More than 50 years ago, Bongard introduced 100 visual concept learning problems as a challenge for artificial vision systems. These problems are now known as Bongard problems. Although they are well known in cognitive science and artificial intelligence, only very little progress has been made toward building systems that can solve a substantial subset of them. In the system presented here, visual features are extracted through image processing and then translated into a symbolic visual vocabulary. We introduce a formal language that allows representing compositional visual concepts based on this vocabulary. Using this language and Bayesian inference, concepts can be induced from the examples that are provided in each problem. We find a reasonable agreement between the concepts with high posterior probability and the solutions formulated by Bongard himself for a subset of 35 problems. While this approach is far from solving Bongard problems like humans, it does considerably better than previous approaches. We discuss the issues we encountered while developing this system and their continuing relevance for understanding visual cognition. For instance, contrary to other concept learning problems, the examples are not random in Bongard problems; instead they are carefully chosen to ensure that the concept can be induced, and we found it helpful to take the resulting pragmatic constraints into account.