Table Of ContentStudies in Universal Logic
Jean-Yves Beziau
Ioannis Vandoulakis
Editors
The Exoteric
Square
of Opposition
The Sixth World Congress on the Square
of Opposition
Studies in Universal Logic
SeriesEditor
Jean-YvesBeziau,FederalUniversityofRiodeJaneiro,RiodeJaneiro,Brazil
EditorialBoardMembers
HajnalAndréka,HungarianAcademyofSciences,Budapest,Hungary
MarkBurgin,UniversityofCalifornia,LosAngeles,CA,USA
Ra˘zvanDiaconescu,RomanianAcademy,Bucharest,Romania
RaffaelaGiovagnoli,PontificalLateranUniversity,Vatican
AndreasHerzig,UniversityPaulSabatier,Toulouse,France
ArnoldKoslow,CityUniversityofNewYork,NewYork,USA
Jui-LinLee,NationalFormosaUniversity,HuweiTownship,Taiwan
LarissaMaksimova,RussianAcademyofSciences,Novosibirsk,Russia
GrzegorzMalinowski,UniversityofLódz,Lódz,Poland
FrancescoPaoli,UniversityofCagliari,Cagliari,Italy
PeterSchröder-Heister,UniversityofTübingen,Tübingen,Germany
VladimirVasyukov,RussianAcademyofSciences,Moscow,Russia
AnnaZamansky,UniversityofHaifa,Haifa,Israel
Jean-Yves Beziau (cid:129) Ioannis Vandoulakis
Editors
The Exoteric Square
of Opposition
The Sixth World Congress on the Square
of Opposition
Editors
Jean-YvesBeziau IoannisVandoulakis
FederalUniversityofRiodeJaneiro TheHellenicOpenUniversity
RiodeJaneiro,Brazil Patras,Greece
ISSN2297-0282 ISSN2297-0290 (electronic)
StudiesinUniversalLogic
ISBN978-3-030-90822-5 ISBN978-3-030-90823-2 (eBook)
https://doi.org/10.1007/978-3-030-90823-2
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Preface
This book gathers 16 papers on the square of opposition, related to the 6th World
CongressontheSquareofOppositionwhichtookplaceattheOrthodoxAcademy
ofCrete,November1–5,2018.
The square of opposition is a logical structure coming from Aristotelian logic.
The topic has been continuously studied for 2000 years. Even Frege, one of the
foundersofmodernmathematicallogic,hascontributedtothetopic.
During the second half of the twentieth century, research on the topic was
revived. New extensions and generalization of the square of opposition were sug-
gested,farbeyondAristotle’straditionallogic.Theseadvanceswereaccompanied
by different configurations (triangle, rhombus, hexagon, octagon, polyhedra, and
multi-dimensionalobjects)thatwereconceivedtoillustratethenewtheories.
These theories found exciting applications in many fields, ranging from met-
alogic to highway code, through economics, music, physics, color theory, and
theology.
Theresearchinthisfieldisinterdisciplinarybutanchoredonaclearandprecise
logical theory, symbolized by the traditional diagram of the square of opposition.
Thetheoryofthesquareofoppositionisatthesametimeatraditionalandoriginal
topicthatincludednewinterpretationsofthepasttheories,discoveriesonthelogical
theory,andmodernapplicationsinavarietyofspheres.
Alltheseaspectsarepresentinthisbook,withchaptersdisplayingnovelaspects
of the theory of opposition and its variations that are embraced in a mainstream
researchprojectthatrevitalizesthetheoryofthesquareofopposition.
We thank all the contributors to this volume and the work of the referees who
havecriticallyanalyzedthepapers,leadingtoimprovedversions.
RiodeJaneiro,Brazil Jean-YvesBeziau
Patras,Greece IoannisVandoulakis
v
Contents
TheSquareofOpposition:Past,Present,andFuture ....................... 1
Jean-YvesBeziauandIoannisVandoulakis
Division of Entities and Foundations of Reality: Aristotle’s
OntologicalSquare............................................................... 15
GianluigiSegalerba
LogicalOppositionsinAvicenna’sHypotheticalLogic...................... 53
SalouaChatti
IncommensurabilityandInapplicabilityoftheSquaresofOpposition.... 89
SumantaSarathiSharma
TheSquareofOppositionasaFrameworkforStephenLangton’s
TheologicalSolutions ............................................................ 109
MarcinTrepczyn´ski
TheLimitsoftheSquare:Hegel’sOppositiontoDiagramsinIts
HistoricalContext................................................................ 123
ValentinPluder
AugustusDeMorgan’sUnpublishedOctagonofOpposition ............... 141
Anna-SophieHeinemannandLorenzDemey
ABitstringSemanticsforCalculusCL........................................ 171
JensLemanskiandFabienSchang
LogicalDiagrams,VisualizationCriteria,andBooleanAlgebras.......... 195
RolandBolz
TurnstileFiguresofOpposition................................................. 225
Jean-YvesBeziau
TheNaturalnessofJacquesLacan’sLogic.................................... 241
HubertMartinSchüler
vii
viii Contents
OnModalOppositionWithinSomeModalDiscussiveLogics.............. 259
KrystynaMruczek-NasieniewskaandMarekNasieniewski
OntheTransformationsoftheSquareofOppositionfromthe
PointofViewofInstitutionModelTheory.................................... 277
YiannisKiouvrekis,PetrosStefaneas,andIoannisVandoulakis
Color-CodedEpistemicModesinaJungianHexagonofOpposition...... 303
JulioMichaelStern
Many-ValuedLogicalHexagonsina3-OppositionalTrisimplex........... 333
RégisAngot-Pellissier
Tri-simplicialContradiction:The“Pascalian3DSimplex”for
theOppositionalTri-segment................................................... 347
AlessioMoretti
Contributors
RégisAngot-PellissierLycéeMontesquieu,LeMans,France
Jean-YvesBeziauFederalUniversityofRiodeJaneiro,RiodeJaneiro,Brazil
RolandBolzHumboldtUniversityBerlin,Berlin,Germany
SalouaChattiFSHST,UniversityofTunis,Tunis,Tunisia
LorenzDemeyCenterforLogicandPhilosophyofScience,KULeuven,Leuven,
Belgium
Anna-SophieHeinemannUniversitätPaderborn,Paderborn,Germany
YiannisKiouvrekisDepartmentofPublicandOneHealth,UniversityofThessaly,
Volos,GreeceandUniversityofNicosia,Nicosia,Cyprus
JensLemanski FernUniversität in Hagen, Institute for Philosophy LG, Hagen,
Germany
AlessioMoretti Université degli Studi eCampus, Novedrate, Italy and SSML di
Varese,Varese,Italy
KrystynaMruczek-Nasieniewska Department of Logic, Institute of Philosophy,
FacultyofPhilosophyandSocialSciences,NicolausCopernicusUniversity,Torun´,
Poland
MarekNasieniewski Department of Logic, Institute of Philosophy, Faculty of
PhilosophyandSocialSciences,NicolausCopernicusUniversity,Torun´,Poland
ValentinPluderUniversityofSiegen,Siegen,Germany
FabienSchangUniversidadeFederaldeGoiás,Goiânia,Brazil
HubertMartinSchülerUniversityofHagen,Hagen,Germany
ix
x Contributors
GianluigiSegalerba aka – Arbeitskreis Kulturanalyse Universität Wien(Working
Group Cultural Analysis University of Vienna), Vienna, Austria and International
MariinskayaAcademynamedafterM.D.Shapovalenko,Moscow,Russia
SumantaSarathiSharmaSchoolofPhilosophyandCulture,FacultyofHumani-
tiesandSocialSciences,ShriMataVaishnoDeviUniversity,JammuandKashmir,
India
PetrosStefaneas Department of Mathematics, National Technical University of
Athens,Athens,Greece
JulioMichaelStern IME-USP Institute of Mathematicsátics and Statisticsístics,
UniversityofSãoPaulo,SãoPaulo,Brazil
MarcinTrepczyn´ski Faculty of Philosophy, Uniwersytet Warszawski, University
ofWarsaw,Warsaw,Poland
IoannisVandoulakisTheHellenicOpenUniversity,Patras,Greece
