Hodes (2021) “looked under the hood” of the familiar versions of the classical propositional modal logic K and its intuitionistic counterpart (see Plotkin & Sterling 1986). This paper continues that project, addressing some familiar classical strengthenings of K (D, T, K4, KB, K5, Dio (the Diodorian strengthening of K) and GL), and their intuitionistic counterparts (see Plotkin & Sterling 1986 for some of these counterparts). Section 9 associates two intuitionistic one-step proof-theoretic systems to each of the just mentioned intuitionistic logics, this by adding for each a new rule to those which generated IK in Hodes (2021). For the systems associated with the intuitionistic counterparts of D and T, these rules are “pure one-step”: their schematic formulations does not use □ or ♢. For the systems associated with the intuitionistic counterparts of K4, etc., these rules meet these conditions: neither □ nor ♢ is iterated; none use both □ and ♢. The join of the two systems associated with each of these familiar logics is the full one-step system for that intuitionistic logic. And further “blended” intuitionistic systems arise from joining these systems in various ways. Adding the 0-version of Excluded Middle to their intuitionistic counterparts yields the one-step systems corresponding to the familiar classical logics. Each proof-theoretic system defines a consequence relation in the obvious way. Section 10 examines inclusions between these consequence relations. Section 11 associates each of the above consequence relations with an appropriate class of models, and proves them sound with respect to their appropriate class. This allows proofs of some failures of inclusion between consequence relations. (Sections 10 and 11 provide an exhaustive study of a variety of intuitionistic modal logics.) Section 12 proves that the each consequence relation is complete or (for those corresponding to GL) weakly complete, that relative to its appropriate class of models. The Appendix presents three further results about some of the intuitionistic consequence relations discussed in the body of the paper.

Hodes（2021）“深入了解”了经典命题模态逻辑 K 及其直觉对应版本的熟悉版本（参见 Plotkin & Sterling 1986）。本文继续该项目，讨论一些熟悉的 K 的经典强化（D、T、K4、KB、K5、Dio（K 的狄奥多里强化）和 GL）及其直觉对应物（有关其中一些，请参阅 Plotkin & Sterling 1986同行）。第 9 节将两个直觉主义一步证明理论系统与刚才提到的每个直觉主义逻辑相关联，这是通过为每个在 Hodes (2021) 中生成 IK 的规则添加新规则。对于与 D 和 T 的直觉对应物相关联的系统，这些规则是“纯一步法”：它们的示意图公式不使用 □ 或 ♢。对于与 K4 等直觉对应物相关联的系统，这些规则满足以下条件：□ 和 ♢ 均不迭代；none 同时使用 □ 和 ♢。与这些熟悉的逻辑中的每一个相关联的两个系统的结合是该直觉逻辑的完整一步系统。进一步的“混合”直觉系统来自于以各种方式连接这些系统。添加0- 版本的 Excluded Middle 与其直觉对应物产生对应于熟悉的经典逻辑的一步系统。每个证明理论系统都以显而易见的方式定义了一个结果关系。第 10 节检查这些后果关系之间的包含。第 11 节将上述每个后果关系与适当的模型类别相关联，并证明它们在适当的类别方面是合理的。这允许证明一些后果关系之间的包含失败。（第 10 节和第 11 节提供了对各种直觉模态逻辑的详尽研究。）第 12 节证明了每个后果关系是完整的或（对于那些对应于 GL 的）弱完整的，相对于其适当的模型类别。