The present paper studies formal properties of so-called modal information logics (MILs)—modal logics first proposed in (van Benthem 1996) as a way of using possible-worlds semantics to model a theory of information. They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states. First proposed in 1996, MILs have been around for some time, yet not much is known: (van Benthem 2017, 2019) pose two central open problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability. The main results of the first part of this paper are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability. In the proof of the latter, an emphasis is put on the method applied as a heuristic for proving decidability ‘via completeness’ for semantically introduced logics; the logics lack the FMP w.r.t. their classes of definition, but not w.r.t. a generalized class. These results are build upon to axiomatize and prove decidable the MILs attained by endowing the language with an ‘informational implication’—in doing so a link is also made to the work of (Buszkowski 2021) on the Lambek Calculus.

本文研究所谓模态信息逻辑（MIL）的形式属性——模态逻辑首次在（van Benthem 1996）中提出，作为使用可能世界语义来建模信息理论的一种方式。他们通过用二元模态扩展命题逻辑语言来实现这一点，该二元模态定义为两个状态的至高无上。MIL 于 1996 年首次提出，已经存在了一段时间，但人们知之甚少：（van Benthem 2017，2019）提出了两个中心开放问题，即（1）分别公理化先序和偏序集上的至上的两个基本 MIL， (2) 证明（不可）可判定性。本文第一部分的主要结果是解决这两个问题：（1）通过提供公理化[具有使两个逻辑相同的完整性证明]，以及（2）通过证明可判定性。在后者的证明中，重点放在作为启发式应用的方法，以“通过完整性”为语义引入的逻辑证明可判定性；逻辑缺乏关于其定义类的 FMP，但不关于广义类。这些结果是建立在通过赋予语言“信息含义”而获得的 MIL 公理化和可判定的基础上的——这样做也与 (Buszkowski 2021) 关于 Lambek 微积分的工作建立了链接。