I explore, from a proof-theoretic perspective, the hierarchy of classical and paraconsistent logics introduced by Barrio, Pailos and Szmuc in (Journal o f Philosophical Logic, 49, 93-120, 2021). First, I provide sequent rules and axioms for all the logics in the hierarchy, for all inferential levels, and establish soundness and completeness results. Second, I show how to extend those systems with a corresponding hierarchy of validity predicates, each one of which is meant to capture “validity” at a different inferential level. Then, I point out two potential philosophical implications of these results. (i) Since the logics in the hierarchy differ from one another on the rules, I argue that each such logic maintains its own distinct identity (contrary to arguments like the one given by Dicher and Paoli in 2019). (ii) Each validity predicate need not capture “validity” at more than one metainferential level. Hence, there are reasons to deny the thesis (put forward in Barrio, E., Rosenblatt, L. & Tajer, D. (Synthese, 2016)) that the validity predicate introduced in by Beall and Murzi in (Journal o f Philosophy, 110(3), 143–165, 2013) has to express facts not only about what follows from what, but also about the metarules, etc.

我从证明理论的角度探索了 Barrio、Pailos 和 Szmuc 在（Journal of Philosophical Logic , 49, 93-120, 2021)。首先，我为层次结构中的所有逻辑、所有推理级别提供了后续规则和公理，并建立健全性和完整性结果。其次，我展示了如何使用相应的有效性谓词层次结构扩展这些系统，每个层次都旨在捕获不同推理级别的“有效性”。然后，我指出这些结果的两个潜在哲学含义。(i) 由于层次结构中的逻辑在规则上彼此不同，我认为每个这样的逻辑都保持着自己独特的身份（与 Dicher 和 Paoli 在 2019 年给出的论点相反）。(ii) 每个有效性谓词不需要在多个元推断级别捕获“有效性”。因此，有理由否认该论文（在 Barrio, E., Rosenblatt, L. & Tajer, D. 中提出（Synthese , 2016)) Beall 和 Murzi 在 ( Journal of Philosophy , 110 (3), 143–165, 2013) 中引入的有效性谓词不仅要表达关于从何而来的事实，还要表达关于元规则的事实，等等。