{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T00:05:21Z","timestamp":1759017921560,"version":"3.44.0"},"publisher-location":"Cham","reference-count":25,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783032060846","type":"print"},{"value":"9783032060853","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We present a non-deterministic semantic framework for all modal logics in\u00a0the modal cube, extending prior works by Kearns and others. Our approach introduces modular and uniform multi-valued non-deterministic matrices (Nmatrices)\u00a0for each logic, where necessitation is captured by the systematic use of\u00a0level valuations. The semantics is grounded in an eight-valued system and provides\u00a0a sound and complete decision procedure for each modal logic, extending\u00a0and refining earlier semantics as particular cases. Additionally, we propose\u00a0a novel model-theoretic perspective that links our framework to relational (Kripke-style) semantics, addressing longstanding questions regarding\u00a0the correspondence between modal axioms and semantic conditions\u00a0in non-deterministic settings. This yields a philosophically robust and technically modular alternative to traditional possible-world semantics.<\/jats:p>","DOI":"10.1007\/978-3-032-06085-3_10","type":"book-chapter","created":{"date-parts":[[2025,9,27]],"date-time":"2025-09-27T10:45:22Z","timestamp":1758969922000},"page":"181-200","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Modal Cube Revisited: Semantics Without Worlds"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8199-1261","authenticated-orcid":false,"given":"Renato","family":"Leme","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7264-7773","authenticated-orcid":false,"given":"Carlos","family":"Olarte","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7113-0801","authenticated-orcid":false,"given":"Elaine","family":"Pimentel","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1807-0520","authenticated-orcid":false,"given":"Marcelo Esteban","family":"Coniglio","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,25]]},"reference":[{"key":"10_CR1","unstructured":"Nmatrices for the Modal Cube. https:\/\/github.com\/nmatrices"},{"issue":"3","key":"10_CR2","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1093\/logcom\/exi001","volume":"15","author":"A Avron","year":"2005","unstructured":"Avron, A., Lev, I.: Non-deterministic multiple-valued structures. J. Log. Comput. 15(3), 241\u2013261 (2005). https:\/\/doi.org\/10.1093\/logcom\/exi001","journal-title":"J. Log. Comput."},{"key":"10_CR3","doi-asserted-by":"publisher","unstructured":"Avron, A., Zamansky, A.: Non-deterministic semantics for logical systems, pp. 227\u2013304. Springer, Dordrecht (2011). https:\/\/doi.org\/10.1007\/978-94-007-0479-4_4","DOI":"10.1007\/978-94-007-0479-4_4"},{"issue":"1","key":"10_CR4","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1080\/11663081.2015.1011543","volume":"25","author":"ME Coniglio","year":"2015","unstructured":"Coniglio, M.E., del Cerro, L.F., Peron, N.M.: Finite non-deterministic semantics for some modal systems. J. Appl. Non Class. Logics 25(1), 20\u201345 (2015). https:\/\/doi.org\/10.1080\/11663081.2015.1011543","journal-title":"J. Appl. Non Class. Logics"},{"key":"10_CR5","doi-asserted-by":"publisher","unstructured":"Coniglio, M.E., del Cerro, L.F., Peron, N.M.: Modal logic with non-deterministic semantics: part I - propositional case. Log. J. IGPL 28(3), 281\u2013315 (2020). https:\/\/doi.org\/10.1093\/jigpal\/jzz027","DOI":"10.1093\/jigpal\/jzz027"},{"key":"10_CR6","doi-asserted-by":"publisher","unstructured":"Coniglio, M.E., Pawlowski, P., Skurt, D.: Modal logics - RNmatrices vs. Nmatrices. In: Indrzejczak, A., Zawidzki, M. (eds.) Proceedings Eleventh International Conference on Non-Classical Logics. Theory and Applications, NCL 2024, \u0141\u00f3d\u017a, Poland, 5\u20138 September 2024. EPTCS, vol.\u00a0415, pp. 138\u2013149 (2024). https:\/\/doi.org\/10.4204\/EPTCS.415.14","DOI":"10.4204\/EPTCS.415.14"},{"key":"10_CR7","doi-asserted-by":"publisher","unstructured":"Coniglio, M.E., Pawlowski, P., Skurt, D.: RNmatrices for Modal Logics. Rev. Symb. Logic (2025, to appear). https:\/\/doi.org\/10.1017\/S1755020325100737","DOI":"10.1017\/S1755020325100737"},{"issue":"4","key":"10_CR8","doi-asserted-by":"publisher","first-page":"150","DOI":"10.2307\/2268175","volume":"5","author":"J Dugundji","year":"1940","unstructured":"Dugundji, J.: Note on a property of matrices for lewis and langford\u2019s calculi of propositions. J. Symb. Log. 5(4), 150\u2013151 (1940). https:\/\/doi.org\/10.2307\/2268175","journal-title":"J. Symb. Log."},{"key":"10_CR9","doi-asserted-by":"publisher","unstructured":"Dur\u00e1n, F., et al.: Programming and symbolic computation in Maude. J. Log. Algebraic Methods Program. 110 (2020). https:\/\/doi.org\/10.1016\/j.jlamp.2019.100497","DOI":"10.1016\/j.jlamp.2019.100497"},{"key":"10_CR10","unstructured":"Feferman, S., Dawson, J.W., Kleene, S.C., Moore, G.H., Solovay, R.M., van Heijenoort, J.: Kurt G\u00f6del: Collected Works. Vol. 1: Publications 1929-1936. Oxford University Press (1986)"},{"issue":"1","key":"10_CR11","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1093\/logcom\/exab068","volume":"32","author":"L Gr\u00e4tz","year":"2022","unstructured":"Gr\u00e4tz, L.: Truth tables for modal logics T and S4, by using three-valued non-deterministic level semantics. J. Log. Comput. 32(1), 129\u2013157 (2022). https:\/\/doi.org\/10.1093\/logcom\/exab068","journal-title":"J. Log. Comput."},{"issue":"3\u20134","key":"10_CR12","first-page":"114","volume":"17","author":"J Ivlev","year":"1988","unstructured":"Ivlev, J.: A semantics for modal calculi. Bull. Sect. Logic 17(3\u20134), 114\u2013121 (1988)","journal-title":"Bull. Sect. Logic"},{"issue":"1","key":"10_CR13","doi-asserted-by":"publisher","first-page":"77","DOI":"10.2307\/2273259","volume":"46","author":"JT Kearns","year":"1981","unstructured":"Kearns, J.T.: Modal semantics without possible worlds. J. Symb. Log. 46(1), 77\u201386 (1981). https:\/\/doi.org\/10.2307\/2273259","journal-title":"J. Symb. Log."},{"key":"10_CR14","doi-asserted-by":"publisher","unstructured":"Lahav, O., Zohar, Y.: Effective semantics for the modal logics K and KT via non-deterministic matrices. In: Blanchette, J., Kov\u00e1cs, L., Pattinson, D. (eds.) Automated Reasoning - 11th International Joint Conference, IJCAR 2022, Haifa, Israel, August 8-10, 2022, Proceedings. Lecture Notes in Computer Science, vol. 13385, pp. 468\u2013485. Springer (2022). https:\/\/doi.org\/10.1007\/978-3-031-10769-6_28","DOI":"10.1007\/978-3-031-10769-6_28"},{"key":"10_CR15","unstructured":"Leme, R., Coniglio, M.E., Lopes, B.: A new decision method for intuitionistic logic by 3-valued non-deterministic truth-tables (2025, submitted)"},{"key":"10_CR16","unstructured":"Leme, R., Olarte, C., Pimentel, E., Coniglio, M.E.: The modal cube revisited: semantics without worlds (technical report) (2025). https:\/\/arxiv.org\/abs\/2505.12824"},{"key":"10_CR17","doi-asserted-by":"publisher","unstructured":"Marin, S., Pereira, L.C., Pimentel, E., Sales, E.: Ecumenical modal logic. In: Martins, M.A., Sedl\u00e1r, I. (eds.) Dynamic Logic. New Trends and Applications - Third International Workshop, DaL\u00ed 2020, Prague, Czech Republic, 9\u201310 October 2020, Revised Selected Papers. Lecture Notes in Computer Science, vol. 12569, pp. 187\u2013204. Springer (2020). https:\/\/doi.org\/10.1007\/978-3-030-65840-3_12","DOI":"10.1007\/978-3-030-65840-3_12"},{"key":"10_CR18","doi-asserted-by":"publisher","unstructured":"Marin, S., Pereira, L.C., Pimentel, E., Sales, E.: A pure view of ecumenical modalities. In: Silva, A., Wassermann, R., de\u00a0Queiroz, R.J.G.B. (eds.) Logic, Language, Information, and Computation - 27th International Workshop, WoLLIC 2021, Virtual Event, 5\u20138 October 2021, Proceedings. Lecture Notes in Computer Science, vol. 13038, pp. 388\u2013407. Springer (2021). https:\/\/doi.org\/10.1007\/978-3-030-88853-4_24","DOI":"10.1007\/978-3-030-88853-4_24"},{"key":"10_CR19","doi-asserted-by":"publisher","DOI":"10.1016\/J.JLAMP.2022.100827","volume":"130","author":"C Olarte","year":"2023","unstructured":"Olarte, C., Pimentel, E., Rocha, C.: A rewriting logic approach to specification, proof-search, and meta-proofs in sequent systems. J. Log. Algebraic Methods Program. 130, 100827 (2023). https:\/\/doi.org\/10.1016\/J.JLAMP.2022.100827","journal-title":"J. Log. Algebraic Methods Program."},{"issue":"1","key":"10_CR20","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1093\/logcom\/exab079","volume":"32","author":"P Pawlowski","year":"2022","unstructured":"Pawlowski, P., Rosa, E.L.: Modular non-deterministic semantics for T, TB, S4, S5 and more. J. Log. Comput. 32(1), 158\u2013171 (2022). https:\/\/doi.org\/10.1093\/logcom\/exab079","journal-title":"J. Log. Comput."},{"issue":"2","key":"10_CR21","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/s10992-023-09733-4","volume":"53","author":"P Pawlowski","year":"2024","unstructured":"Pawlowski, P., Skurt, D.: 8 valued non-deterministic semantics for modal logics. J. Philos. Log. 53(2), 351\u2013371 (2024). https:\/\/doi.org\/10.1007\/s10992-023-09733-4","journal-title":"J. Philos. Log."},{"key":"10_CR22","doi-asserted-by":"publisher","unstructured":"Pawlowski, P., Skurt, D.: $$\\diamond $$ and $$\\Box $$ in eight-valued non-deterministic semantics for modal logics. J. Log. Comput. 35(2) (2025). https:\/\/doi.org\/10.1093\/logcom\/exae010","DOI":"10.1093\/logcom\/exae010"},{"key":"10_CR23","doi-asserted-by":"publisher","unstructured":"Pimentel, E., Pereira, L.C.: A tour on ecumenical systems (invited talk). In: Baldan, P., de\u00a0Paiva, V. (eds.) 10th Conference on Algebra and Coalgebra in Computer Science, CALCO 2023, 19\u201321 June 2023, Indiana University Bloomington, IN, USA. LIPIcs, vol.\u00a0270, pp. 3:1\u20133:15. Schloss Dagstuhl - Leibniz-Zentrum f\u00fcr Informatik (2023). https:\/\/doi.org\/10.4230\/LIPICS.CALCO.2023.3","DOI":"10.4230\/LIPICS.CALCO.2023.3"},{"issue":"5","key":"10_CR24","first-page":"815","volume":"3","author":"D Skurt","year":"2016","unstructured":"Skurt, D., Omori, H.: More modal semantics without possible worlds. FLAP 3(5), 815\u2013846 (2016)","journal-title":"FLAP"},{"key":"10_CR25","unstructured":"W\u00f3jcicki, R.: Lectures on Propositional Calculi. Pub. House of the Polish Academy of Sciences, Ossolineum [Poland] (1984)"}],"container-title":["Lecture Notes in Computer Science","Automated Reasoning with Analytic Tableaux and Related Methods"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-032-06085-3_10","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,27]],"date-time":"2025-09-27T10:45:24Z","timestamp":1758969924000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-032-06085-3_10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,25]]},"ISBN":["9783032060846","9783032060853"],"references-count":25,"URL":"https:\/\/doi.org\/10.1007\/978-3-032-06085-3_10","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,25]]},"assertion":[{"value":"25 September 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"TABLEAUX","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Automated Reasoning with Analytic Tableaux and Related Methods","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Reykjavik","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Iceland","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2025","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"27 September 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"29 September 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"34","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"tableaux2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/icetcs.github.io\/frocos-itp-tableaux25\/tableaux\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}