The Logical Legacy of Nikolai Vasiliev and Modern Logic

221 Pages·2017·5.44 MB·English

by  Vladimir Markin

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
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BeritBrogaard,UniversityofMiami,USA
AnjanChakravartty,UniversityofNotreDame,USA
StevenFrench,UniversityofLeeds,UK
CatarinaDutilhNovaes,UniversityofGroningen,TheNetherlands
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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
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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