Type Theory with Opposite Types: A Paraconsistent Type Theory

A version of intuitionistic type theory is extended with opposite types, allowing a different formalization of negation and obtaining a paraconsistent type theory ($\textsf{PTT} $). The rules for opposite types in $\textsf{PTT} $ are based on the rules of the so-called constructible falsity. A propositions-as-types correspondence between the many-sorted paraconsistent logic $\textsf{PL}_\textsf{S} $ (a many-sorted extension of López-Escobar’s refutability calculus presented in natural deduction format) and $\textsf{PTT} $ is proven. Moreover, a translation of $\textsf{PTT} $ into intuitionistic type theory is presented and some properties of $\textsf{PTT} $ are discussed.