Table Of ContentTrends in Logic 44
Johannes Stern
Toward
Predicate
Approaches
to Modality
Trends in Logic
Volume 44
TRENDS IN LOGIC
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VOLUME 44
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Johannes Stern
Toward Predicate
Approaches to Modality
123
JohannesStern
MunichCenterforMathematicalPhilosophy,
FakultätfürPhilosophie,Wissenschaftstheorie
undReligionswissenschaften
Ludwig-Maximilians-UniversitätMünchen
München
Germany
ISSN 1572-6126 ISSN 2212-7313 (electronic)
Trends inLogic
ISBN978-3-319-22556-2 ISBN978-3-319-22557-9 (eBook)
DOI 10.1007/978-3-319-22557-9
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Acknowledgments
This monograph is a revised version of my doctoral thesis I defended at the
University of Geneva and for which I received the Paul Bernays Award by the
Swiss Society of Logic and Philosophy of Science (SSLPS). I was supervised by
FabriceCorreiaandKarl-Georg Niebergall,andIthankbothfortheirsupportover
the years.
IbenefitedalotfromthehelpfuldiscussionsIenjoyedwithVolkerHalbachwho
was kind enough to welcome me in Oxford for a nine-month research stay.
I thank Kevin Mulligan for his support during my years in Geneva and his
always refreshing views on philosophical questions.
Leon Horsten and Hannes Leitgeb provided very useful comments on my
doctoral dissertation, which helped me in preparing the present monograph.
I owe a lot to my friends and colleagues at Geneva, Oxford, Munich and else-
wherefortheirphilosophicalandnon-philosophicalsupport.Iwouldespeciallylike
to thank Martin Fischer for his help. Martin also provided detailed comments on
drafts of this work. Finally, I want to thank my parents for their support and, of
course, Catrin.
On the more institutional level my research was made possible by a three-year
Ph.D. scholarship within the Pro*Doc “Mind, Normativity, Self and Properties”
funded by the Swiss National Science Foundation (SNF) at the University of
Geneva. After these three years I benefited from a two-month Ph.D. scholarship
awarded by the “Faculté des Lettres” of the University of Geneva, before leaving
for Oxford funded by a nine-month SNF prospective researcher fellowship. In
November 2011 I joined the Munich Center for Mathematical Philosophy. Since
thenmyresearchwassupportedbytheAlexandervonHumboldtFoundationanda
research project on “Syntactical Treatments of Interacting Modalities” funded by
the German Research Foundation (DFG).
Partsofthematerialpresentedinthismonographhavebeenpublishedasarticles
invariousjournalsandvolumes,andarereprintedwithpermission.Letmemention
the relevant articles and how they overlap with the present monograph: Chap. 2,
Sect. 2.2. entitled “Montague’s Theorem and Modal Logic” has been published,
v
vi Acknowledgments
slightlymodified,underthesamenameinErkenntnis,79(3):551–570,2014.There
is also some overlap between the paper “The Truth about Predicates and
Connectives” published in A. Reboul (ed.), Mind, Values, and Metaphysics.
Philosophical Essays in Honor of Kevin Mulligan—Volume 1, Springer, 2014 and
Chap. 2, Sect. 2.3.2.
A variant of Chap. 3, Sects. 3.1 and 3.2 is published as joint work with Martin
Fischer under the title “Paradoxes of Interaction?” in Journal of Philosophical
Logic,44(3):287–308,2015.However,thepublishedpaperdiffersfromthepresent
presentation of the material in that the central result is proved in a different way.
Finally,Chap.4isaslightlymoredetailedpresentationofthematerialpublished
inthetwopapers“ModalityandAxiomaticTheoriesofTruthI:Friedman-Sheard.”
and“ModalityandAxiomaticTheoriesofTruthII:Kripke-Feferman.”,whichboth
appeared in The Review of Symbolic Logic, 7(2).
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Modal Predicates and Language. . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Objects of Modal Ascriptions and Lewy’s Argument . . . . . . 7
1.3 Modal Predicates and De Re Modality . . . . . . . . . . . . . . . . . . . 11
1.4 Technical Prerequisites and Notation. . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Possible World Semantics . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 Arithmetic and Arithmetization . . . . . . . . . . . . . . . . . . . 15
1.4.3 Fixed-Points and Monotone Operations. . . . . . . . . . . . . . 20
2 Modality and Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 History and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Montague’s Theorem and Modal Logic. . . . . . . . . . . . . . . . . . . 31
2.2.1 A Classical Model for Truth . . . . . . . . . . . . . . . . . . . . . 35
2.2.2 Models for Modalities Conceived as Predicates . . . . . . . . 42
2.2.3 Adequacy of the Predicate Approach to Modalities. . . . . . 47
2.3 Operators and Predicates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3.1 Predicates and Quantification. . . . . . . . . . . . . . . . . . . . . 53
2.3.2 Operators, Quantification, and the Paradoxes
of Indirect Discourse. . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.3 Modalities, Reification and Self-reference . . . . . . . . . . . . 62
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 Consistencies and Inconsistencies in Modal Logic. . . . . . . . . . . . . . 69
3.1 Simple and Diagonal Modal Logic. . . . . . . . . . . . . . . . . . . . . . 70
3.1.1 Simple Modal Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1.2 Normal Modal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.1.3 Diagonal Modal Logic . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.1.4 Interpretations for DML . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 (In)Consistencies in Normal Modal Logic . . . . . . . . . . . . . . . . . 90
vii
viii Contents
3.3 (In)Consistencies in Non-normal Modal Logic. . . . . . . . . . . . . . 94
3.3.1 (In)Consistencies in Classical Modal Logic . . . . . . . . . . . 95
3.3.2 Closure and Non-normal Modal Logic . . . . . . . . . . . . . . 97
3.4 DML and ω-Inconsistency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.5 (In)Consistencies in Multimodal Logic . . . . . . . . . . . . . . . . . . . 106
3.5.1 Multimodal Logic and DML . . . . . . . . . . . . . . . . . . . . . 106
3.5.2 Inconsistencies in Multimodal Logic. . . . . . . . . . . . . . . . 107
3.5.3 Consistencies in Multimodal Logic. . . . . . . . . . . . . . . . . 113
4 Modality and Axiomatic Theories of Truth . . . . . . . . . . . . . . . . . . 121
4.1 Friedman-Sheard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.1.1 Modal Friedman-Sheard . . . . . . . . . . . . . . . . . . . . . . . . 128
4.1.2 Modal Operator Logic and MFS. . . . . . . . . . . . . . . . . . . 134
4.1.3 The Kripke Reduction and MFS. . . . . . . . . . . . . . . . . . . 137
4.1.4 Modal Revision Semantics and MFS. . . . . . . . . . . . . . . . 141
4.2 Kripke-Feferman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.2.1 Modal Extensions of Kripke-Feferman . . . . . . . . . . . . . . 151
4.2.2 Modal Operator Logic and Modal Extensions of KF. . . . . 154
4.2.3 Modal Fixed-Point Semantics and MKF . . . . . . . . . . . . . 155
4.2.4 The Kripke Reduction and MKF . . . . . . . . . . . . . . . . . . 166
4.3 MFS, MKF and Some Limitations of MKF . . . . . . . . . . . . . . . . 171
5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Chapter 1
Introduction
Inthisessaywestudypredicateapproachestomodality.Thatis,wewanttodiscuss
andexploretheviewaccordingtowhichmodalnotionsarebestconceivedaspredi-
catesapplicabletonamesofsentencesorpropositions.Thiscontraststowhatmight
betakentobethestandardview,atleastinlogic,thatconceivesofmodalnotionsas
sententialoperators.Asententialoperatorasopposedtoapredicatetakesasentence
asargumentandyieldsanewsentence.Ultimately,ourmainmotivationforexplor-
ingpredicateapproachestomodalityis,orsoweargue,thatthestandardoperator
approachestomodalityareunsatisfactoryfromanaturallanguageperspective.
Inanutshellwehavetworeasonsforourassessment.Ontheonehandwethinkthat
truthandthemodalnotionsareexpressionsofthesamegrammaticalcategoryand,
inabsenceofanargumenttothecontrary,shouldthereforebetreatedinauniform
way.Butasthingsare,truthisusuallytreatedasapredicateandthustheuniformity
requirementisviolated,ifmodalnotionsaretreatedasoperators.Ontheotherhand,
innaturallanguagewefrequentlyquantifyintotheargumentpositionofthemodal
notions, refer to other sentences and expressions and talk about substitutions of
theseexpressions.Wearguethattheoperatorapproachtomodality,asitisusually
conceived,doesnotprovidetheexpressiveresourcestoaptlyformalizetheseaspects
ofnaturallanguagewhichwetaketoplayanimportantrolewithinphilosophy.We
donotdenythattheoperatorapproachcanbeenrichedtothiseffect,butwedothink
thatitismorestraightforwardtoaccountfortheseaspectsofnaturallanguagewithin
apredicateapproachtomodality.
Optingforapredicateapproachtomodalityistantamounttoanalyzingthelogical
formofmodalascriptionstobeofsubject-predicateformandthisanalysismayand
hasbeencriticizedondifferentgrounds.Argumentsthatpurporttoshowthatmodal
ascriptionsarenotofsubject-predicateformmaystemfromthreedifferentareasand
theirinterfaces:syntax,semanticsandlogic.Argumentsfromsyntaxholdthatitis
grammatically confused to analyze the modal notion as a predicate whereas argu-
mentsfromsemanticsholdthattakingmodalascriptionstobeofsubject-predicate
formgivesafaultyaccountoftheircontentormeaning.Finally,argumentsfromlogic
usuallypurporttoshowthatnosatisfactoryformaltreatmentofthemodalnotionsis
possible,ifweoptforapredicateapproachtomodality.
©SpringerInternationalPublishingSwitzerland2016 1
J.Stern,TowardPredicateApproachestoModality,
TrendsinLogic44,DOI10.1007/978-3-319-22557-9_1
