We present nested sequent systems for propositional G\"odel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these G\"odel-Dummett logics, we introduce a new structural rule, called the "linearity rule," which (bottom-up) operates by linearizing branching structure in a given nested sequent. In addition, an interesting feature of our calculi is the inclusion of reachability rules, which are special logical rules that operate by propagating data and/or checking if data exists along certain paths within a nested sequent. Such rules require us to generalize our nested sequents to include signatures (i.e. finite collections of variables) in the first-order cases, thus giving rise to a generalization of the usual nested sequent formalism. Our calculi exhibit favorable properties, admitting the height-preserving invertibility of every logical rule and the (height-preserving) admissibility of a large collection of structural and reachability rules. We prove all of our systems sound and cut-free complete, and show that syntactic cut-elimination obtains for the intuitionistic systems. We conclude the paper by discussing possible extensions and modifications, putting forth an array of structural rules that could be used to provide a sizable class of intermediate logics with cut-free nested sequent systems.

翻译：本文提出命题哥德尔-杜梅特逻辑及其带非常域和常域的一阶扩展的嵌套矢列系统，这些系统构建于直觉主义逻辑的嵌套演算之上。为得到这些哥德尔-杜梅特逻辑的嵌套系统，我们引入一种新的结构规则，称为“线性规则”，该规则（自底向上）通过线性化给定嵌套矢列中的分支结构来运作。此外，我们演算的一个有趣特征是包含了可达性规则，这些特殊逻辑规则通过沿嵌套矢列内特定路径传播数据或检查数据是否存在来运作。此类规则要求我们在一阶情形下将嵌套矢列推广至包含签名（即变量的有限集合），从而产生对常规嵌套矢列形式的一种推广。我们的演算展现出优良性质，允许每个逻辑规则的高度保持可逆性，以及大量结构和可达性规则的（高度保持）可容纳性。我们证明所有系统是可靠的且无割完备的，并表明直觉主义系统可获得语法割消除。最后，我们通过讨论可能的扩展与修改来总结本文，提出一系列结构规则，这些规则可用于为一大类中间逻辑提供无割嵌套矢列系统。