The Road to Universal Logic: Festschrift for the 50th Birthday of Jean-Yves Béziau Volume II

607 Pages·2015·9.836 MB·English

by  Arnold Koslow, Arthur Buchsbaum (eds.)

Studies in Universal Logic
Arnold Koslow
Arthur Buchsbaum
Editors
The Road
to Universal Logic
Festschrift for the 50th Birthday of Jean-Yves Béziau
Volume II

StudiesinUniversal Logic
SeriesEditor
Jean-YvesBéziau(FederalUniversityofRiodeJaneiroandBrazilianResearchCouncil,
RiodeJaneiro,Brazil)
EditorialBoardMembers
HajnalAndréka(HungarianAcademyofSciences,Budapest,Hungary)
MarkBurgin(UniversityofCalifornia,LosAngeles,USA)
RazvanDiaconescu(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
andframeworksforthestudyoflogics,inparticularlogicalmatrices,Kripkestructures,
combinationoflogics,categoricallogic,abstractprooftheory,consequenceoperators,
andalgebraiclogic. Itincludesalsobookswithhistoricalandphilosophicaldiscussions
aboutthenatureandscopeoflogic. Threetypesofbookswillappearintheseries:
graduatetextbooks,researchmonographs,andvolumeswithcontributedpapers.
(cid:2)
Arnold Koslow Arthur Buchsbaum
Editors
The Road to Universal Logic
Festschrift for the 50th Birthday
of Jean-Yves Beziau
Volume II
Editors
ArnoldKoslow ArthurBuchsbaum
CityUniversityofNewYork FederalUniversityofSantaCatarina
NewYork,USA Florianópolis,SC,Brazil
ISSN2297-0282 ISSN2297-0290(electronic)
StudiesinUniversalLogic
ISBN978-3-319-15367-4 ISBN978-3-319-15368-1(eBook)
DOI10.1007/978-3-319-15368-1
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014953260
MathematicsSubjectClassification(2010):03B22,03B42,03B62,03G30,03A05,01A99
Birkhäuser
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Preface
Thesetwovolumesgathertogetherthetributesofadistinguishedgroupofcolleaguesand
friendsinhonorofProfessorJean-YvesBeziauonhis50thbirthday.
The chapters in each of the two volumes (of which this is the second) fall, broadly
speaking,intofourcategories:
1. thoseconcernedwithuniversallogic;
2. thoseconcernedwithhexagonalandothergeometricaldiagramsofopposition;
3. thoseconcernedwithparaconsistency,and
4. currentworknotdirectlyconnectedtotheworkofJean-YvesBeziau.
With these contributed chapters, we want to express our gratitude for the intellectual
andorganizationalworkofJean-Yvesinuncoveringagoldentraditionoflogicalthought,
and his constant encouragement to all of us to ensure that tradition will continue and
flourish.Manythanks,Jean-Yves.Ourheartfeltthanksonyour50thbirthday.
Withthepossibleexceptionofthelastcategory,therearethreesubdivisionsofuniver-
sal logic as conceived by Jean-Yves Beziau. In order to understand this project, we can
donobetter thantorecallthewayin whichuniversallogicwascompactlydescribedby
Beziau in the preface to what is probably the defining collection on the subject,1 and to
expanduponit,briefly:
(i) BeyondParticularLogicalSystems “Universallogicisa generalstudy of logical
structures.Theideaistogobeyondparticularlogicalsystemstoclarifyfundamental
conceptsoflogicandtoconstructgeneralproofs.”(p.v)
(ii) ComparisonofLogics“Comparisonoflogicsisacentralfeatureofuniversallogic.”
(p.v)
(iii) Abstraction and the CentralNotion ofConsequence “Buttheabstraction rise is
not necessarily progressive, there are also some radical jumps into abstraction. In
logic, we find such jumps in the work of Paul Hertz on Satzsysteme (Part 1), and
of AlfredTarskionthenotionof aconsequenceoperator(Part3).Whatis primary
in these theories are not the notions of logical operators or logical constants (con-
nectivesandquantifiers),butamorefundamentalnotion:arelationofconsequence
1Beziau[2].
v
vi Preface
definedonundeterminedabstractobjectsthatcanbepropositionsofanyscience,but
alsodata,acts,events.”(p.vi)
(iv) BeyondSyntaxandSemantics“Inuniversallogic,consequenceisthecentralcon-
cept.Butthisconsequencerelationisneithersyntactical(proof-theoretical),norse-
mantical(model-theoretical).Wearebeyondthedichotomysyntax/semantics(proof
theory/modeltheory.”(p.vi)
ThereareofcourseotherthemesthatarecharacteristicofUniversalLogic,butitseems
evidentto usthatthefirst observation– (i) Beyondparticular LogicalSystems – indi-
catesclearlythatuniversallogicdoesnotadvocateauniquelogicalsystemthatistheone
correct, most expressive, accurate, and useful logical structure. Universal logic includes
in its domain a host of logical structures in all their variety. But universal logic is not
simply acatalog of alladvocated orimagined logicalstructures, alllogicalpossibilities,
asitwouldhavealltheutilityofatelephonebookthatisusefulforcertainproblems,but
cognitivelydumb.
Itisthesecondobservation–(ii)ComparisonofLogics–whichaddsintellectualcon-
tenttotheproject.Comparisonisindeedcentraltouniversallogic,butnotcomparisonsof
avalidkind.Whatisintendedarecomparisonsthatnotonlynotethedifferencebetween
logicalstructures,butexplanationsofwhytherearethosedifferencesinawaythatreveal
their differentlogicalcharacter. Thesecond observationsuggests that notonly arecom-
parisonsoffered,buttheremaybealsomanydifferentwaysoforderingthoselogics,and
onecannottakeforgrantedthatthoseorderingsorcomparisonsarecoherentwhentaken
together.Thiskindofissueisnicelyillustratedwhenwethinkofapapernowcommonly
referredtoas“Beziau’stranslationparadox”.2Simplyput,twologicalsystemsK (classi-
calpropositionallogic),andK=2aredescribed.Twoorderingsorrelationsareprovedto
hold:K isanextensionofK=2andalsothatthereisafaithfultranslationofK intoK=2.
Sotherearetwoorderings.ThefirstseemstoindicatethatK isclearlythestrongerlogic,
yetthesecondresultseemstosayotherwise(thatthereiswithinK=2afaithfultranslation
ofclassicalpropositionallogic).Eachofthetwoorderingsseemstomeasurethestrength
ofonelogicoveranother.AccordingthentoBeziau’sconceptofuniversallogic,compar-
isonsareacentraltask,butitisalsoataskofuniversallogictofigureoutwhattodowhen
the orderings seem to go in different directions. Beziau has suggested that it is like the
so-calledGalileanparadox,whichnotesthattherearemoresquarenaturalnumbersthan
therearenaturalnumbers,andalsonotesthatthosetwocollectionsareevenlymatched.It
isnotthatGalileo’ssolutionisrecommendedfortheBeziauexample.Thatisnotapossi-
blewayout,sinceGalileothoughtthat,inthecaseofinfinitecollections,thenotionof“is
largerthan”justdoesn’tapply.Theintendedsimilarity,asweseeit,isthatinbothcases
therearetwowaysofexplainingthenotionofonecollectionhavingmoremembersthan
another, and one logic being more powerful than another. The two ways give opposing
verdicts,andtheresolutionofthissituation,Beziaumaintains,isataskthatliessquarely
withintheprovinceofuniversallogic.
WementionedthatthestudyofHexagonallogicsofoppositionfallssquarelywithinthe
provinceofuniversallogic,fortheyprovideagoodexampleoffinitelogicalsystems,with
2Beziau[1].
Preface vii
aspecifiedparticularimplicationrelationbetweentheirsentences(takenpairwise).Infact
thereisagrowingliteraturewhichconsidersconsequencerelationsonfinitegeometrical
arraysofdifferentdimension.Allbelongcomfortablywithintheprojectthatisuniversal
logic.
Wealsomentionedthatparaconsistentlogicsareincludedintheprogram.Thatshould
beobviousifoneconsidersthevariousconsequencerelationstobefoundinthatbranchof
logic.AlsoweneedtomentionthebeautifulstudiesofDovGabbayinwhichheproposed
the study of restrictive access logics as an alternative to paraconsistent logics that is an
extensionofclassicallogic.3
Theserestrictiveaccesslogicscanbedescribedbyusingasubstructuralconsequence
relation,wherethereisamodificationoftheGentzenstructuralconditionsonimplication.
It then becomes an interesting problem to see what features the logical operators have
willhaveasaconsequence.4Theexamplesofparaconsistentandrestrictivelogicsliewell
withintheprovinceofpresentdaylogic.
Incontrast,whatisinterestingandnovelisthatBeziau’sobservationsin(iv)Beyond
SyntaxandSemanticspermitstheextensionoftheprogrambeyondthemoretraditional
range of contemporary logical systems. As he stated, not only can we have the notion
of consequence for scientific propositions, and nonpropositional, nonsentential objects
including,data,acts,andevents,butwedonowaddpictures(perhapsmathematicaldia-
grams),andeventheepistemicnotionofstatesofbeliefforwhichconsequencerelations
exist,andthepossibilityoflogicaloperatorsactingonpicturesaswellasstatesofbelief.
We are concerned with consequence relations that are beyond the semantical or proof-
theoretical.
Thecaseforaconsequencerelationbetweenpictureshasrecentlybeenforcefullymade
byJanWesterhoff.Here,compactly,istheclaim:
“I will describe an implication relation between pictures. It is then possible to give
precisedefinitionsofconjunctions,disjunctions,negations,etc.ofpictures.Itwillturnout
thattheselogicaloperationsarecloselyrelated to,orevenidenticalwith basiccognitive
relationswenaturallyemploywhenthinkingaboutpictures.”5
Thisexamplewithitsparticularconsequencerelation,andthepicturesitrelates,isan
extensionwellbeyondtheusualrestrictionoflogictosyntaxandsemantics.Itillustrates
the broad implications of Beziau’s observations in (iv) and the fertility of the project of
universallogic.Itisnotbusinessasusual.
Finally,wewillbrieflydescribeanothercasePeterGärdenfors,6whodevelopedalogic
ofpropositionsonthebasisofatheoryaboutbeliefrevision.Hisresultscanberecastin
suchawaythattheyalsofollowasacasewherehedefinespropositionsasspecialkinds
of functions, and also defines a special relation among those functions that turns out to
beaconsequencerelation.Theresultisfascinating:theconjunctionoffunctionsturnsout
tobethefunctionalcompositionoffunctions,andGärdenfors’specialrelationamongthe
3GabbayandHunter[4].
4PrivatecommunicationfromD.Gabbay,2005.
5Westerhoff[6].TheimplicationrelationproposedforpicturesissimilartoonethatCorcoran[3]pro-
posedforpropositions,asnotedbyWesterhoff.
6Gärdenfors[5].
viii Preface
functionsisaconsequencerelationprovidedthatfunctionalcompositioniscommutative
andidempotent.
More specifically, (1) let S be a set of states of belief of some person. (2) Let P be
a set of functions from S to S (called propositions) which is closed under functional
composition.(3)Foranymembersf ;f ;:::;f andginP,let(G)betheconditionthat
1 2 n
f ;f ;:::;f )g ifandonlyif gf f :::f Df f :::f
1 2 n 1 2 n 1 2 n
(theconcatenationoftwofunctionshereindicatestheirfunctionalcomposition).
In particular,for anytwo propositions(functions)f and g, f implies g (f ) g) if
andonlyifgf D f.Itiseasytoprovethattherelation(G)isaconsequencecondition
ifandonlyiffunctionalcompositioniscommutativeandidempotent.Thelogicofthese
propositionshasbeenshownbyGärdenforstobeIntuitionistic,andhisconsequencerela-
tion(G)isclearlyepistemic.Again,itisnotlogicasusual,butitisjustonemorecaseof
thefruitfulnessoftheideasthattheprojectofuniversallogicembodies.
References
1. Beziau, J-Y.: Classical negation can be expressed by one of its halves. Log. J. IGPL
7(2),145–151(1999)
2. Beziau, J-Y.: Universal Logic: An Anthology, from Paul Hertz to Dov Gabbay.
Birkhäuser,Boston(2012)
3. Corcoran,J.: Information-theoreticlogic. In:Martinez, C. (ed.) Truthin Perspective,
pp.113–135.Ashgate,Aldershot(1998)
4. Gabbay,D.M.:Restrictiveaccesslogicsforinconsistentinformation.In:ECCSQARU.
LectureNotesinComputerScience,pp.137–144.Springer,Berlin(1993)
5. Gärdenfors, P.: The dynamics of belief as a basis for logic. Br. J. Philos. Sci. 1–10
(1984)
6. Westerhoff,J.:Logicalrelationsbetweenpictures.J.Philos.102(12),603–623(2005)
NewYork,USA ArnoldKoslow
Florianópolis,Brazil ArthurBuchsbaum
Contents
PersonalRecollectionsAboutJYBbyNewtondaCostaandOthers . . . . . . . 1
K.Gan-Krzywoszyn´ska
LogicalAutobiography50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
J.-Y.Béziau
AQuantitative-InformationalApproachtoLogicalConsequence . . . . . . . . . 105
M.A.AlvesandI.M.LoffredoD’Ottaviano
Finite-VariableLogicsDonotHaveWeakBethDefinabilityProperty . . . . . . 125
H.AndrékaandI.Németi
Peirce’sRoleintheHistoryofLogic:LinguaUniversalisand
CalculusRatiocinator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
I.H.Anellis
TheMeaning(s)of“Is”:Normativevs.NaturalisticViewsofLanguage . . . . . 171
I.Angelelli
TheRelationBetweenLogic,SetTheoryandToposTheoryasItIsUsed
byAlainBadiou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
R.Angot-Pellissier
PotentialityandContradictioninQuantumMechanics . . . . . . . . . . . . . . . . 201
J.R.BeckerArenhartandD.Krause
Two,Many,andDifferentlyMany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
D.Batens
LogicsandTheirGalaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
H.Bensusanetal.
ix