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, 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 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.

翻译：本文将为双直觉主义逻辑发展一种λ-项演算λ-2Int，并探讨其对双边主义框架下推导的意义与指称概念的启示。为此，我将运用柯里-霍华德对应——该对应已在简单类型λ-演算与直觉主义逻辑的自然演绎系统之间得到充分确立——并将其应用于一种展示两种可推导关系（分别用于证明与反驳）的双边主义证明系统。该研究基于万辛格双直觉主义逻辑2Int的自然演绎系统，我将为其赋予项标注形式。因此，我们需要一种扩展为双类型带类型λ-演算的类型理论。我将为2Int提出这样一种项标注证明系统，并证明关于该系统中证明与反驳的对偶定理。基于这些形式化结果，我将论证：从双边主义视角出发，这为我们理解关于意义与指称、以及证明的同义性与同一性问题提供了富有启发性的洞见。