This paper has two major parts. The first is an extensive analysis of the problem of induction, and the second part is a detailed study of selective induction. Throughout the paper we integrate a number of notions, mainly from artificial intelligence, but also from pattern recognition and cognitive psychology. The result is a synthetic view which exploits uncertainty, task-guidance, and biases such as language restriction. Some of the main themes and contributions are as follows. (1) Practical induction is really a problem of efficacy and efficiency (power). (2) Search in a space of hypothetical concepts is governed by a credibility function which combines various knowledge sources in a single subjective probability or belief measure μ. (3) The amount of knowledge supplied by various sources can often be quantifieds these sources include various biases and the learning system itself. (4) Induction is equivalent to discovery of a utility function u, which captures the purpose or goal of induction. (5) The difficulty of induction may be characterized by the form of u. Smooth or coherent functions mean selective induction, which has had the most attention in machine learning. (6) Systems for selective induction are more similar than commonly understood. By juxtaposing them we can discover similarities and improvements. (7) Our analysis suggests a number of incipient principles for powerful induction.