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Outstanding Contributions to Logic 23 Nikolaos Galatos Kazushige Terui Editors Hiroakira Ono on Substructural Logics Outstanding Contributions to Logic Volume 23 Editor-in-Chief SvenOveHansson,DivisionofPhilosophy,KTHRoyalInstituteofTechnology, Stockholm,Sweden Outstanding Contributions to Logic puts focus on important advances in modern logical research. Each volume is devoted to a major contribution by an eminent logician. The series will cover contributions to logic broadly conceived, including philosophicalandmathematicallogic,logicincomputerscience,andtheapplication oflogicinlinguistics,economics,psychology,andotherspecializedareasofstudy. AtypicalvolumeofOutstandingContributionstoLogiccontains: (cid:129) Ashortscientificautobiographybythelogiciantowhomthevolumeisdevoted (cid:129) Thevolumeeditor’sintroduction.Thisisasurveythatputsthelogician’scontribu- tionsincontext,discussesitsimportanceandshowshowitconnectswithrelated workbyotherscholars (cid:129) Themainpartofthebookwillconsistofaseriesofchaptersbydifferentscholars thatanalyze,developorconstructivelycriticizethelogician’swork (cid:129) Responsetothecomments,bythelogiciantowhomthevolumeisdevoted (cid:129) Abibliographyofthelogician’spublicationsOutstandingContributionstoLogic ispublishedbySpringeraspartoftheStudiaLogicaLibrary. Thisbookseries,isalsoasisterseriestoTrendsinLogicandLogicinAsia:Studia Logica Library. All books are published simultaneously in print and online. This bookseriesisindexedinSCOPUS. Proposalsfornewvolumesarewelcome.Theyshouldbesenttotheeditor-in-chief [email protected] Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/10033 · Nikolaos Galatos Kazushige Terui Editors Hiroakira Ono on Substructural Logics Editors NikolaosGalatos KazushigeTerui DepartmentofMathematics ResearchInstituteforMathematical UniversityofDenver Sciences Denver,CO,USA KyotoUniversity Kyoto,Japan ISSN2211-2758 ISSN2211-2766 (electronic) OutstandingContributionstoLogic ISBN978-3-030-76919-2 ISBN978-3-030-76920-8 (eBook) https://doi.org/10.1007/978-3-030-76920-8 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface ThisvolumeoftheseriesOutstandingContributionsisdedicatedtoHiroakiraOno. ProfessorOnohasbeenaninfluentialfigureinlogicformanyyearsandhiscontri- butions are long, deep and foundational. From the very beginning, in our initial discussions with Professor Ono, he made it very clear that he had a strong prefer- encethatthisisbookwouldnotbeacelebratoryvolumewherethecontributorspay tributetohisresultsintheareaorevenacollectionofpapersonsomeofthemany topicshehaspioneered.Instead,heconveyedtoushisstrongdesirethatthisvolume becomes a useful tool for graduate students, young researchers and researchers in neighboringfields;thatthisbookisinformativeandfacilitatesfutureresearch.We wereveryhappythatourcallforcontributionswasansweredpositivelybyleaders inthefieldandweareveryproudofthecontentsofthisvolume.Itcontainsamixof bothsurvey/introductorychaptersandalsooriginalpapersonimportanttopics.Our hopeisthatthisvolumewillcomeclosetoProfessorOno’svision. Professor Ono has worked on many areas of Mathematical Logic and his work hasbeenpioneeringandarea-forming.Inverygeneraltermshehasworkedininter- mediatelogics,modallogicsandsubstructurallogics.Hehasemployedbothproof- theoreticandsemanticaltools(algebraicandrelational)andheisoneoftheleading expertsinProofTheory.Hehaswritten7books,edited17volumesandauthored81 papers.Hehasdeliveredcountlessinfluentialtalksandhasservedcontinuouslyin numerousprogramcommitteesforconferences.Thevolumecontainsafullbibliog- raphyofhiscontributions.Italsocontainsanautobiographywritteninacaptivating and easy-to-read style. Both of the editors have been fortunate to have interacted extensivelywithProfessorOno;KazushigeTeruiisbasedinJapanandNickGalatos hasworkedasapostdoctoralresearcherinProfessorOno’sgroupandco-authored paperswithhim.Weareverygratefulforalltheinteractionswehadwithhim,both onaprofessionalandapersonallevel;hehasplayedaninfluentialroleinourcareers. ProfessorOnoisconsideredasoneofthefoundersofsubstructurallogics.Full Lambek calculus, usually formulated in sequent style, does not contain the three structural rules of intuitionistic logic (exchange, weakening and contraction), thus serving as the basic substructural logic; substructural logics are defined as exten- sionsofFullLambekcalculus.Theseincludeamongothersclassical,intuitionistic, relevance, linear and many-valued logic. The algebraic semantics of substructural v vi Preface logicsareresiduatedlatticesandtheyhaveanindependenthistoryanddevelopment. They include structures such as lattices of ideals of a ring, lattice ordered groups, algebras of relations, as well as Boolean, Heyting and MV-algebras. Substructural logicsadmitrelationalsemantics,whichhaveanamenableforminthedistributive case.Duetothevarietyofwaysofapproachingandstudyingsubstructurallogics, tools from various areas are employed in their study. The introductory chapters of thisvolumeprovideextensivesurveystotheseareas. Inparticular,thefirstchapter,contributedbyJamesRaftery,providesadetailed surveyofUniversalAlgebra,anindispensabletoolforthealgebraicstudyofsubstruc- tural logics. The survey starts with the basic notions of a variety of algebras, free algebras and the characterization of varieties as equational classes, as well as the crucial role of subdirectly irreducible algebras and the study of lattices of subva- rietiesand ofother class operators.Itquickly touches onmoreadvanced topics of particular importance to algebraic logic, such as (equationally) definable principal congruences,subtractivetermsandpointregularity,semisimplicity,amalgamation, finite embeddability property and the characterization of categorical equivalence amongothers.Thechapterseamlesslyincorporatesmoregeneraluniversal-algebraic notions and results, such as elements of commutator theory, finite basis theorems, the investigation of residual finiteness, many structure theorems and the extension ofresultstothecongruencemodularcase.Manyconclusionsofexistingresultsare giveninaconcisemannerandplethoraofreferencesisprovidedforfurtherreading. The general study of algebra-based semantics for logical systems, known as AbstractAlgebraicLogic,notonlyguidesandfacilitatestheinvestigationofsubstruc- turallogics,buttheconverseinteractionisalsobeneficial;forexample,substructural logicsprovidenaturalexamplesofparameterizedandlocaldeductiontheorems.The chaptercontributedbyJosep-MariaFontprovidesanoverviewofthegeneraltheory in a very gentle and accessible style. The reader is reminded of the completeness theorem,equationallogicandstandardalgebraizabilityandthenguidedintosucces- sivelyhigherlevelsofabstraction.ThisincludesalgebraizablelogicsviatheTarski- Lindenbaumprocess,butformulatedusinglogicalmatricesandintroducingtheLeib- nitzcongruence,thecaseofimplicativelogics,thenotionoforder-algebraizability, and the introduction of the Leibnitz hierarchy. Further the Suszko and the Frege operatoraredefinedandfinallyevenmoregeneralnotionsofsemanticsarediscussed. Thefirsttwochaptersputthestudyofsubstructurallogicinthecontextofgeneral algebraic logic. Relational-topological semantics form the other most prominent semanticaltoolusedinthestudyofsubstructurallogicsandtheyendupbeingthe (categorical) dual structures to thealgebraic ones. MaiGehrke provides a detailed account of this duality for lattices with residuated operators, presented in its algebraic incarnation in the spirit of Jónsson, Tarski and Birkhoff. The chapter gives a comprehensive account of completions in general, stratified by different density conditions,discussesindetailtherelationshipbetweentheintermediatestructureand the Dedekind-MacNeille completion (and the canonical extension), and explains the two prominent dualities for lattices in terms of the canonical extension. The chapterfurtherdiscussescanonicityandprovidesexamplesofcanonicalequationsfor residuatedlatticesandpreservationunderclassoperators;morespecializedsubjects Preface vii of interest to substructural logics, such as the relationship between closure under differentcompletions,arediscussed. Oneofthestrongforces,amongmany,drivingtheinterestinsubstructurallogics aretheconnectionstovariouspartsofcomputerscience.PeterJipsenandTadeusz Litak provide a survey of the applications of substructural logics to the theory of memory allocation and pointer arithmetic in computer science. In particular they introduceseparationlogicanditsHoarelogic,bunchedimplicationlogic.Thelatter isasubstructurallogicwithadistributivebaseandwithtwoimplications,oneintu- itionisticandtheothersubstructural.Thestudyismotivatedbyconcretemodelsused incomputerscience(forexample,theheapmodel)andtheKripke-styleframesare linkedtopartialalgebras,suchaseffectalgebras.Thechapterexpandsintothestudy ofsubvarietiesofthecorrespondingsemantics,anaccountforthedualitybetweenthe twotypesofsemantics,theproof-theoreticviewandquestionsabout(un)decidability ofthesystems.Theauthorsmanagetobringveryspecificstructuresusedincomputer sciencetoalogicalandmathematicalaudienceinaverysuccessfulway. Theseexpositorychapterssetthestagefortheoriginalcontributionspresentedin thesubsequentchapters.ConstantineTsinakisandJoseGil-Ferézextendthenotionof arecognizableformallanguagetothesettingofarecognizableelementofaresiduated lattice, using the notion of a module over a residuated lattice. The authors further characterizerecognizabilityintermsoftheimageofstructuralclosureoperatorsand exploreanumberofinterestingquestionsrelatedtorecognizabilityrelativetocertain typesofmodules. RostistavHorcˇík,alsoexploringthemesconnectingtoformallanguages,provides anovelcharacterizationofthefiniteembeddabilitypropertyforresiduatedlattices intermsofregularityconditionsofformallanguages,usingresiduatedframes.This characterization is used to derive both positive and negative results for concrete varietiesofresiduatedlattices. Robert Goldblatt expands on the notion of a modal FL-algebra introduced by HiroakiraOnoandrecastsitinthecontextofcoveringsemantics.Inparticular,he provesarepresentationtheoremthateverymodalFL-algebracanbeembeddedinto the FL-algebra of a strong modal cover system and the embedding preserves all existingjoinsandmeets.ThisembeddingisreminiscentoftheDedekind-MacNeille completionofanFL-algebra.Kripke-stylesemanticsareextractedandalsoapplied totheinvolutivecase. Nobu-YukiSuzukisolvesaproblemposedbyHiroakraOnomorethanaquarter- centruryago.Inparticularhegivesanegativeanswertowhethertheexistenceprop- erty(thatexistentialstatementsmustbewitnessed)impliesthedisjunctionproperty (thatdisjunctionsarewitnessed)forintermediatepredicatelogics.Healsoshowsthat, incontrast,forthemostnaturalintermediatelogics(includingtheonespossessing Kripkebasis)thequestionhasapositiveanswer. In the final contribution, Jacopo Amidei, Rodolpho Ertola-Biraben and Franco MontagnastudytheconservativityofexpansionsofFullLambekcalculuswithweak- eningandexchange.Inparticulartheyconsiderexpansionsobtainedbyaddingthe Baaz-Monteiro operator delta, the residuals of the lattice operations, propositional quantificationandthesuccessoroperator. viii Preface WehopethatthisvolumewillbeusefultoresearchersexactlyasProfessorOno envisioned. Denver,USA NikolaosGalatos Kyoto,Japan KazushigeTerui April2020 Contents AScientificAutobiography ......................................... 1 HiroakiraOno ExpositoryandSurveyChapters UniversalAlgebraicMethodsforNon-classicalLogics ................ 27 JamesG.Raftery AbstractAlgebraicLogic ........................................... 71 JosepMariaFont TopologicalDualityandAlgebraicCompletions ...................... 143 MaiGehrke AnAlgebraicGlimpseatBunchedImplicationsandSeparation Logic ............................................................. 185 PeterJipsenandTadeuszLitak SpecialTopics RecognizabilityinResiduatedLattices ............................... 245 JoséGil-FérezandConstantineTsinakis FiniteEmbeddabilityPropertyforResiduatedLatticesviaRegular Languages ........................................................ 273 RostislavHorcˇík CoverSystemsfortheModalitiesofLinearLogic ..................... 299 RobertGoldblatt A Negative Solution to Ono’s Problem P52: Existence andDisjunctionPropertiesinIntermediatePredicateLogics .......... 319 Nobu-YukiSuzuki ConservativeExpansionsofSubstructuralLogics .................... 339 JacopoAmidei,RodolfoC.Ertola-Biraben,andFrancoMontagna Bibliography ...................................................... 369 ix