Proof Irrelevance in Type-Theoretical Semantics

Abstract

Type theories have been used as foundational languages for formal semantics. Under the propositions-as-types principle, most modern type systems have explicit proof objects which, however, cause problems in obtaining correct identity criteria in semantics. Therefore, it has been proposed that some principle of proof irrelevance should be enforced in order for a type theory to be an adequate semantic language. This paper investigates how proof irrelevance can be enforced, particularly in predicative type systems. In an impredicative type theory such as UTT, proof irrelevance can be imposed directly since the type Prop in such a type theory represents the totality of logical propositions and helps to distinguish propositions from other types. In a predicative type theory, however, such a simple approach would not work; for example, in Martin-Löf’s type theory (MLTT), propositions and types are identified and, hence, proof irrelevance would have implied the collapse of all types. We propose that Martin-Löf’s type theory should be extended with h-logic, as proposed by Veovodsky and studied in the HoTT project, where proof irrelevance is built-in in the notion of logical proposition. This amounts to MLTT\(_h\), a predicative type system that can be adequately employed for formal semantics.