Barendregt gave a sound semantics of the simple type assignment system by generalizing Tait’s proof of the strong normalization theorem. In this paper, we aim to extend the semantics so that the completeness theorem holds.
"Term-Space Semantics of Typed Lambda Calculus." Notre Dame J. Formal Logic 61 (4) 591 - 600, November 2020. https://doi.org/10.1215/00294527-2020-0028