Abstract
We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive instead of being formulated via the propositions-as-types representation. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the double-negation translation.
Citation
Peter Aczel. Nicola Gambino. "The generalised type-theoretic interpretation of constructive set theory." J. Symbolic Logic 71 (1) 67 - 103, March 2006. https://doi.org/10.2178/jsl/1140641163
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