and its variant ProWL-IF. Both ProWL and ProWL-IF can replenish the missing functions of proteins. In addition, ProWL-IF makes use of the knowledge that a protein cannot have certain functions, which can further boost the performance of protein function prediction. Our experimental results on protein-protein interaction networks and gene expression benchmarks validate the effectiveness of both ProWL and ProWL-IF.