  Frege gave the outline of a theory of intensionality, but no intensional logic in any formal sense. There have been attempts to fill in his outline. Alonzo Church (1951) went at it quite directly. In this paper there is a formal logic in which terms have both senses and denotations. These are simply taken to be different sorts, and minimal requirements are placed on them. Nonetheless the logic is quite complex. The formal logic that Frege had created for his work on the foundations of mathematics was type free. Russell showed his famous paradox applied to Frege’s system, so it was inconsistent. As a way out of this problem, Russell developed the type theory that was embodied in Principia Mathematica. Church had given an elegant and precise formulation of the simple theory of types (Church 1940), and that was incorporated into his work on intensionality, which is one of the reasons for its formal complexity.