Much effort has been devoted to understanding learning and reasoning in artificial intelligence. However, very few models attempt to integrate these two complementary processes. Rather, there is a vast body of research in machine learning, often focusing on inductive learning from examples, quite isolated from the work on reasoning in artificial intelligence. Though these two processes may be different, they are very much interrelated. The ability to reason about a domain of knowledge is often based on rules about that domain, that must be learned somehow. And the ability to reason can often be used to acquire new knowledge, or learn. This paper introduces an Incremental Learning Algorithm (ILA) that attempts to combine inductive learning with prior knowledge and reasoning. ILA has many important characteristics useful for such a combination, including: 1) incremental, self-organizing learning, 2) non-uniform learning, 3) inherent non-monotonicity, 4) extensional and intensional capabilities, and 5) low order polynomial complexity. The paper describes ILA, gives simulation results for several applications, and discusses e ach of the above characteristics in detail.