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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 \u2018via completeness\u2019 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 \u2018informational implication\u2019\u2014in doing so a link is also made to the work of (Buszkowski 2021) on the Lambek Calculus.<\/jats:p>","DOI":"10.1007\/s10992-023-09724-5","type":"journal-article","created":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T09:02:59Z","timestamp":1699002179000},"page":"1723-1766","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Modal Information Logics: Axiomatizations and Decidability"],"prefix":"10.1007","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-9835-4195","authenticated-orcid":false,"given":"S\u00f8ren Brinck","family":"Knudstorp","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,11,3]]},"reference":[{"key":"9724_CR1","unstructured":"Adriaans, P. (2020). Information. In E.N. 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