Barrio et al. (Journal of Philosophical Logic, 49(1), 93–120, 2020) and Pailos (Review of Symbolic Logic, 2020(2), 249–268, 2020) develop an approach to define various metainferential hierarchies on strong Kleene models by transferring the idea of distinct standards for premises and conclusions from inferences to metainferences. In particular, they focus on a hierarchy named the \(\mathbb {S}\mathbb {T}\)-hierarchy where the inferential logic at the bottom of the hierarchy is the non-transitive logic ST but where each subsequent metainferential logic ‘says’ about the former logic that it is transitive. While Barrio et al. (2020) suggests that this hierarchy is such that each subsequent level ‘in some intuitive sense, more classical than’ the previous level, Pailos (2020) proposes an extension of the hierarchy through which a ‘fully classical’ metainferential logic can be defined. Both Barrio et al. (2020) and Pailos (2020) explore the hierarchy in terms of semantic definitions and every proof proceeds by a rather cumbersome reasoning about those semantic definitions. The aim of this paper is to present and illustrate the virtues of a proof-theoretic tool for reasoning about the \(\mathbb {S}\mathbb {T}\)-hierarchy and the other metainferential hierarchies definable on strong Kleene models. Using the tool, this paper argues that each level in the \(\mathbb {S}\mathbb {T}\)-hierarchy is non-classical to an equal extent and that the ‘fully classical’ metainferential logic is actually just the original non-transitive logic ST ‘in disguise’. The paper concludes with some remarks about how the various results about the \(\mathbb {S}\mathbb {T}\)-hierarchy could be seen as a guide to help us imagine what a non-transitive metalogic for ST would tell us about ST. In particular, it teaches us that ST is from the perspective of ST as metatheory not only non-transitive but also transitive.

巴里奥等人。( Journal of Philosophical Logic , 49 (1), 93–120, 2020) 和 Pailos ( Review of Symbolic Logic , 2020 (2), 249–268, 2020) 开发了一种方法来定义强 Kleene 模型上的各种元推断层次结构从推理到元推理的前提和结论的不同标准的想法。特别是，他们关注名为\(\mathbb {S}\mathbb {T}\)-层次结构，其中层次结构底部的推理逻辑是非传递逻辑 ST，但是每个后续的元推理逻辑都“说”前一个逻辑它是传递的。而巴里奥等人。(2020) 建议这个层次结构使得每个后续级别“在某种直观意义上，比”前一个级别更经典，Pailos (2020) 提出了层次结构的扩展，通过它可以定义“完全经典的”元推断逻辑。巴里奥等人。(2020) 和 Pailos (2020) 在语义定义方面探索了层次结构，每个证明都是通过对这些语义定义进行相当繁琐的推理来进行的。本文的目的是展示和说明用于推理\(\mathbb {S}\mathbb {T}\)-层次结构和其他可在强 Kleene 模型上定义的元推断层次结构。使用该工具，本文认为\(\mathbb {S}\mathbb {T}\) -hierarchy中的每个级别在同等程度上都是非经典的，并且“完全经典”的元推论逻辑实际上只是原始的非传递逻辑 ST '变相'。这篇论文的结尾是关于\(\mathbb {S}\mathbb {T}\)层次结构的各种结果如何被视为指导，帮助我们想象 ST 的非传递元逻辑会告诉我们什么关于 ST。特别是，它告诉我们，从 ST 作为元理论的角度来看，ST 不仅是非传递性的，而且是传递性的。