Deep learning and symbolic reasoning are complementary techniques for an intelligent system. However, principled combinations of these techniques have limited scalability, rendering them ill-suited for real-world applications. We propose Scallop, a system that builds upon probabilistic deductive databases, to bridge this gap. The key insight underlying Scallop is a provenance framework that introduces a tunable parameter to specify the level of reasoning granularity. Scallop thereby i) generalizes exact probabilistic reasoning, ii) asymptotically reduces computational cost, and iii) provides relative accuracy guarantees. On a suite of tasks that involve mathematical and logical reasoning, Scallop scales significantly better without sacrificing accuracy compared to DeepProbLog, a principled neural logic programming approach. We also create and evaluate on a real-world Visual Question Answering (VQA) benchmark that requires multi-hop reasoning. Scallop outperforms two VQA-tailored models, a Neural Module Networks based and a transformer based model, by 12.42% and 21.66% respectively.