{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T02:14:27Z","timestamp":1790820867657,"version":"4.1.0"},"reference-count":103,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2019,7,4]],"date-time":"2019-07-04T00:00:00Z","timestamp":1562198400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2019,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), \u201cSyntactic aspects of modal incompleteness theorems,\u201d and a longstanding open question: whether every normal modal logic can be characterized by a class of<jats:italic>completely additive<\/jats:italic>modal algebras, or as we call them,<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1755020317000259_inline1\"\/><jats:tex-math>${\\cal V}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-<jats:sc>bao<\/jats:sc>s. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem\u2019s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1755020317000259_inline2\"\/><jats:tex-math>${\\cal V}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-<jats:sc>bao<\/jats:sc>s, namely the provability logic<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1755020317000259_inline3\"\/><jats:tex-math>$GLB$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>(Japaridze, 1988; Boolos, 1993). We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1755020317000259_inline4\"\/><jats:tex-math>${\\cal V}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1755020317000259_inline5\"\/><jats:tex-math>${\\cal V}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-incompleteness. In the end, we return to van Benthem\u2019s theme of syntactic aspects of modal incompleteness.<\/jats:p>","DOI":"10.1017\/s1755020317000259","type":"journal-article","created":{"date-parts":[[2019,7,3]],"date-time":"2019-07-03T10:53:42Z","timestamp":1562151222000},"page":"487-535","source":"Crossref","is-referenced-by-count":8,"title":["COMPLETE ADDITIVITY AND MODAL INCOMPLETENESS"],"prefix":"10.1017","volume":"12","author":[{"given":"WESLEY H.","family":"HOLLIDAY","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"TADEUSZ","family":"LITAK","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2019,7,4]]},"reference":[{"key":"S1755020317000259_ref14","unstructured":"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":"S1755020317000259_ref31","doi-asserted-by":"publisher","DOI":"10.2307\/2274183"},{"key":"S1755020317000259_ref58","doi-asserted-by":"publisher","DOI":"10.2307\/2586754"},{"key":"S1755020317000259_ref52","first-page":"239","volume-title":"Algebraic Methods in Logic and in Computer Science","volume":"Vol. 28","author":"Jipsen","year":"1993"},{"key":"S1755020317000259_ref50","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-006-8297-1"},{"key":"S1755020317000259_ref27","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/bxp004"},{"key":"S1755020317000259_ref49","doi-asserted-by":"publisher","DOI":"10.7551\/mitpress\/9055.001.0001"},{"key":"S1755020317000259_ref48","volume-title":"Introduction to Automata Theory, Languages, and Computation","author":"Hopcroft","year":"2003"},{"key":"S1755020317000259_ref42","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(96)00048-6"},{"key":"S1755020317000259_ref47","first-page":"367","volume-title":"Advances in Modal Logic","volume":"Vol. 12","author":"Holliday","year":"2018"},{"key":"S1755020317000259_ref5","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/978-94-017-8860-1_10","volume-title":"Leo Esakia on Duality in Modal and Intuitionistic Logics","volume":"Vol. 4","author":"Beklemishev","year":"2014"},{"key":"S1755020317000259_ref41","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)E0084-2"},{"key":"S1755020317000259_ref21","unstructured":"ten Cate B. & Litak T. (2007). The importance of being discrete. Technical Report PP-2007-39, Institute for Logic, Language and Computation, University of Amsterdam."},{"key":"S1755020317000259_ref40","doi-asserted-by":"publisher","DOI":"10.1007\/BF01054038"},{"key":"S1755020317000259_ref33","doi-asserted-by":"publisher","DOI":"10.1007\/BF02770513"},{"key":"S1755020317000259_ref83","first-page":"289","volume-title":"Advances in Modal Logic","volume":"Vol. 7","author":"Shapirovsky","year":"2008"},{"key":"S1755020317000259_ref29","doi-asserted-by":"publisher","DOI":"10.2168\/LMCS-2(1:5)2006"},{"key":"S1755020317000259_ref43","doi-asserted-by":"publisher","DOI":"10.1023\/A:1012491022267"},{"key":"S1755020317000259_ref26","first-page":"63","volume-title":"Advances in Modal Logic","volume":"Vol. 1","author":"Chagrova","year":"1998"},{"key":"S1755020317000259_ref24","doi-asserted-by":"publisher","DOI":"10.2307\/2275108"},{"key":"S1755020317000259_ref32","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-2807-2"},{"key":"S1755020317000259_ref23","first-page":"71","volume-title":"Advances in Modal Logic","volume":"Vol. 4","author":"Chagrov","year":"2003"},{"key":"S1755020317000259_ref36","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093891703"},{"key":"S1755020317000259_ref57","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004900300438"},{"key":"S1755020317000259_ref19","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19860321307"},{"key":"S1755020317000259_ref46","unstructured":"Holliday W. H. (2015). Possibility frames and forcing for modal logic. UC Berkeley Working Paper in Logic and the Methodology of Science. February 2018 version. Available at: https:\/\/escholarship.org\/uc\/item\/0tm6b30q."},{"key":"S1755020317000259_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/BF00649480"},{"key":"S1755020317000259_ref6","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1978.tb00830.x"},{"key":"S1755020317000259_ref13","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107050884"},{"key":"S1755020317000259_ref25","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198537793.001.0001","volume-title":"Modal Logic","author":"Chagrov","year":"1997"},{"key":"S1755020317000259_ref4","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1515\/9783110324907.135","volume-title":"Ways of Proof Theory","volume":"Vol. 2","author":"Beklemishev","year":"2010"},{"key":"S1755020317000259_ref39","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-009-9169-2"},{"key":"S1755020317000259_ref64","unstructured":"Litak T. (2005a). An Algebraic Approach to Incompleteness in Modal Logic. Ph.D. Thesis, Japan Advanced Institute of Science and Technology."},{"key":"S1755020317000259_ref69","first-page":"1","article-title":"Model theory and proof theory of coalgebraic predicate logic","volume":"14","author":"Litak","year":"2018","journal-title":"Logical Methods in Computer Science"},{"key":"S1755020317000259_ref20","first-page":"13","article-title":"A representation theorem for co-diagonalizable algebras","volume":"38","author":"Buszkowski","year":"2004","journal-title":"Reports on Mathematical Logic"},{"key":"S1755020317000259_ref96","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1052669066"},{"key":"S1755020317000259_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-73859-6_10"},{"key":"S1755020317000259_ref3","first-page":"1","volume-title":"Logic, Language, and Computation: 8th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2009, Bakuriani, Georgia, September 21\u201325, 2009. Revised Selected Papers","author":"Beklemishev","year":"2011"},{"key":"S1755020317000259_ref76","first-page":"1","article-title":"Decidability of second-order theories and automata on infinite trees","volume":"141","author":"Rabin","year":"1969","journal-title":"Transactions of the American Mathematical Society"},{"key":"S1755020317000259_ref68","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-31585-5_29"},{"key":"S1755020317000259_ref77","volume-title":"An Algebraic Approach to Non-classical Logics","author":"Rasiowa","year":"1974"},{"key":"S1755020317000259_ref22","doi-asserted-by":"publisher","DOI":"10.1007\/BF02215288"},{"key":"S1755020317000259_ref37","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-006-7196-9"},{"key":"S1755020317000259_ref54","doi-asserted-by":"publisher","DOI":"10.2307\/2372074"},{"key":"S1755020317000259_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-016-0401-4"},{"key":"S1755020317000259_ref45","first-page":"313","volume-title":"Advances in Modal Logic","volume":"Vol. 10","author":"Holliday","year":"2014"},{"key":"S1755020317000259_ref51","first-page":"16","volume-title":"Intensional Logics and the Logical Structure of Theories: Proceedings of the Fourth Soviet-Finnish Symposium on Logic, Telavi, May 1985","author":"Japaridze","year":"1988"},{"key":"S1755020317000259_ref28","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-06025-5_36"},{"key":"S1755020317000259_ref38","doi-asserted-by":"publisher","DOI":"10.1023\/A:1024621922509"},{"key":"S1755020317000259_ref8","volume-title":"Modal Logic and Classical Logic","author":"van Benthem","year":"1983"},{"key":"S1755020317000259_ref7","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1979.tb00794.x"},{"key":"S1755020317000259_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.indag.2019.01.001"},{"key":"S1755020317000259_ref60","first-page":"243","volume-title":"Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974)","volume":"Vol. 1","author":"Kuznetsov","year":"1975"},{"key":"S1755020317000259_ref16","volume-title":"Algebraizable Logics","volume":"no. 396","author":"Blok","year":"1989"},{"key":"S1755020317000259_ref63","doi-asserted-by":"publisher","DOI":"10.1023\/B:STUD.0000032102.67838.f2"},{"key":"S1755020317000259_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-0454-0_4"},{"key":"S1755020317000259_ref62","doi-asserted-by":"publisher","DOI":"10.1007\/BF00257488"},{"key":"S1755020317000259_ref35","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00076.x"},{"key":"S1755020317000259_ref44","doi-asserted-by":"publisher","DOI":"10.1016\/S1570-8683(03)00008-9"},{"key":"S1755020317000259_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-0452-6_3"},{"key":"S1755020317000259_ref15","doi-asserted-by":"publisher","DOI":"10.2307\/2273660"},{"key":"S1755020317000259_ref70","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-005-3702-8"},{"key":"S1755020317000259_ref74","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139084673"},{"key":"S1755020317000259_ref71","first-page":"30","article-title":"Untersuchungen \u00fcber den Aussagenkalk\u00fcl","volume":"23","author":"\u0141ukasiewicz","year":"1930","journal-title":"Comptes Rendus des s\u00e9ances de la Societ\u00e9 des Sciences et des Lettres de Varsovie"},{"key":"S1755020317000259_ref72","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093894226"},{"key":"S1755020317000259_ref73","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-007-9052-4"},{"key":"S1755020317000259_ref85","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757006"},{"key":"S1755020317000259_ref78","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-006-8296-2"},{"key":"S1755020317000259_ref67","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-008-2062-4"},{"key":"S1755020317000259_ref79","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1953-0053041-6"},{"key":"S1755020317000259_ref34","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":"S1755020317000259_ref81","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/exn096"},{"key":"S1755020317000259_ref82","volume-title":"An Essay in Classical Modal Logic","volume":"Vol. 13","author":"Segerberg","year":"1971"},{"key":"S1755020317000259_ref95","doi-asserted-by":"publisher","DOI":"10.4064\/-9-1-341-345"},{"key":"S1755020317000259_ref98","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-003-0180-4"},{"key":"S1755020317000259_ref56","doi-asserted-by":"publisher","DOI":"10.2307\/2275487"},{"key":"S1755020317000259_ref86","doi-asserted-by":"publisher","DOI":"10.2307\/2695098"},{"key":"S1755020317000259_ref87","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020310000201"},{"key":"S1755020317000259_ref101","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19960420113"},{"key":"S1755020317000259_ref88","volume-title":"Logic, Semantics, Metamathematics: Papers from 1923 to 1938","author":"Tarski","year":"1956"},{"key":"S1755020317000259_ref30","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/exs016"},{"key":"S1755020317000259_ref84","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581007"},{"key":"S1755020317000259_ref89","doi-asserted-by":"publisher","DOI":"10.2307\/2272558"},{"key":"S1755020317000259_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2014.07.005"},{"key":"S1755020317000259_ref59","doi-asserted-by":"publisher","DOI":"10.2168\/LMCS-8(3:14)2012"},{"key":"S1755020317000259_ref90","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00077.x"},{"key":"S1755020317000259_ref80","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2007.09.023"},{"key":"S1755020317000259_ref91","doi-asserted-by":"publisher","DOI":"10.2307\/2272895"},{"key":"S1755020317000259_ref92","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00674.x"},{"key":"S1755020317000259_ref93","doi-asserted-by":"publisher","DOI":"10.2307\/2272167"},{"key":"S1755020317000259_ref94","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19750210114"},{"key":"S1755020317000259_ref97","doi-asserted-by":"publisher","DOI":"10.1016\/S1570-2464(07)80009-7"},{"key":"S1755020317000259_ref99","unstructured":"Wolter F . (1993). Lattices of Modal Logics. Ph.D. Thesis, Fachbereich Mathematik, Freien Universit\u00e4t Berlin."},{"key":"S1755020317000259_ref100","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19960420140"},{"key":"S1755020317000259_ref102","first-page":"427","volume-title":"Handbook of Modal Logic","author":"Wolter","year":"2006"},{"key":"S1755020317000259_ref55","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(99)80001-9"},{"key":"S1755020317000259_ref66","first-page":"333","volume-title":"Advances in Modal Logic","volume":"Vol. 6","author":"Litak","year":"2006"},{"key":"S1755020317000259_ref75","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1145\/2893582.2893594","article-title":"Nominal techniques","volume":"3","author":"Pitts","year":"2016","journal-title":"ACM SIGLOG News"},{"key":"S1755020317000259_ref61","volume-title":"The \u201cLemmon Notes\u201d: An Introduction to Modal Logic","author":"Lemmon","year":"1977"},{"key":"S1755020317000259_ref17","volume-title":"The Logic of Provability","author":"Boolos","year":"1993"},{"key":"S1755020317000259_ref103","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-0454-0_2"},{"key":"S1755020317000259_ref65","first-page":"149","volume-title":"Advances in Modal Logic","volume":"Vol. 5","author":"Litak","year":"2005b"},{"key":"S1755020317000259_ref53","doi-asserted-by":"publisher","DOI":"10.2307\/2372123"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020317000259","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,20]],"date-time":"2024-07-20T15:37:17Z","timestamp":1721489837000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020317000259\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,4]]},"references-count":103,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,9]]}},"alternative-id":["S1755020317000259"],"URL":"https:\/\/doi.org\/10.1017\/s1755020317000259","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,7,4]]}}}