Table Of ContentOutstanding 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 alge-
braic 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
