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Sara Ayhan, Meaning and identity of proofs in a bilateralist setting: A two-sorted typed Lambda-calculus for proofs and refutations, Journal of Logic and Computation, Volume 35, Issue 2, March 2025, exae014, https://doi.org/10.1093/logcom/exae014
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Abstract
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.
