A definition of a typed language is said to be "intrinsic" if it assigns<br />meanings to typings rather than arbitrary phrases, so that ill-typed<br />phrases are meaningless. In contrast, a definition is said to be "extrinsic"<br />if all phrases have meanings that are independent of their typings,<br />while typings represent properties of these meanings.<br />For a simply typed lambda calculus, extended with recursion, subtypes,<br />and named products, we give an intrinsic denotational semantics<br />and a denotational semantics of the underlying untyped language. We<br />then establish a logical relations theorem between these two semantics,<br />and show that the logical relations can be "bracketed" by retractions<br />between the domains of the two semantics. From these results, we<br />derive an extrinsic semantics that uses partial equivalence relations.