Type theories in the sense of Martin-Löf and the NuPRL system are based on taking as primitive a type-free programming language given by an operational semantics, and defining types as partial equivalence relations on the set of closed terms. The construction of a type system is based on a general form of inductive definition that may either be taken as acceptable in its own right, or further explicated in terms of other patterns of induction. One such account, based on a general theory of inductively-defined relations, was given by Allen. An alternative account, based on an essentially set-theoretic argument, is presented.