Around and Beyond the Square of Opposition

391 Pages·2012·4.29 MB·English

by  Jean-Yves Béziau

StudiesinUniversalLogic
SeriesEditor
Jean-YvesBéziau(FederalUniversityofRiodeJaneiroandBrazilianResearchCouncil,
RiodeJaneiro,Brazil)
EditorialBoardMembers
HajnalAndréka(HungarianAcademyofSciences,Budapest,Hungary)
MarkBurgin(UniversityofCalifornia,LosAngeles,USA)
Ra˘zvanDiaconescu(RomanianAcademy,Bucharest,Romania)
JosepMariaFont(UniversityofBarcelona,Barcelona,Spain)
AndreasHerzig(CentreNationaldelaRechercheScientifique,Toulouse,France)
ArnoldKoslow(CityUniversityofNewYork,NewYork,USA)
Jui-LinLee(NationalFormosaUniversity,HuweiTownship,Taiwan)
LarissaMaksimova(RussianAcademyofSciences,Novosibirsk,Russia)
GrzegorzMalinowski(UniversityofŁódz´,Łódz´,Poland)
DarkoSarenac(ColoradoStateUniversity,FortCollins,USA)
PeterSchröder-Heister(UniversityTübingen,Tübingen,Germany)
VladimirVasyukov(RussianAcademyofSciences,Moscow,Russia)
Thisseriesisdevotedtotheuniversalapproachtologicandthedevelopmentofageneral
theoryoflogics.Itcoverstopicssuchasglobalset-upsforfundamentaltheoremsof
logicandframeworksforthestudyoflogics,inparticularlogicalmatrices,Kripke
structures,combinationoflogics,categoricallogic,abstractprooftheory,consequence
operators,andalgebraiclogic.Itincludesalsobookswithhistoricalandphilosophical
discussionsaboutthenatureandscopeoflogic.Threetypesofbookswillappearinthe
series:graduatetextbooks,researchmonographs,andvolumeswithcontributedpapers.
Jean-Yves Béziau (cid:2) Dale Jacquette
Editors
Around and Beyond
the Square of
Opposition
Editors
Jean-YvesBéziau DaleJacquette
InstituteofPhilosophy InstituteofPhilosophy
FederalUniversityofRiodeJaneiro UniversityofBern
RiodeJaneiro,RJ,Brazil Bern,Switzerland
ISBN978-3-0348-0378-6 ISBN978-3-0348-0379-3(eBook)
DOI10.1007/978-3-0348-0379-3
SpringerBaselHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012938884
MathematicsSubjectClassification: 03-XX,03Axx,03Bxx
©SpringerBasel2012
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Preface
Thetheoryofinferencesandoppositionsamongcategoricalpropositions,basedonAris-
totelian term logic, is pictured in a striking square diagram. The graphic representation
of contradictories, contraries, subcontraries and subalterns intended as a foundation for
syllogistic logic can be understood and applied in many different ways with interesting
implications for various disciplines, notably including epistemology, linguistics, math-
ematics, psychology. The square can also be generalized in other two-dimensional and
multi-dimensional graphic depictions of logical and other relations, extending in breath
anddepththeoriginalAristoteliantheory.Thesquareofoppositionisaccordinglyavery
attractivethemewhichhaspersisteddownthroughthecenturieswithnosignsofdisap-
pearing or even diminishing in fascination. For the last 10 years, there has been a new
growing interest for the square due to new discoveries and challenging interpretations.
This book presents a collection of previously unpublished papers by well-regarded spe-
cialists on the theory and interpretation of the concept and application of the square of
oppositionfromallovertheworld.Wethankalltheauthorswhohavecontributedapaper
to this book, and the referees who have analyzed, commented on, and made invaluable
recommendationsforimprovingtheessays.
RiodeJaneiro,Brazil Jean-YvesBéziau
Bern,Switzerland DaleJacquette
v

Contents
PartI HistoricalandCriticalAspectsoftheSquare
TheNewRisingoftheSquareofOpposition . . . . . . . . . . . . . . . . . . . 3
Jean-YvesBéziau
LogicalOppositionsinArabicLogic:AvicennaandAverroes . . . . . . . . . . 21
SalouaChatti
BoethiusontheSquareofOpposition . . . . . . . . . . . . . . . . . . . . . . . 41
ManuelCorreia
Leibniz,ModalLogicandPossibleWorldSemantics:TheApuleanSquareas
aProcrusteanBedforHisModalMetaphysics . . . . . . . . . . . . . . . . 53
Jean-PascalAlcantara
ThinkingOutsidetheSquareofOppositionBox . . . . . . . . . . . . . . . . . 73
DaleJacquette
JohnBuridan’sTheoryofConsequenceandHisOctagonsofOpposition . . . 93
StephenRead
WhytheFregean“SquareofOpposition”MattersforEpistemology . . . . . . 111
RaffaelaGiovagnoli
PartII PhilosophicalDiscussionAroundtheSquareofOpposition
TwoConceptsofOpposition,MultipleSquares . . . . . . . . . . . . . . . . . . 119
JohnT.Kearns
DoesaLeakingO-CornerSavetheSquare? . . . . . . . . . . . . . . . . . . . . 129
PieterA.M.Seuren
TheRightSquare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
HartleySlater
OppositionsandOpposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
FabienSchang
vii
viii Contents
PluralisminLogic:TheSquareofOpposition,Leibniz’PrincipleofSufficient
ReasonandMarkov’sPrinciple . . . . . . . . . . . . . . . . . . . . . . . . 175
AntoninoDrago
PartIII TheSquareofOppositionandNon-classicalLogics
TheSquareofOppositioninOrthomodularLogic . . . . . . . . . . . . . . . . 193
H.Freytes,C.deRonde,andG.Domenech
No Group of Opposition for Constructive Logics: The Intuitionistic and
LinearCases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
BaptisteMélès
TheSquareofOppositionandGeneralizedQuantifiers . . . . . . . . . . . . . 219
DuilioD’Alfonso
Privations,NegationsandtheSquare:BasicElementsofaLogicofPrivations 229
StamatiosGerogiorgakis
FuzzySyllogisms,NumericalSquare,TriangleofContraries,Inter-bivalence . 241
FerdinandoCavaliere
PartIV ConstructionsGeneralizingtheSquareofOpposition
GeneralPatternsofOppositionSquaresand2n-gons. . . . . . . . . . . . . . . 263
Ka-fatChow
The Cube Generalizing Aristotle’s Square in Logic of Determination of
Objects(LDO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Jean-PierreDesclésandAncaPascu
HypercubesofDuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
ThierryLibert
PartV ApplicationsoftheSquareofOpposition
HowtoSquareKnowledgeandBelief . . . . . . . . . . . . . . . . . . . . . . . 305
WolfgangLenzen
StructuresofOppositionsinPublicAnnouncementLogic . . . . . . . . . . . . 313
LorenzDemey
LogicalOppositionandCollectiveDecisions. . . . . . . . . . . . . . . . . . . . 341
Srec´koKovacˇ
AMetamathematicalModelforA/OOppositioninScientificInquiry . . . . . 357
MarkWeinstein