{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T15:01:07Z","timestamp":1790780467092,"version":"4.1.0"},"reference-count":32,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T00:00:00Z","timestamp":1790726400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004281","name":"Narodowe Centrum Nauki","doi-asserted-by":"publisher","award":["2023\/49\/N\/HS1\/04070"],"award-info":[{"award-number":["2023\/49\/N\/HS1\/04070"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"abstract":"<jats:title>Astract<\/jats:title>\n                  <jats:p>We construct an uncountable sequence of minimal varieties of modal algebras which do not contain any modal algebra whose Boolean reduct forms a powerset algebra. By algebraizabilty and duality between modal algebras and neighborhood frames, this yields a result stating that there exists an uncountable set of Post complete and neighborhood unsound (strongly incomplete) congruential modal logics. Our result resolves an open problem posed by Peter Fritz in his paper from 2016. Furthermore, it shows a sharp contrast with the lattice of normal modal logics (NMLs), since every NML is known to be Kripke (and therefore neighborhood) sound.<\/jats:p>","DOI":"10.1017\/s1755020326101294","type":"journal-article","created":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T14:47:33Z","timestamp":1790779653000},"page":"1-17","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["MAXIMAL NEIGHBORHOOD UNSOUND CONGRUENTIAL MODAL LOGICS"],"prefix":"10.1017","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4367-4796","authenticated-orcid":false,"given":"KRZYSZTOF ALEKSANDER","family":"KRAWCZYK","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03bqmcz70","id-type":"ROR","asserted-by":"publisher"}],"name":"JAGIELLONIAN UNIVERSITY"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2026,9,30]]},"reference":[{"key":"S1755020326101294_r2","first-page":"59","volume-title":"Structural Completeness in Algebra and Logic","volume":"54","author":"Bergman","year":"1988"},{"key":"S1755020326101294_r15","doi-asserted-by":"publisher","DOI":"10.2307\/2271893"},{"key":"S1755020326101294_r5","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198537793.001.0001"},{"key":"S1755020326101294_r7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511621192"},{"key":"S1755020326101294_r9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02770513"},{"key":"S1755020326101294_r25","unstructured":"[25] Litak, T. (2018). A continuum of incomplete intermediate logics. Preprint, https:\/\/arxiv.org\/abs\/1808.06284, pp. 1\u201311."},{"key":"S1755020326101294_r13","first-page":"288","volume-title":"Proceedings of the 11th Conference (AIML 2016), Budapest, Hungary, August 30\u2013September 2, 2016","author":"Fritz","year":"2016"},{"key":"S1755020326101294_r27","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-67149-9"},{"key":"S1755020326101294_r14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02121664"},{"key":"S1755020326101294_r18","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020317000259"},{"key":"S1755020326101294_r11","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00076.x"},{"key":"S1755020326101294_r12","unstructured":"[12] Fornasiere, D. , & Moraschini, T. (2025). Degrees of incompleteness of implicative logics: the trichotomy theorem. Online preprint, pp. 1\u201335. https:\/\/moraschini.github.io\/files\/submitted\/2025-Trichotomy-Theorem.pdf."},{"key":"S1755020326101294_r19","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-3315588"},{"key":"S1755020326101294_r20","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10850"},{"key":"S1755020326101294_r21","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-10670082"},{"key":"S1755020326101294_r10","first-page":"185","article-title":"A note on incompleteness of modal logics with respect to neighbourhood semantics","volume":"7","author":"Dziobiak","year":"1978","journal-title":"Bulletin of the Section of Logic"},{"key":"S1755020326101294_r26","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093894226"},{"key":"S1755020326101294_r32","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1052669066"},{"key":"S1755020326101294_r1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1978.tb00830.x"},{"key":"S1755020326101294_r17","doi-asserted-by":"publisher","DOI":"10.48550\/arXiv.2603.17724"},{"key":"S1755020326101294_r3","unstructured":"[3] Blok, W. J. (1978). On the degree of incompleteness of modal logic and the covering relation in the lattice of modal logics. Technical Report 78-07, University of Amsterdam."},{"key":"S1755020326101294_r23","doi-asserted-by":"publisher","DOI":"10.1023\/B:STUD.0000032102.67838.f2"},{"key":"S1755020326101294_r4","doi-asserted-by":"publisher","DOI":"10.2307\/2273184"},{"key":"S1755020326101294_r31","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00077.x"},{"key":"S1755020326101294_r8","unstructured":"[8] Ding, Y. , & Holliday, W. H. (2020). In Olivetti, N. , and Verbrugge, R. , editors. Another Problem in Possible World Semantics, Advances in Modal Logic , 13, London: College Publications, pp. 149\u2013168."},{"key":"S1755020326101294_r24","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-008-2062-4"},{"key":"S1755020326101294_r6","first-page":"63","volume-title":"On the Degree of Neighborhood Incompleteness of Normal Modal Logics","author":"Chagrova","year":"1998"},{"key":"S1755020326101294_r22","first-page":"131","article-title":"A continuum of incomplete intermediate logics","author":"Litak","year":"2002","journal-title":"Reports on Mathematical Logic"},{"key":"S1755020326101294_r29","first-page":"985","article-title":"On incomplete propositional logics","volume":"18","author":"Shetman","year":"1977","journal-title":"Soviet Mathematics Doklady"},{"key":"S1755020326101294_r30","doi-asserted-by":"publisher","DOI":"10.2307\/2272558"},{"key":"S1755020326101294_r28","volume-title":"An Essay in Classical Modal Logic","author":"Segerberg","year":"1971"},{"key":"S1755020326101294_r16","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-2025-0030"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020326101294","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T14:47:37Z","timestamp":1790779657000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020326101294\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,30]]},"references-count":32,"alternative-id":["S1755020326101294"],"URL":"https:\/\/doi.org\/10.1017\/s1755020326101294","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,30]]},"assertion":[{"value":"\u00a9 The Author(s), 2026. Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}