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Outstanding Contributions to Logic 15 Sergei Odintsov Editor Larisa Maksimova on Implication, Interpolation, and Definability Outstanding Contributions to Logic Volume 15 Editor-in-chief Sven Ove Hansson, Royal Institute of Technology, Stockholm, Sweden Editorial Board Marcus Kracht, Universität Bielefeld Lawrence Moss, Indiana University Sonja Smets, Universiteit van Amsterdam Heinrich Wansing, Ruhr-Universität Bochum More information about this series at http://www.springer.com/series/10033 Sergei Odintsov Editor Larisa Maksimova on Implication, Interpolation, fi and De nability 123 Editor SergeiOdintsov SobolevInstitute ofMathematics Novosibirsk Russia ISSN 2211-2758 ISSN 2211-2766 (electronic) Outstanding Contributions to Logic ISBN978-3-319-69916-5 ISBN978-3-319-69917-2 (eBook) https://doi.org/10.1007/978-3-319-69917-2 LibraryofCongressControlNumber:2017956323 ©SpringerInternationalPublishingAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Several scientific affairs in 2012 and 2013 were dedicated to Prof. Larisa Maksimova who celebrated her anniversary in November 2013. Among them are the special session on the conference Advances in Modal Logic 2012 (22–25 August 2012, Copenhagen, Denmark) and the conference Mal’tsev Meeting 2013 (12–15 August 2013, Novosibirsk, Russia). Mal’tsev Meeting is an annual con- ferenceofNovosibirkalgebraicandlogicalschool,anoutstandingrepresentativeof whichisLarisaMaksimova.ProfessorHeinrichWansing,amemberoftheeditorial board of the series Outstanding Contributions to Logic, participated Mal’tsev Meeting2013,wherehepresentedaninvitedtalk.Theideatopublishavolumeof thisseriesdevotedtoLarisaMaksimovabelongsnamelytoHeinrichWansing.The field of interests of Larisa Maksimova can be characterized as applied algebraic logic. She intensively used in her investigations various algebraic methods; establishedalgebraicanalogsofnumerouslogicalpropertiessuchasdifferentkinds of interpolation and definability properties, disjunction property; and carefully studied the relationship between relational and algebraic semantics for various kinds of logics. However, her investigations were aimed all the time at obtaining concrete logical results. In many cases, it was an exhaustive description of logics with one or another remarkable property ina big class of logics. The present book collects original contributions written by colleagues, collaborators, and former Ph. D.studentsofMaksimovathatarerelatedtooneoranothertopicofherresearch.It contains a “historical” part (Chaps. 1–3) including a scientific autobiography of ProfessorMaksimova,asurveyofherworksonrelevantlogics,andasurveyofher first famous results on classes of superintuitionistic logics and of normal modal logics extending S4. The rest of the book constitutes research papers as well as a few surveys. Novosibirsk, Russia Sergei Odintsov September 2017 v Acknowledgements I would like to thank Prof. Maksimova for her kind attention to my work on this book. As a customer of the seminar “Non-standard logics” in Novosibirsk State University,IamgratefultoProf.Maksimovaforcreatingandsupportingthisform of intellectual communication, which caused, in particular, the change of my field ofscientificinterestsfromtheclassicalcomputabilitytheorytonon-classicallogics. I would like to thank all authors of this volume for valuable contributions and all anonymous referees for useful reviews. My special thanks to Veta Yun for com- prisingthelistofpublicationsbyLarisaMaksimovaandtoSergeiDrobyshevichfor hisessentialhelpintheworkonthecamera-readycopyonthebook.Finally,Iam grateful to Heinrich Wansing for the suggestion to edit this volume. vii Contents 1 Maksimova,RelevanceandtheStudyofLatticesofNon-classical Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Sergei Odintsov 2 A Short Scientific Autobiography . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Larisa Maksimova 3 Larisa Maksimova’s Early Contributions to Relevance Logic. . . . . 33 Katalin Bimbó and J. Michael Dunn 4 A Note on Strictly Positive Logics and Word Rewriting Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Lev Beklemishev 5 Characteristic Formulas Over Intermediate Logics. . . . . . . . . . . . . 71 Alex Citkin 6 A Generalization of Maksimova’s Criterion for the Disjunction Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Alex Citkin 7 Rasiowa–Sikorski Sets and Forcing. . . . . . . . . . . . . . . . . . . . . . . . . 117 Janusz Czelakowski 8 l-Levels of Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Giovanna D’Agostino 9 Decidability of Some Interpolation Properties for Weakly Transitive Modal Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Anastasia Karpenko 10 Lattice NExtS4 from the Embedding Theorem Viewpoint . . . . . . . 185 Alexei Muravitsky ix x Contents 11 LinearTemporalLogicwithNon-transitiveTime,Algorithmsfor Decidability and Verification of Admissibility . . . . . . . . . . . . . . . . . 219 Vladimir V. Rybakov 12 Segerberg Squares of Modal Logics and Theories of Relation Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Valentin Shehtman 13 On Algebraisation of Superintuitionistic Predicate Logics. . . . . . . . 297 Dmitry Tishkovsky 14 Dummett Logic, Irreflexive Modality and Novikov Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Alexander Yashin 15 On Linear Logic of Knowledge and Time. . . . . . . . . . . . . . . . . . . . 339 Veta F. Yun Appendix A: Larisa Maksimova. List of Publications .. .... ..... .... 351 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 367 Contributors Lev Beklemishev Steklov Mathematical Institute, RAS, Moscow, Russia; Moscow M.V. Lomonosov State University, Moscow, Russia; National Research University Higher School of Economics, Moscow, Russia KatalinBimbó DepartmentofPhilosophy,UniversityofAlberta,Edmonton,AB, Canada Alex Citkin Metropolitan Telecommunications, Warren, NJ, USA; Metropolitan Telecommunications, New York, NY, USA Janusz Czelakowski Insitute of Mathematics and Informatics, Opole University, Opole, Poland GiovannaD’Agostino DepartmentofMathematicsandComputerScience,Udine, Italy J. Michael Dunn School of Informatics, Computing, and Engineering and Department of Philosophy, Indiana University, Bloomington, IN, USA Anastasia Karpenko Novosibirsk State University, Novosibirsk, Russia Larisa Maksimova Sobolev Institute of Mathematics, Novosibirsk, Russia Alexei Muravitsky Louisiana Scholars’ College, Northwestern State University, Natchitoches, LA, USA Sergei Odintsov Sobolev Institute of Mathematics, Novosibirsk, Russia VladimirV.Rybakov Institute ofMathematicsandInformatics, Siberian Federal University Krasnoyarsk, and (part time) Institute of Informatics Systems (Siberian Division RAS), Krasnoyarsk, Russia ValentinShehtman KharkevichInstituteforInformationTransmissionProblems, RAS,Moscow,Russia;NationalResearchUniversityHigherSchoolofEconomics, Moscow, Russia; Moscow State University, Moscow, Russia xi