{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T17:38:00Z","timestamp":1782927480728,"version":"3.54.5"},"reference-count":28,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2024,3,20]],"date-time":"2024-03-20T00:00:00Z","timestamp":1710892800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,3,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper we study several extensions of the minimal modal logic M. This minimal modal logic is formulated in the language of classical propositional logic together with two modal operators $\\Box $ and $\\Diamond $, which have no deductive power. By extending the Hilbert calculus for M with various axioms for $\\Box $ and $\\Diamond $ and\/or the rule of necessitation, we obtain several well-known normal modal logics, as well as systems that are of pure theoretical interest. Those systems are shown to be sound and complete wrt to eight-valued semantics. Those semantics are obtained by refinements of an eight-valued semantics for M. Furthermore, we will briefly discuss some limitations of the method presented in this article.<\/jats:p>","DOI":"10.1093\/logcom\/exae010","type":"journal-article","created":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T18:15:50Z","timestamp":1711044950000},"source":"Crossref","is-referenced-by-count":5,"title":["\u2610 and \u25c7 in eight-valued non-deterministic semantics for modal logics"],"prefix":"10.1093","volume":"35","author":[{"given":"Pawel","family":"Pawlowski","sequence":"first","affiliation":[{"name":"Centre for Logic and Philosophy of Science , Ghent University, Ghent 9000, Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Skurt","sequence":"additional","affiliation":[{"name":"Ruhr University Bochum , Institute for Philosophy I, Bochum 44801, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,3,20]]},"reference":[{"key":"2025030912422493400_ref1","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1093\/jigpal\/jzi030","article-title":"Multi-valued calculi for logics based on non-determinism","volume":"13","author":"Avron","year":"2005","journal-title":"Logic Journal of IGPL"},{"key":"2025030912422493400_ref2","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-007-0479-4_4","article-title":"Non-deterministic semantics for logical systems","volume-title":"Handbook of Philosophical Logic","author":"Avron","year":"2011"},{"key":"2025030912422493400_ref3","doi-asserted-by":"crossref","first-page":"1355","DOI":"10.1016\/B978-044450813-3\/50022-9","article-title":"Automated deduction for many-valued logics","volume-title":"Handbook of Automated Reasoning","author":"Baaz","year":"2001"},{"key":"2025030912422493400_ref4","volume-title":"Paraconsistent Logic: Consistency, Contradiction and Negation","author":"Carnielli","year":"2016"},{"key":"2025030912422493400_ref5","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1093\/jigpal\/jzy072","article-title":"Non-deterministic algebraization of logics by swap structures","volume":"28","author":"Coniglio","year":"2020","journal-title":"Logic Journal of the IGPL"},{"key":"2025030912422493400_ref6","doi-asserted-by":"crossref","first-page":"2243","DOI":"10.1007\/s00500-018-03707-4","article-title":"Swap structures semantics for Ivlev-like modal logics","volume":"23","author":"Coniglio","year":"2019","journal-title":"Soft Computing"},{"key":"2025030912422493400_ref7","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1080\/11663081.2015.1011543","article-title":"Finite non-deterministic semantics for some modal systems","volume":"25","author":"Coniglio","year":"2015","journal-title":"Journal of Applied Non-Classical Logics"},{"key":"2025030912422493400_ref8","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1080\/11663081.2017.1300436","article-title":"Errata and addenda to \u2018finite non-deterministic semantics for some modal systems\u2019","volume":"26","author":"Coniglio","year":"2016","journal-title":"Journal of Applied Non-Classical Logics"},{"key":"2025030912422493400_ref9","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1093\/jigpal\/jzz027","article-title":"Modal logic with non-deterministic semantics: Part I-propositional case","volume":"28","author":"Coniglio","year":"2019","journal-title":"Logic Journal of the IGPL, pages"},{"key":"2025030912422493400_ref10","first-page":"695","article-title":"Modal logic with non-deterministic semantics: Part II-quantified case","author":"Coniglio","year":"2021","journal-title":"Logic Journal of the IGPL"},{"key":"2025030912422493400_ref11","volume-title":"On Six-Valued Logics of Evidence and Truth Expanding Belnap-Dunn Four-Valued Logic.","author":"Coniglio","year":"2022"},{"key":"2025030912422493400_ref12","volume-title":"SQEMA - 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