A model is presented for the class of inductive inference problems that are solved by refinement algorithms - that is, algorithms that modify a hypothesis by making it more general or more specific in response to examples. The separate effects of the syntax (rule space) and semantics, and the relevant orderings on these, are precisely specified. Relations called refinement operators are defined, one for generalization and one for specialization. General and particular properties of these relations are considered, and algorithm schemas for top-down and bottom-up inference are given. Finally, difficulties common to refinement algorithms are reviewed.