Table Of ContentStudies in Universal Logic
Jean-Yves Béziau
Gianfranco Basti
Editors
The Square of
Opposition: A
Cornerstone
of Thought
Studies inUniversalLogic
SeriesEditor
Jean-YvesBéziau(FederalUniversityofRiodeJaneiroandBrazilianResearchCouncil,
RiodeJaneiro,Brazil)
EditorialBoardMembers
HajnalAndréka(HungarianAcademyofSciences,Budapest,Hungary)
MarkBurgin(UniversityofCalifornia,LosAngeles,USA)
Ra˘zvanDiaconescu(RomanianAcademy,Bucharest,Romania)
JosepMariaFont(UniversityofBarcelona,Spain)
AndreasHerzig(CentreNationaldelaRechercheScientifique,Toulouse,France)
ArnoldKoslow(CityUniversityofNewYork,USA)
Jui-LinLee(NationalFormosaUniversity,HuweiTownship,Taiwan)
LarissaMaksimova(RussianAcademyofSciences,Novosibirsk,Russia)
GrzegorzMalinowski(UniversityofŁódz´,Poland)
DarkoSarenac(ColoradoStateUniversity,FortCollins,USA)
PeterSchröder-Heister(UniversityTübingen,Germany)
VladimirVasyukov(RussianAcademyofSciences,Moscow,Russia)
Thisseriesisdevotedtotheuniversalapproachtologicandthedevelopmentofageneral
theoryoflogics.Itcoverstopicssuchasglobalset-upsforfundamentaltheoremsoflogic
and frameworksforthe study of logics, in particularlogicalmatrices, Kripke structures,
combinationoflogics,categoricallogic,abstractprooftheory,consequenceoperators,and
algebraiclogic.Itincludesalsobookswithhistoricalandphilosophicaldiscussionsabout
the nature and scope of logic. Three types of books will appear in the series: graduate
textbooks,researchmonographs,andvolumeswithcontributedpapers.
Moreinformationaboutthisseriesathttp://www.springer.com/series/7391
Jean-Yves Béziau (cid:129) Gianfranco Basti
Editors
The Square of Opposition:
A Cornerstone of Thought
Editors
Jean-YvesBéziau GianfrancoBasti
ResearchCouncil FacultyofPhilosophy
UniversityofBrazil PontificalLateranUniversity
RiodeJaneiro,Brazil Rome,Italy
ISSN2297-0282 ISSN2297-0290 (electronic)
StudiesinUniversalLogic
ISBN978-3-319-45061-2 ISBN978-3-319-45062-9 (eBook)
DOI10.1007/978-3-319-45062-9
LibraryofCongressControlNumber:2016961939
MathematicsSubjectClassification(2010):03Axx
©SpringerInternationalPublishingSwitzerland2017
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Contents
PartI Introduction
TheSquareofOpposition:ACornerstoneofThought............................. 3
Jean-YvesBéziauandGianfrancoBasti
Welcome Address to the Participants of the IV International
Congresson:TheSquareofOppositionVaticanCity,PUL,May5–9,2014 ..... 13
MonsignorEnricoDalCovolo
PartII HistoricalPerspectivesontheSquare
TheProto-expositionofAristotelianCategoricalLogic ............................ 21
ManuelCorreia
TheModalOctagonandJohnBuridan’sModalOntology......................... 35
SpencerJohnston
From Aristotle’s Square of Opposition to the “Tri-unity’s
Concordance”:Cusanus’Non-classicalReasoning ................................. 53
AntoninoDrago
PartIII ReinterpretationsoftheSquare
SymmetricPropertiesoftheSyllogisticSystemInheritedfromthe
SquareofOpposition ................................................................... 81
Bora˙I.Kumova
TheSquareofOppositionInterpretedwithaDecidableModalLogic............ 105
PaulWeingartner
TwoStandardandTwoModalSquaresofOpposition.............................. 119
JiˇríRaclavský
v
vi Contents
PartIV PhilosophicalPerspectivesontheSquare
TheManyFacesofInconsistency ..................................................... 145
BobenriethM.Andrés
Aristotle,Fregeand“SecondNature”................................................ 169
RaffaelaGiovagnoliandPhilipLarrey
ThereIsNoCubeofOpposition....................................................... 179
Jean-YvesBéziau
PartV TheoreticalInvestigationsontheSquare
TheUnreasonableEffectivenessofBitstringsinLogicalGeometry............... 197
HansSmessaertandLorenzDemey
AnArithmetizationofLogicalOppositions.......................................... 215
FabienSchang
Groups,NotSquares:ExorcizingaFetish ........................................... 239
WalterCarnielli
PartVI ExpansionsandVariationsoftheSquare
FromtheSquaretoOctahedra........................................................ 253
JoséDavidGarcía-Cruz
IconicandDynamicModelstoRepresent“Distinctive”Predicates:
TheOctagonalPrismandtheComplexTetrahedronofOpposition.............. 273
FerdinandoCavaliere
TheExactIntuitionisticMeaningoftheSquareofOpposition.................... 291
JosephVidal-Rosset
PartVII ApplicationsoftheSquare
TheOntologicalModalCollapseasaCollapseoftheSquareofOpposition..... 307
ChristophBenzmüllerandBrunoWoltzenlogelPaleo
FuzzyEubouliaticLogic:AFuzzyVersionofAnderson’sLogicofPrudence ... 315
Gert-JanC.Lokhorst
Why Care beyondthe Square? Classicaland ExtendedShapes
ofOppositionsinTheirApplicationto“IntrospectiveDisputes”.................. 325
SaschaBenjaminFink
Part I
Introduction
The Square of Opposition: A Cornerstone
of Thought
Jean-YvesBéziau andGianfrancoBasti
Abstract We first describe how after having started in Montreux, Switzerland in 2007,
the congresson the squareof oppositionmovedto the AmericanUniversityof Beirutin
Lebanonin2012afterastopattheUniversityPasqualePaoliinCorsicain2010.Wethen
describe the square congress at the Pontifical Lateran University in the Vatican in 2014
andtheresultingpublications.
Keywords Fuzzylogic (cid:129) Interdisciplinarity (cid:129) Intuitionisticlogic (cid:129) Modallogic (cid:129) Para-
consistentlogic (cid:129) Squareofopposition (cid:129) Syllogistic
MathematicsSubjectClassification(2000) Primary00B25;Secondary00A66,03A05,
03B22,03B45;03B53
1 From Montreux to Beirut, via Corsica
The World Congress on the Square of Opposition—SQUARE—is an interdisciplinary
event.TheideaoftheSQUARE congressisto promoteinterdisciplinarityaroundavery
simpletheorythateverybodycanunderstand,developandapply.
ThefirstcongressonthesquareofoppositionwasorganizedinMontreux,Switzerland,
June1–3,2007,bythefirstauthorofthispaper(hereafterJYB)atthetimehewasworking
attheUniversityofNeuchâtel.NeuchâtelisthetownofJeanPiaget(1896–1980).Hewas
bornthereandhisfather,ArthurPiaget(1865–1952),wasthefirstRectoroftheUniversity
ofNeuchâtel.JeanPiagethadthespiritofresearchsincehewasquiteyoung.Hestarted
bystudyingthemollusksofLakeNeuchâtel.Hisstudiesthenevolveduptovertebrates,in
particularrationalanimals.Interestedinintelligencehenaturallyworkedinlogicincluding
thetheoryofopposition.Piagetwasastrongpromoterofinterdisciplinarity.Hecoinedthe
word“transdisciplinarity”,thathethoughtwasbetter,duringacongressin1970:
Finally,wehopetoseesucceeding tothestageofinterdisciplinaryrelationsofasuperior stage,
whichshouldbe“transdisciplinary”,i.e.whichwillnotbelimitedtorecognizetheinteractionsand
orreciprocitiesbetweenthespecializedresearches,butwhichwilllocatetheselinksinsideatotal
systemwithoutstableboundariesbetweenthedisciplines[29].
Inspired by Piaget, JYB organized two interdisciplinary events at the University of
Neuchâtel where he was working. One in 2005 on symbolic thinking and one in 2008
on imagination. The 1st SQUARE was however not organized in Neuchâtel but by the
©SpringerInternationalPublishingSwitzerland2017 3
J.-Y.Béziau,G.Basti(eds.),TheSquareofOpposition:ACornerstoneofThought,
StudiesinUniversalLogic,DOI10.1007/978-3-319-45062-9_1
4 J.-Y.BéziauandG.Basti
Fig.1 TerenceParsons—1stSQUAREinMontreuxin2007
banks of another lake, Lake Geneva, at the Hotel Helvétie in Montreux, a nice Hotel
wherewaspreviouslyorganizedthefirstWorldCongressandSchoolonUniversalLogic—
UNILOG—in 2005. Participantsof the 1st UNILOG liked very much this location so it
was decidedto organizeagainan eventthere.Moreoverthe SQUARE was developedin
the same spirit as UNILOG. In both cases the idea is not to construct a big totalitarian
system,buttopromoteexchangeofideasandopennessofminds.
The1stSQUAREgatheredpeoplefromallovertheworldfrommanydifferentfields.
AmongspeakerswerePascalEngel,JanWolen´ski,LaurenceHorn,PeterSchröder-Heister,
TerenceParsons(Fig.1),SieghardBeller,DagWesterståhl.Therewasasquarejazzshow
andtheprojectionofthemovieSalomé,aremakeofthebiblicalstorybasedonthesquare.
Fortheresultingpublicationssee[12]and[13].
The 2nd SQUARE was organized in Corte, at the University of Corsica Pasquale
Paoli,June17–20,2010.TherelationbetweenCorsicaandSwitzerlandisnotnecessarily
obvious. Let us however remember that Jean-Jacques Rousseau was asked to write a
constitution for Corsica and that JYB had lived in Corsica in his youth (cf. [4]). The
proposal to organize the event in Corsica was made at the final round square table in
Montreux by Pierre Simonnet, a computer scientist working at the University Pasquale
Paoli.Hesubsequentlysucceededtoconvincehiscolleaguesofthisuniversitytoorganize
theeventthere.Theygavetheirfullsupport,inparticularJean-FrançoisSantucci.
