Quantifiers, Propositions and Identity: Admissible Semantics for Quantified Modal and Substructural Logics

283 Pages·2011·1.16 MB·English

by  Robert Goldblatt

Quantifiers,PropositionsandIdentity
ManysystemsofquantifiedmodallogiccannotbecharacterisedbyKripke’s
well-knownpossibleworldssemanticanalysis.Thisbookshowshowtheycanbe
characterisedbyamoregeneral‘admissiblesemantics’,usingmodelsinwhichthere
isarestrictiononwhichsetsofworldscountaspropositions.Thisrequiresanew
interpretationofquantifiersthattakesintoaccounttheadmissibilityofpropositions.
TheauthorshedsnewlightonthecelebratedBarcanFormula,whoserolebecomes
thatoflegitimisingtheKripkeaninterpretationofquantification.Thetheoryisworked
outforsystemswithquantifiersrangingoveractualobjects,andoverallpossibilia,
andforlogicswithexistenceandidentitypredicatesanddefinitedescriptions.The
finalchapterdevelopsanewadmissible‘coversemantics’forpropositionaland
quantifiedrelevantlogic,adaptingideasfromtheKripke-Joyalsemanticsfor
intuitionisticlogicintopostheory.
Thisbookisformathematicalorphilosophicallogicians,computerscientistsand
linguists.
ROBERT GOLDBLATTisProfessorofPureMathematicsattheVictoria
UniversityofWellington,NewZealand,andaFellowoftheRoyalSocietyofNew
Zealand.HehasservedastheCo-ordinatingEditorofTheJournalofSymbolLogic,
andhasbeenaManagingEditorofStudiaLogicaforthepasttwodecades.

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Quantifiers, Propositions
and Identity
AdmissibleSemanticsforQuantifiedModal
andSubstructuralLogics
ROBERTGOLDBLATT
VictoriaUniversityofWellington,
NewZealand
associationforsymboliclogic
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
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PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
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Informationonthistitle:www.cambridge.org/9781107010529
AssociationforSymbolicLogic
RichardA.Shore,Publisher
DepartmentofMathematics,CornellUniversity,Ithaca,NY14853
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(cid:2)c AssociationforSymbolicLogic2011
Thispublicationisincopyright.Subjecttostatutoryexception
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Firstpublished2011
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ISBN978-1-107-01052-9Hardback
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CONTENTS
IntroductionandOverview......................................... ix
Chapter1. LogicswithActualistQuantifiers.................... 1
1.1. Syntax.................................................... 1
1.2. Logics.................................................... 4
1.3. IncompletenessandAdmissibility.......................... 11
1.4. SomeHistoryoftheQuantifiers............................ 19
1.5. ModelStructures.......................................... 22
1.6. PremodelsandModels.................................... 25
1.7. Soundness................................................ 33
1.8. InfinitelyManyConstants................................. 43
1.9. CanonicalModelsandCompleteness....................... 45
1.10. CompletenessandCanonicityforQS....................... 53
1.11. KindsofIncompleteness................................... 59
1.11.1. IncompletenessforKripkeanModels....................... 59
1.11.2. KripkeanS-frameIncompleteness.......................... 61
1.11.3. Non-CanonicalS-frameIncompleteness.................... 63
Chapter2. TheBarcanFormulas................................. 67
2.1. LogicswithCBF.......................................... 68
2.2. ContractingDomainsforAll.............................. 70
2.3. ConstantDomainsforCBF............................... 74
2.4. OneUniversalDomain.................................... 79
2.5. TheDeductiveRoleofCommutingQuantifiers............. 84
2.6. CompletenesswithCBFandBF........................... 89
2.7. CompletenesswithUIandBF............................. 95
2.8. S-frameIncompletenessRevisited.......................... 98
Chapter3. TheExistencePredicate............................... 103
3.1. AxiomatisingExistence....................................104
3.2. CompletenessforKripkeanE-Models......................108
3.3. Necessityof(Non)Existence...............................116
3.4. IndependenceofBFfromNNE............................119
vii
viii Contents
3.5. WhatistheRoleoftheBarcanFormula?................... 122
Chapter4. PropositionalFunctionsandPredicateSubstitution. 127
4.1. FunctionalModelStructures.............................. 128
4.2. PredicateSubstitution: NotationandTerminology..........136
4.3. TheAnatomyofPredicateSubstitution.....................141
4.3.1. ChangingBoundVariables.................................141
4.3.2. SufficientFreeness.........................................141
4.3.3. Definingϕ((cid:3) /Px :P ∈Π)inGeneral...................142
P P
4.3.4. StronglyFreeSubstitution.................................143
4.3.5. ParameterlessSubstitution.................................145
4.4. SoundnessofPredicateSubstitution........................146
4.5. FunctionalCanonicity.....................................152
Chapter5. Identity................................................159
5.1. IntensionversusExtension.................................159
5.2. Syntax....................................................161
5.3. IdentityandRigidFunctions.............................. 163
5.4. ModelStructuresandModels..............................165
5.5. ValidityofSubstitutivityofIdenticals...................... 169
5.6. AxiomatisationandCompleteness......................... 176
5.7. KripkeanModels......................................... 187
5.8. DefiniteDescriptions......................................193
Chapter6. CoverSemanticsforRelevantLogic.................. 203
6.1. Routley-MeyerModelsforR.............................. 203
6.2. AdmissibleSemanticsforRQ.............................. 209
6.3. LocalTruthandCovers................................... 214
6.4. RelevantCoverSystems................................... 218
6.5. CoverSystemCompletenessforR..........................225
6.6. ModellingFusion......................................... 230
6.7. CoverSystemCharacterisationofRQ...................... 231
6.8. HeytingImplicationforFullSystems.......................239
6.9. LocalicCoverSystemsforHRandHRQ................... 245
References.......................................................... 251
Index................................................................261
INTRODUCTIONANDOVERVIEW
Thisbookisaboutthepossible-worldssemanticanalysisofsystemsoflogic
thathavequantifiersbindingindividualvariables. Ourapproachisbasedona
notionof“admissible”modelthatplacesarestrictiononwhichsetsofworlds
canserveaspropositions. Weshowthatadmissiblemodelsprovidesemantic
characterisationsofawiderangeoflogicalsystems,includingmanyforwhich
thewell-knownmodeltheoryofKripke[1963b]isincomplete. Thekeytothis
isaninterpretationofquantificationthattakesintoaccounttheadmissibility
ofpropositions.
Thisisasubjectthatbristleswithchoicesandchallenges. Shouldtermsbe
treated as rigid designators, or should their denotations vary from world to
world? Shouldindividualconstantsandvariablesbetreatedthesameinthis
respect, or differently? Should each world have its own domain of existing
individualsoverwhichthequantifiablevariablesrange,orshouldtherebejust
asingledomainofindividuals? Iftherearevaryingdomains,howshouldthey
relatetoeachother? Cananyfunctionfromworldstoindividualsberegarded
asthe“meaning”ofsomeindividualconcept? Shouldanarbitrarymapping
from individuals to propositions be admissible as a propositional function?
Canwedeductivelyaxiomatisetheclassofvalidformulasdeterminedbyeach
answertothesequestions?
One striking phenomenon arises in quantified modal logic: there are ax-
iomaticallydefinedlogicsthatcannotbecharacterisedbythekindofpossible-
worldssemanticsintroducedbyKripke, eventhoughthepropositionalfrag-
mentsofthoselogicsarecharacterisedbytheirmodelsasin[Kripke1963a].
Thesamecanhappenwithrelationalsemanticsfornon-modallogics,includ-
ingquantifiedrelevantlogic[Fine1989]andsomeextensionsofintuitionistic
first-order logic [Ono 1973]. This failure of the completeness under Kripke
semantics to lift from the propositional to the quantificational level raises a
natural question. Is there some similar but more general kind of relational
modeltheorythatcancharacterisethese“incomplete”logics?
We give a positive answer to that question, using two key ideas. The first
is to take seriously the view that, whereas any proposition can be identified
with a set of worlds—namely the set of all worlds in which it is true—the
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