An interpretation of Martin-löf's type theory in a type-free theory of propositions

Journal of Symbolic Logic 49 (3):730-753 (1984)

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We present a formal theory of propositions and combinator terms, and in this theory we give an interpretation of Martin-Löf's type theory. The construction of the interpretation is inspired by the semantics for type theory, but it can also be viewed as a formalized realizability interpretation
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DOI 10.2307/2274128
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An Expressive First-Order Logic with Flexible Typing for Natural Language Semantics.Chris Fox & Shalom Lappin - 2004 - Logic Journal of the Interest Group in Pure and Applied Logics 12 (2):135--168.

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