We present a family of minimal modal logics (namely, modal logics based on minimal propositional logic) corresponding each to a different classical modal logic. The minimal modal logics are defined based on their classical counterparts in two distinct ways: (1) via embedding into fusions of classical modal logics through a natural extension of the G\"odel-Johansson translation of minimal logic into modal logic S4; (2) via extension to modal logics of the multi- vs. single-succedent correspondence of sequent calculi for classical and minimal logic. We show that, despite being mutually independent, the two methods turn out to be equivalent for a wide class of modal systems. Moreover, we compare the resulting minimal version of K with the constructive modal logic CK studied in the literature, displaying tight relations among the two systems. Based on these relations, we also define a constructive correspondent for each minimal system, thus obtaining a family of constructive modal logics which includes CK as well as other constructive modal logics studied in the literature.

翻译：本文提出了一系列最小模态逻辑（即基于最小命题逻辑的模态逻辑），每种逻辑对应于不同的经典模态逻辑。最小模态逻辑通过两种不同方式基于其经典对应物定义：（1）通过将最小逻辑到模态逻辑S4的Gödel-Johansson翻译自然扩展，嵌入到经典模态逻辑的融合中；（2）通过将经典逻辑与最小逻辑的后件多/单对应关系扩展到模态逻辑的相继式演算中。我们证明，尽管两种方法相互独立，但对于广泛的模态系统而言，它们是等价的。此外，我们将所得K系统的最小版本与文献中研究的构造性模态逻辑CK进行比较，揭示了两系统之间的紧密关系。基于这些关系，我们还为每个最小系统定义了构造性对应物，从而得到一系列构造性模态逻辑，其中包括CK及文献中研究的其他构造性模态逻辑。