{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:42:36Z","timestamp":1775839356997,"version":"3.50.1"},"reference-count":46,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T00:00:00Z","timestamp":1638403200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This article provides an algebraic study of the propositional system <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S175502032100054X_inline2.png\"\/><jats:tex-math>\n$\\mathtt {InqB}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of inquisitive logic. We also investigate the wider class of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S175502032100054X_inline3.png\"\/><jats:tex-math>\n$\\mathtt {DNA}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S175502032100054X_inline4.png\"\/><jats:tex-math>\n$\\mathtt {DNA}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-varieties. We prove that the lattice of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S175502032100054X_inline5.png\"\/><jats:tex-math>\n$\\mathtt {DNA}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-logics is dually isomorphic to the lattice of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S175502032100054X_inline6.png\"\/><jats:tex-math>\n$\\mathtt {DNA}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff\u2019s classic variety theorems. We also introduce locally finite <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S175502032100054X_inline7.png\"\/><jats:tex-math>\n$\\mathtt {DNA}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S175502032100054X_inline8.png\"\/><jats:tex-math>\n$\\mathtt {InqB}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is dually isomorphic to the ordinal <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S175502032100054X_inline9.png\"\/><jats:tex-math>\n$\\omega +1$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and give an axiomatisation of these logics via Jankov <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S175502032100054X_inline10.png\"\/><jats:tex-math>\n$\\mathtt {DNA}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].<jats:sup>1<\/jats:sup><\/jats:p>","DOI":"10.1017\/s175502032100054x","type":"journal-article","created":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T09:26:15Z","timestamp":1638437175000},"page":"950-990","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":6,"title":["AN ALGEBRAIC APPROACH TO INQUISITIVE AND -LOGICS"],"prefix":"10.1017","volume":"15","author":[{"given":"NICK","family":"BEZHANISHVILI","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1631-3648","authenticated-orcid":false,"given":"GIANLUCA","family":"GRILLETTI","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"DAVIDE EMILIO","family":"QUADRELLARO","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2021,12,2]]},"reference":[{"key":"S175502032100054X_r17","unstructured":"[17] de Jongh, D. (1968). Investigations on the Intuitionistic Propositional Calculus. PhD Thesis, University of Wisconsin."},{"key":"S175502032100054X_r43","doi-asserted-by":"publisher","DOI":"10.2307\/2274142"},{"key":"S175502032100054X_r23","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129519000203"},{"key":"S175502032100054X_r9","unstructured":"[9] Ciardelli, I. (2009). Inquisitive Semantics and Intermediate Logics. MSc Thesis, University of Amsterdam."},{"key":"S175502032100054X_r40","doi-asserted-by":"publisher","DOI":"10.1007\/s10849-015-9219-2"},{"key":"S175502032100054X_r15","unstructured":"[15] Citkin, A. (2014). Characteristic formulas 50 years later (an algebraic account). arXiv:1407.5823 [math.LO]."},{"key":"S175502032100054X_r28","unstructured":"[28] Ilin, J. (2018). Filtration Revisited: Lattices of Stable Non-Classical Logics. PhD Thesis, University of Amsterdam."},{"key":"S175502032100054X_r46","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2016.03.003"},{"key":"S175502032100054X_r4","unstructured":"[4] Bezhanishvili, N. (2006). Lattices of Intermediate and Cylindric Modal Logics. PhD Thesis, ILLC, University of Amsterdam."},{"key":"S175502032100054X_r14","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-010-9142-6"},{"key":"S175502032100054X_r11","unstructured":"[11] Ciardelli, I. (2016). Questions in Logic. PhD Thesis, University of Amsterdam."},{"key":"S175502032100054X_r45","first-page":"39","article-title":"Intermediate logics and the disjunction property","volume":"1","author":"Wronski","year":"1973","journal-title":"Reports on Mathematical Logic"},{"key":"S175502032100054X_r31","first-page":"806","article-title":"The construction of a sequence of strongly independent superintuitionistic propositional calculi","volume":"9","author":"Jankov","year":"1968","journal-title":"Soviet Mathematics Doklady"},{"key":"S175502032100054X_r8","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198537793.001.0001","volume-title":"Modal Logic","author":"Chagrov","year":"1997"},{"key":"S175502032100054X_r36","first-page":"227","article-title":"Finite problems","volume":"3","author":"Medvedev","year":"1962","journal-title":"Soviet Mathematics Doklady"},{"key":"S175502032100054X_r32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-76920-8_5"},{"key":"S175502032100054X_r34","doi-asserted-by":"publisher","DOI":"10.1007\/BF01881550"},{"key":"S175502032100054X_r13","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198814788.001.0001"},{"key":"S175502032100054X_r33","first-page":"229","article-title":"Eine Unableitbarkeitsbeweismethode F\u00fcr den Intuitionistischen Aussagenkalk\u00fcl","volume":"23","author":"Kreisel","year":"1958","journal-title":"Journal of Symbolic Logic"},{"key":"S175502032100054X_r27","unstructured":"[27] Groenendijk, J. , & Roelofsen, F. (2009). Inquisitive semantics and pragmatics. In Larrazabal, J. M. , & Zubeldia, L. , editors. Meaning, Content, and Argument: Proceedings of the ILCLI International Workshop on Semantics, Pragmatics, and Rhetoric. San Sebasti\u00e1n: Universidad del Pa\u00eds Vasco, Servicio Editorial, pp. 41\u201372."},{"key":"S175502032100054X_r1","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-008-9415-6"},{"key":"S175502032100054X_r19","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00081.x"},{"key":"S175502032100054X_r25","doi-asserted-by":"publisher","DOI":"10.1007\/BF00351052"},{"key":"S175502032100054X_r7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8130-3"},{"key":"S175502032100054X_r21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-52921-8_14"},{"key":"S175502032100054X_r39","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020319000017"},{"key":"S175502032100054X_r16","volume-title":"Introduction to Lattices and Orders","author":"Davey","year":"1990"},{"key":"S175502032100054X_r24","unstructured":"[24] Grilletti, G. , & Quadrellaro, D. E. (2020). Lattices of intermediate theories via Ruitenburg\u2019s theorem. arXiv:2004.00989 [math.LO]."},{"key":"S175502032100054X_r44","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-85820-8"},{"key":"S175502032100054X_r20","volume-title":"Abstract Algebraic Logic. An Introductory Textbook","author":"Font","year":"2016"},{"key":"S175502032100054X_r3","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-011-9348-9"},{"key":"S175502032100054X_r5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-59533-6_3"},{"key":"S175502032100054X_r18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-12096-2"},{"key":"S175502032100054X_r12","doi-asserted-by":"crossref","unstructured":"[12] Ciardelli, I. , Groenendijk, J. , & Roelofsen, F. (2011). Attention! Might in inquisitive semantics. In Cormany, E., Ito, S., and Lutz, D., editors. 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Joint PhD Thesis, University of Amsterdam."},{"key":"S175502032100054X_r37","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093635238"},{"key":"S175502032100054X_r6","volume-title":"Modal Logic","author":"Blackburn","year":"2002"},{"key":"S175502032100054X_r10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-31803-5_8"},{"key":"S175502032100054X_r22","volume-title":"Residuated Lattices: An Algebraic Glimpse at Substructural Logics","author":"Galatos","year":"2007"},{"key":"S175502032100054X_r30","doi-asserted-by":"publisher","DOI":"10.1070\/IM1968v002n05ABEH000690"},{"key":"S175502032100054X_r41","unstructured":"[41] Quadrellaro, D. E. 2019. Lattices of $DNA$ -Logics and Algebraic Semantics of Inquisitive Logic. 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