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Modality without reference [Elektronische Ressource] : an alternative semantics for substitutional quantified modal logic and its philosophical significance / vorgelegt von Bartosz Wiȩckowski

322 pages
MODALITY WITHOUT REFERENCEAn Alternative Semantics for SubstitutionalQuantified Modal Logic and its PhilosophicalSignificanceInaugural-Dissertationzur Erlangung des Grades einesDoktors der Philosophieder Fakult¨at fur¨ Philosophie und Geschichteder Eberhard Karls Universit¨at Tubingen¨vorgelegtvonBartosz WięckowskiausWrocław (Breslau)im September 2005Hauptberichterstatter: Prof. Dr. Peter Schroeder-HeisterMitberichterstatter: Prof. Dr. Thomas Ede ZimmermannDekan: Prof. Dr. Anton SchindlingTag der mundlic¨ hen Prufung:¨ 8. Februar 2006gedrucktmitGenehmigungderFakult¨atfur¨ PhilosophieundGeschichteder Universit¨at Tubing¨ en.herausgegeben im EigenverlagiiAbstractThisdissertationdevelopsasubstitutionalsemanticsforfirst-order(modal)logicwhich,unliketruth-valuesemantics,allowsafine-grainedanalysisofthesemantical behaviour of the terms and predicates from which atomic formu-lae are composed. Moreover, it proposes a nondenotational philosophicalfoundation for the semantics of substitutional quantified (modal) logic.Keywords: modality, predication, quantified modal logic, substitutionalquantification, substitutional semantics.iiiivFor Mariola & JurekvviContentsAbstract iiiPreface xiiiIntroduction xv1 Motivation 11.1 Denotational Semantics . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Nonmodal Denotational Semantics . . . . . . . . . . . 11.1.2 Modal Denotational Semantics . . . . . . . . . . . . . . 71.1.
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MODALITY WITHOUT REFERENCE
An Alternative Semantics for Substitutional
Quantified Modal Logic and its Philosophical
Significance
Inaugural-Dissertation
zur Erlangung des Grades eines
Doktors der Philosophie
der Fakult¨at fur¨ Philosophie und Geschichte
der Eberhard Karls Universit¨at Tubingen¨
vorgelegt
von
Bartosz Więckowski
aus
Wrocław (Breslau)
im September 2005Hauptberichterstatter: Prof. Dr. Peter Schroeder-Heister
Mitberichterstatter: Prof. Dr. Thomas Ede Zimmermann
Dekan: Prof. Dr. Anton Schindling
Tag der mundlic¨ hen Prufung:¨ 8. Februar 2006
gedrucktmitGenehmigungderFakult¨atfur¨ PhilosophieundGeschichte
der Universit¨at Tubing¨ en.
herausgegeben im Eigenverlag
iiAbstract
Thisdissertationdevelopsasubstitutionalsemanticsforfirst-order(modal)
logicwhich,unliketruth-valuesemantics,allowsafine-grainedanalysisofthe
semantical behaviour of the terms and predicates from which atomic formu-
lae are composed. Moreover, it proposes a nondenotational philosophical
foundation for the semantics of substitutional quantified (modal) logic.
Keywords: modality, predication, quantified modal logic, substitutional
quantification, substitutional semantics.
iiiivFor Mariola & Jurek
vviContents
Abstract iii
Preface xiii
Introduction xv
1 Motivation 1
1.1 Denotational Semantics . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Nonmodal Denotational Semantics . . . . . . . . . . . 1
1.1.2 Modal Denotational Semantics . . . . . . . . . . . . . . 7
1.1.3 Summary: Doubts about Denotational Semantics . . . 28
1.2 Substitutional Semantics . . . . . . . . . . . . . . . . . . . . . 29
1.2.1 Nonmodal Substitutional Semantics . . . . . . . . . . . 29
1.2.2 Modal Substitutional Semantics . . . . . . . . . . . . . 43
1.2.3 Summary: Doubts about Substitutional Semantics . . . 46
1.3 Mixed Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3.1 Nonmodal Mixed Semantics . . . . . . . . . . . . . . . 48
1.3.2 Modal Mixed Semantics . . . . . . . . . . . . . . . . . 54
1.3.3 Summary: Doubts about Mixed Semantics . . . . . . . 57
1.4 Concluding Remarks: Semantic Intuitions . . . . . . . . . . . 57
vii2 Associative Substitutional Semantics 63
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 Substitutional Language . . . . . . . . . . . . . . . . . . . . . 64
2.2.1 Basic Language L . . . . . . . . . . . . . . . . . . . . 640
2.2.2 Language L . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3 Associative Substitutional Semantics . . . . . . . . . . . . . . 67
2.3.1 Associative Substitutional Models . . . . . . . . . . . . 67
2.3.2 Associativenal Models: Comments . . . . . 69
2.3.3 Truth at an Index in a Model . . . . . . . . . . . . . . 71
2.3.4 Truth at an Index in a Model: Comments . . . . . . . 72
2.3.5 Truth-Conditions Theorem for L-Sentences . . . . . . . 76
2.3.6 Truth in a Model . . . . . . . . . . . . . . . . . . . . . 79
2.3.7 Validity in a Frame . . . . . . . . . . . . . . . . . . . . 79
2.3.8 Validity in a Frame: Illustrations . . . . . . . . . . . . 79
2.4 Associative Substitutional Semantics with Variable Assignments 84
2.4.1 Assignments to Nominal Variables . . . . . . . . . . . . 84
2.4.2 Truth at an Index in a Model with Variable Assignments 86
2.4.3 Truth at an Index in a Model with Variable Assign-
ments: Comments . . . . . . . . . . . . . . . . . . . . . 86
2.4.4 Truth-Conditions Theorem for L-Formulae . . . . . . . 88
2.4.5 Truth in a Model with Variable Assignments . . . . . . 88
2.4.6 Validity in a Frame with Variablets . . . . . 89
2.4.7 The Syntactical de nomine/de dicto Distinction . . . . 89
2.5 Axiom Systems with SBF . . . . . . . . . . . . . . . . . . . . 90
2.5.1 Substitutability . . . . . . . . . . . . . . . . . . . . . . 90
2.5.2 Relettering, Agreement, and Replacement . . . . . . . 90
2.5.3 Axiomatization . . . . . . . . . . . . . . . . . . . . . . 95
2.5.4 Soundness of Λ+SBF=¨ . . . . . . . . . . . . . . . . . . 97
2.5.5 of Λ+SBF=¨: Comments . . . . . . . . . . . 100
2.5.6 Completeness of Λ+SBF=¨ . . . . . . . . . . . . . . . . 102
2.5.7 The Model-Theoretic de nomine/de dicto Distinction . 107
viii2.6 Axiom Systems without SBF . . . . . . . . . . . . . . . . . . . 113
2.6.1 Axiomatization . . . . . . . . . . . . . . . . . . . . . . 113
2.6.2 Associative Semantics without SBF . . . . . . . . . . . 113
2.6.3 Soundness without SBF . . . . . . . . . . . . . . . . . 114
2.6.4 Completeness without SBF . . . . . . . . . . . . . . . 115
2.7 Axiom Systems without CSBF . . . . . . . . . . . . . . . . . . 118
2.7.1 Semantics without CSBF . . . . . . . . . . . . . . . . . 118
2.7.2 Semantics without CSBF: Comments . . . . . . . . . . 120
2.7.3 Axiomatization . . . . . . . . . . . . . . . . . . . . . . 121
2.7.4 Soundness without CSBF . . . . . . . . . . . . . . . . 122
2.7.5 Completeness without CSBF . . . . . . . . . . . . . . . 124
3 Constrained Associative Semantics 127
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.2 Admissibility Constraints . . . . . . . . . . . . . . . . . . . . . 127
3.2.1 Auxiliary Notions . . . . . . . . . . . . . . . . . . . . . 127
3.2.2 Notions: Comments . . . . . . . . . . . . . . 129
3.2.3 Admissible Assignments . . . . . . . . . . . . . . . . . 131
3.2.4 Admissible Assignments: Comments . . . . . . . . . . 132
3.2.5 Admissible Models . . . . . . . . . . . . . . . . . . . . 136
3.3 Asymmetrical Essential Relations . . . . . . . . . . . . . . . . 136
3.3.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . 137
3.3.2 A Proposal . . . . . . . . . . . . . . . . . . . . . . . . 139
3.3.3 A Note on Fine’s Essentialist Operator . . . . . . . . . 144
4 Applied Associative Semantics 149
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.2 Truth With Respect to the Level of Sense . . . . . . . . . . . 154
4.2.1 The Referential and the Nonreferential Mode of Eval-
uation . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.2.2 The Meanings of Names . . . . . . . . . . . . . . . . . 157
ix4.2.3 The Meanings of Names: Comments . . . . . . . . . . 162
4.2.4 The Meanings of Predicates . . . . . . . . . . . . . . . 165
4.2.5 The Sense-Extensions of Names . . . . . . . . . . . . . 165
4.2.6 The Sense of Names: Comments . . . . . . 170
4.2.7 The Sense-Extensions of Predicates . . . . . . . . . . . 172
4.2.8 The Sense ofes: Comments . . . . 174
4.2.9 The Level of Sense . . . . . . . . . . . . . . . . . . . . 176
4.2.10 Describing and Reflecting . . . . . . . . . . . . . . . . 177
4.2.11 Truth with Respect to the Level of Sense . . . . . . . . 178
4.2.12 From Sentences to Propositions . . . . . . . . . . . . . 180
4.2.13 Definitional Necessity . . . . . . . . . . . . . . . . . . . 182
4.3 Truth at an Index in an Associative Model and Truth With
Respect to the Level of Sense . . . . . . . . . . . . . . . . . . 185
∗4.3.1 Natural Language Proto-Interpretations of L . . . . . 188
∗4.3.2 Natural Language Proto-Interpretations of L : Nota-
tion and Explanations . . . . . . . . . . . . . . . . . . 189
∗4.3.3 NaturalLanguageProto-InterpretationsofL andSense-
Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.3.4 Representation via Bijections . . . . . . . . . . . . . . 191
4.3.5ntation via Bijections: Conventions . . . . . . . 192
4.3.6 Natural Language Interpretations . . . . . . . . . . . . 193
4.3.7 Truth With Respect to the Level of Sense Restated . . 195
4.3.8 A Chihara-Style Connecting Theorem . . . . . . . . . . 198
4.4 Truth With Respect to the Level of Sense and Truth With
Respect to the Level of Reference . . . . . . . . . . . . . . . . 200
4.4.1 The Semantics of a Substitutional First-Order Language200
4.4.2 The Semantics of a Denotational First-Order Language 202
4.4.3 The Level Connecting Theorem . . . . . . . . . . . . . 204
4.4.4 Modal Contexts and Modal Environments . . . . . . . 210
4.5 Some Philosophical Consequences . . . . . . . . . . . . . . . . 211
4.5.1 On Analysing Modality . . . . . . . . . . . . . . . . . . 212
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