The Logical Legacy of Nikolai Vasiliev and Modern Logic PDF
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Synthese Library 387 Studies in Epistemology, Logic, Methodology, and Philosophy of Science Vladimir Markin Dmitry Zaitsev E ditors The Logical Legacy of Nikolai Vasiliev and Modern Logic Synthese Library Studies in Epistemology, Logic, Methodology, and Philosophy of Science Volume 387 Editor-in-Chief OtávioBueno,UniversityofMiami,DepartmentofPhilosophy,USA EditorialBoard BeritBrogaard,UniversityofMiami,USA AnjanChakravartty,UniversityofNotreDame,USA StevenFrench,UniversityofLeeds,UK CatarinaDutilhNovaes,UniversityofGroningen,TheNetherlands The aim of Synthese Library is to provide a forum for the best current work in themethodologyandphilosophyofscienceandinepistemology.Awidevarietyof different approaches have traditionally been represented in the Library, and every effortismadetomaintainthisvariety,notforitsownsake,butbecausewebelieve thattherearemanyfruitfulandilluminatingapproachestothephilosophyofscience andrelateddisciplines. Specialattentionispaidtomethodologicalstudieswhichillustratetheinterplay of empirical and philosophical viewpoints and to contributions to the formal (logical,set-theoretical,mathematical,information-theoretical,decision-theoretical, etc.) methodology of empirical sciences. Likewise, the applications of logical methodstoepistemologyaswellasphilosophicallyandmethodologicallyrelevant studiesinlogicarestronglyencouraged.Theemphasisonlogicwillbetemperedby interestinthepsychological,historical,andsociologicalaspectsofscience. BesidesmonographsSyntheseLibrarypublishesthematicallyunifiedanthologies and edited volumes with a well-defined topical focus inside the aim and scope of the book series. The contributions in the volumes are expected to be focused and structurally organized in accordance with the central theme(s), and should be tied togetherbyanextensiveeditorialintroductionorsetofintroductionsifthevolume isdividedintoparts.Anextensivebibliographyandindexaremandatory. Moreinformationaboutthisseriesathttp://www.springer.com/series/6607 Vladimir Markin (cid:129) Dmitry Zaitsev Editors The Logical Legacy of Nikolai Vasiliev and Modern Logic 123 Editors VladimirMarkin DmitryZaitsev MSU,FacultyofPhilosophy MSU,FacultyofPhilosophy LomonosovMoscowStateUniversity LomonosovMoscowStateUniversity Moscow,Russia Moscow,Russia SyntheseLibrary ISBN978-3-319-66160-5 ISBN978-3-319-66162-9 (eBook) DOI10.1007/978-3-319-66162-9 LibraryofCongressControlNumber:2017954946 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This volume grew out of the international workshop “Nikolai Vasiliev’s Logical Legacy and the Modern Logic” (Moscow, Russia, October 2013) co-hosted by the Department of Logic of the Faculty of Philosophy, Lomonosov Moscow State University,andtheInstituteofLogic,CognitiveSciencesandDevelopmentofPer- sonality.Theworkshop focused ontheimpactof NikolaiVasiliev,theoutstanding Russianlogicianandphilosopher,onmodernlogicandmarkedthecentenaryofhis famouspaper“Imaginary(Non-Aristotelian)Logic”(1912). Nikolai Aleksandrovich Vasiliev (1880–1940), professor of Kazan University (Russia), was a seminal thinker, philosopher, rhyme maker, and logician, who outpaced his time and influenced the development of logical ideas far outside his own country (time and place). He is widely acknowledged for having pioneered logicwithaconsistentlogicalsystemintheformofsyllogisticstheorybasingupon acontradictoryontology,thushavingemergedasthefatherofparaconsistentlogic. What is more, though short (four papers published in 1910–1913), his scientific legacyunderpinsfruitfulideasofmodernlogic.Someresearchersalsotraceinhis writings elements of the many-valued and modal logic as well. All of the above compels one to regard Nikolai Vasiliev along with Jan Łukasiewicz as one of the precursorsofnonclassicallogic. It is his imaginary (non-Aristotelian) logic that attracts the most interest of modernresearches.SincetheseminalV.A.Smirnov’spaperandD.Comey’sreview onitintheJournalofSymbolicLogic,therehavebeenalotofattemptstopresent formally imaginary logic. A. Arruda proposed a number of propositional calcu- luses of paraconsistent type whose language contains atomic formulas expressing contradictory statements. Though none of these systems cannot be considered as anadequateformalizationofVasiliev’slogic(becausehisoriginalimaginarylogic wasconstructedasasyllogistictype),theyrepresentaninitiatorystepontheroad to further development of Vasiliev’s ideas in modern symbolic logic. The task to formallyreconstructVasiliev’soriginalsyllogisticlogicwassetbySovietlogician V.A.SmirnovwhostartedtheworkanalogoustothatexecutedbyŁukasiewiczw.r.t. Aristotelian syllogistics. He proposed topological interpretation of propositions in imaginarylogic.Insodoingheformulatedatranslationofimaginarylogicintofirst- v vi Preface order language enriched with operators of closure, interior, and complementation andproposedapossibleaxiomatizationofresultingtheoryasasyllogistictheoryof aspecialkindbuiltoverapredicatecalculus.ThisworkwasresumedbyV.Markin, T. Kostyuk, and D. Zaitsev. They considered all the variants of imaginary logic (the main version, investigated by Vasiliev in detail, and three “interpretations” of imaginarylogicwithdifferentsetsoflaws)andprovidedadequateaxiomatizations forthem.ItwasevidentthatVasilievwasnotonlyoneofthefirsttoproposealogical alternativetoclassicalonesbutthathealsopointedoutamanifoldofnonclassical logical systems, formulated in the same formal language but with different sets of laws.Itishisdoubtlessunrecognizedmerit. The logical legacy of Nikolai Vasiliev can serve as a promising source for developing an impressive range of philosophical interpretations, as it marries promising technical innovations with challenging philosophical insights. Equally importantisthefactthatthenoveltyofhislogicalprojecthasopenedupprospects for modern logic as well as for nonclassical science in general. Thus, a volume containingaselectionofpaperswrittenbyprominentmodernspecialistsinthefield dealingwithvariousaspectsofVasiliev’slogicalideasispresented. The editors take this occasion to acknowledge with gratitude the support of ProfessorandChairOtávioBuenooftheDepartmentofPhilosophyattheUniversity ofMiamiwhoisalsotheeditorinchiefofSynthese. Moscow,Russia DmitryZaitsev Moscow,Russia VladimirMarkin Contents 1 NikolaiA.Vasiliev,HisLifeandImaginaryLogicLegacy............. 1 ValentinA.Bazhanov 2 N.A. Vasiliev in the Context of Philosophical and Methodological Disputes of the Early Twentieth Century...................................................................... 9 G.V.Sorina 3 IsModernLogicNon-Aristotelian? ...................................... 19 Jean-YvesBeziau 4 VasilievandtheFoundationsofLogic.................................... 43 OtávioBueno 5 Vasiliev’sIdeasforNon-AristotelianLogics:InsightTowards Paraconsistency............................................................. 59 ItalaMariaLoffredoD’OttavianoandEvandroLuísGomes 6 Epicurus,VasilievandAristotleontheLawoftheExcludedThird.. 79 JoséVeríssimoTeixeiradaMata 7 ModernReconstructionofVasiliev’sLogicalSystems ................. 83 VladimirMarkin 8 NikolaiVasiliev’sImaginaryLogicandStableMeaning............... 97 IvanMikirtumov 9 VasilievandtheFoundationsofLogicalLaws .......................... 127 ElenaD.Smirnova 10 TheImpactofN.A.Vasiliev’sImaginaryLogiconEpistemic andRelevanceLogic........................................................ 135 WernerStelzner 11 GeneralizedVasiliev-StylePropositions.................................. 159 DmitryZaitsev vii viii Contents 12 PlurivalentLogics .......................................................... 169 GrahamPriest 13 OnSublogicsinVasilievFragmentoftheLogicDefinablewith A.Arruda’sCalculusV1................................................... 181 VladimirM.PopovandVasilyO.Shangin 14 Vasiliev’sCluetoMourdoukhay-Boltovskoy’sHypersyllogistic ...... 189 VladimirL.Vasyukov 15 ExperimentingwithConsistency ......................................... 199 JulianaBueno-SolerandWalterCarnielli Chapter 1 Nikolai A. Vasiliev, His Life and Imaginary Logic Legacy ValentinA.Bazhanov Abstract The paper deals with intellectual legacy of Kazan University professor N.A.Vasiliev(June29,1880–December31,1940),theforerunnerofmodernnon- classical logic. A thinker with a wide range of interests, he worked in poetry, philosophy, ethics, psychology, and history. N.A. Vasiliev’s path to imaginary logic is presented, along with the revival of his ideas in the early 1960s and their contemporarydevelopment. Keywords Imaginary logic (cid:129) Non-classical logic (cid:129) Paraconsistent logic (cid:129) Metalogic Atthebeginningofthetwentiethcentury,KazanUniversityprofessorofphilosophy NikolaiVasiliev(June29,1880–December31,1940)createdanovellogicalsystem thatbrokeradicallyawayfromAristotle’sthousand-yeartraditionofclassicallogic. Inhisyouth,N.A.Vasilievwasfondofpoetry.Asifanticipatingthefateofhis ownideas,heoncewrote: Weareaquicklydyingflame. Andagainaburningfire. Indeed,Vasiliev’sgroundbreakingworkinlogicfrom1910to1913established himasathinkerandanticipatedthedevelopmentofmanybranchesofmodernnon- classicallogic(see:Arruda1977,1980). Vasiliev’saccomplishmentsinlogicparallelthoseofLobachevskyingeometry. Lobachevsky’s ideas laid the foundation for non-Euclidean—and in that respect “non-classical”—geometry. Lobachevsky’s approach to geometry was called “imaginary”; Vasiliev’s considered his approach to logic also “imaginary.” Lobachevskyopenednewhorizonsforthedevelopmentofmathematics,asVasiliev didforformallogic. Despite Vasiliev’s half century of interesting work in psychology, ethics, phi- losophyofhistory,literarycriticism,andRussianSymbolistpoetry,hisworkwent V.A.Bazhanov((cid:2)) DepartmentofPhilosophyChairperson,UlyanovskStateUniversity,Ulyanovsk,Russia e-mail:[email protected] ©SpringerInternationalPublishingAG2017 1 V.Markin,D.Zaitsev(eds.),TheLogicalLegacyofNikolaiVasilievandModern Logic,SyntheseLibrary387,DOI10.1007/978-3-319-66162-9_1
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