Meaning and identity of proofs in a bilateralist setting: A two-sorted typed lambda-calculus for proofs and refutations

In this paper I will develop a lambda-term calculus, lambda-2Int, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry-Howard correspondence, which has been well-established between the simply typed lambda-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural deduction system of Wansing's bi-intuitionistic logic 2Int (2016a; 2017), which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed lambda-calculus. I will present such a term-annotated proof system for 2Int and prove some properties and results for it, most importantly for this paper a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view.