Philosophy of Logic (eBook)

Front Cover 1
Philosophy of Logic 4
Copyright Page 5
Table of Contents 8
General Preface 6
Preface 12
List of Contributors 14
Introduction: Philosophy of Logic Today 18
Logic and philosophy 18
In the logic candy store 19
Understanding the plurality of logics 22
Future directions in philosophical logic and philosophy of logic 26
What is Logic? 30
1 Logic and inference 30
2 Deductive inference and information 31
3 Analytic inferences 31
4 Ampliative inference 33
5 Logical vs. extralogical systematization 33
6 Syntax vs. semantics 34
7 The limits of syntactical approaches 35
8 The limits of semantical approaches 36
9 Strategic vs. definitory rules 37
10 What is first-order logic? 38
11 Presuppositions of the received first-order logic 39
12 The meaning of quantifiers 40
13 Semantical games 42
14 The implications of independence-friendly logic: Axiomatizability 43
15 Negation and its significance 44
16 Higher-order logics 45
17 A distinction without ontological difference? 47
18 Alternative logics? 48
19 Ampliative inference and acquisition of information 49
Bibliography 51
The Scope and Limits of Logic 58
1 The broad sweep 58
2 Definitions of logic 59
3 Concepts that belong to logic 60
4 The syllogism claim 61
5 Patterns of reasoning 64
6 Reduction to logic 67
7 Quine’s ‘scope of logic’ 69
8 Logical notions 72
9 How we ought to think? 74
Bibliography 78
Logic in Philosophy 82
1 The century that was 82
2 A joy-ride through history 85
3 Logical form and natural logic: From syllogistics to generalized quantifiers 89
4 Reasoning styles: From bolzano to conditional logic and AI 93
5 Mechanisms of semantic interpretation: From tarski to ‘dynamics’ 97
6 Theories and belief revision: From neo-positivism to computer science 102
7 Dynamic logic, communication, action, and games 105
8 Further themes 110
9 Conclusion: Logic and philosophy once more 111
Acknowledgements 111
Bibliography 112
Informal Logic and The Concept of Argument 118
Technical and everyday senses of “argument” 118
Argument as discourse supporting a point of view by offering one or more reasons 120
Arguments as invitations to inference 122
Extensions: Potential arguments and equivalence classes of arguments 124
Complex direct arguments 125
Suppositional arguments 127
First summary 131
Competing conceptions of argument 133
Exclusions from the class of arguments 137
Second summary 138
Postscript 138
Acknowledgements 144
Bibliography 144
On the Relation of Informal to Symbolic Logic 148
1 A strange dichotomy 148
2 Toward a definition of formal and informal logic 150
3 Formal and informal logic in partnership 153
4 Conclusion 169
Acknowledgements 170
Bibliography 170
Vagueness and the Logic of Ordinary Language 172
1 The epistemic diagnosis 172
2 Does logic apply to the ordinary world? 173
3 Vagueness in the platonic heavens 174
4 Does logic apply to ordinary language? 176
5 Vagueness as a pre-logical phenomenon 176
6 Does logic only apply loosely? 179
7 Truth-value gaps 180
8 Logical pluralism 181
9 Deviant logic 182
10 Many-valued logic 183
11 Family resemblance 184
12 Prototypes 187
Bibliography 188
Logic and Semantic Analysis 190
1 Introduction 190
2 Some classic arguments 192
3 Doing semantic analysis by analogy to logic 200
4 Doing semantic analysis in logic 210
5 Open problems and further reading 218
Bibliography 219
Justificatory Irrelevance of Formal Semantics 222
1 Background and overview 222
2 The problem of justifying a formal logic 224
3 System E of entailment 225
4 On use and interpretation of E-model structures - Ems 227
5 On world-line matrices for two and three “points” 235
Bibliography 243
A Brief History of Truth 244
Brief introduction 244
1 Early views 245
2 Middle views 271
3 Later views 296
4 Brief conclusion 331
Bibliography 333
Truth and Paradox: A Philosophical Sketch 342
1 Aims and structure of this essay 342
2 Dtruth and truth 343
3 Liars: Broad picture and projects 345
4 Paracomplete 346
5 Paraconsistent 383
6 Dtruth, validity, and truth-preservation 394
7 Leaving dtruth: Parameters and contextual shifts 398
8 Revision theory 409
9 Set-theoretic paradoxes? 413
10 Revenge –– At last 415
11 Further reading 419
Acknowledgements 422
Bibliography 423
Hilbert’s Program Then and Now 428
1 Introduction 428
2 Hilbert’s program then 429
3 Philosophical interpretation of Hilbert’s program 436
4 Hilbert’s program now 450
5 Conclusion 457
Acknowledgements 458
Bibliography 458
Logicism and its Contemporary Legacy 466
1 Frege 469
2 Context, number and neo-fregean logicism 475
3 Russell’s logicism and the “no-class” theory 488
4 Carnap’s internal logicism 490
5 Contemporary criticisms of logicism 494
6 Structuralist alternatives to logicism 500
7 Logicism and the theory of types 504
Bibliography 511
Classical Logic’s Coming of Age 514
1 Introduction 514
2 The rise to ascendency of first-order logic 515
3 Semantic decidability and semantic completeness 517
4 Formally undecidable statements, the undefinability of truth, and the existence of nonstandard models 520
5 The second incompleteness theorem, and later extensions of G& ouml
6 Consistency proofs in the wake of the second incompleteness theorem 525
7 The emergence of recursion theory 527
8 The relative consistency of the axiom of choice and the generalized continuum hypothesis 533
9 Culmination and codification 536
Bibliography 536
Infinity 540
1 Introduction 540
2 Actual infinity 543
3 Potential infinity 563
4 Physical infinity 581
Bibliography 596
L& ouml
1 Introduction 604
2 The L& ouml
3 L& ouml
4 From “very large” to “very small” 613
5 Building cardinalities into the logic 615
6 Transfer in general logic systems 618
7 General aspects 624
8 Transfer in the finite 627
Acknowledgements 629
Bibliography 629
The Mathematics of Skolem’s Paradox 632
1 Skolem’s Paradox 633
2 A quick technical solution 644
3 The virtues of quantification 648
4 The vices of quantification I 653
5 The vices of quantification II 656
6 A few concluding remarks 660
Bibliography 665
Objectual and Substitutional Interpretations of the Quantifiers 666
Introduction 666
Quine and ontological commitment 669
Referential and nonreferential base semantics 671
The “substitutional strategy” 673
Kripke on substitutional quantification 680
The Lycan-Van Inwagen-Fine objection(s) 684
The problem of too many names 688
Natural language substitutionalism 690
Bibliography 690
Many-Valued Logics 692
1 Basic ideas 692
2 Outline of the history 697
3 Basic systems of many-valued logics 700
4 Standard and algebraic semantics 703
5 Particular three- and four-valued systems 706
6 Logics with t-norm based connectives 714
7 Residuated implications versus S-implications 716
8 Continuous t-norms 717
9 The logic of continuous t-norms 719
10 The logic of left continuous t-norms 722
11 Some generalizations 724
12 Pavelka style extensions 726
13 Gerla‘s general approach 729
14 Some recent applications 730
Bibliography 734
Relevance Logics 740
1 A problem with classical “implication” 742
2 First-degree entailment 746
3 T - ticket entailment in focus 748
4 Other relevance logics: B, L, E and R 791
Acknowledgements 802
Bibliography 802
Paraconsistent Logics and Paraconsistency 808
1 Introduction 808
2 The C-logics 813
3 Paraconsistent set theory 847
4 Ja& sacute
5 Annotated logics 875
6 Developments in paraconsistent logic 887
7 Applications 891
8 Concluding remarks 919
Acknowledgments 920
Bibliography 920
Extensional vs. Intensional Logic 930
1 Frege’s bedeutung 930
2 Carnap’s extension and extensional logic 935
3 Failures of extensionality and Carnap’s intension 941
4 Modal logic 943
5 Modal predicate logic 945
6 Montague’s ‘locally’ intensional logic 947
7 ‘Globally’ intensional logic 951
8 Quine’s extensionalist program 953
9 Beyond intensions 955
Acknowledgement 957
Bibliography 957
Logically Possible Worlds and Counterpart Semantics for Modal Logic 960
Introduction 960
Part I Worlds without objects 963
1 Basic concepts of modal propositional logic 963
2 Translation into classical logic 964
3 Ontology and duality theory 965
4 Possible worlds as an analytic tool 969
5 Accessibility 971
Part II The world of objects 975
6 Modal predicate logic 975
7 Counterpart semantics 984
8 Individual concepts 990
9 Objects in counterpart frames 993
10 Dual ontologies, or, the semantical impact of haecceitism 996
11 Metaframes 1000
12 Essence and identity 1001
13 On the status of the modal language 1002
Bibliography 1005
Modal Realism and its Roots in Mathematical Realism 1014
Introduction 1014
1 David Lewis’s possible worlds 1015
2 Plantinga’s theory of possible worlds 1022
3 An examination of argument T 1029
4 Concluding comment 1038
Acknowledgements 1039
Bibliography 1039
Free Logics 1040
1 The benefits of freedom 1040
2 The empty domain 1042
3 Proof theory 1043
4 Single vs. dual domains 1045
5 Positive semantics 1046
6 Negative semantics 1049
7 Nonvalent semantics 1052
8 Supervaluations 1054
9 Which semantics? 1055
10 Definite descriptions 1056
11 Set theory 1061
12 Definedness logics and programming languages 1063
13 Modal and tense logics 1064
14 Intuitionistic logics 1068
15 Failure of substitutivity of the biconditional 1070
16 An expressive limitation 1071
17 What is a free logic? 1073
18 Conclusion 1074
Bibliography 1075
Fictions and their Logic 1078
1 Logic 1078
2 Targets for a logic of fiction 1083
3 Intuitions 1085
4 Examining the basic conceptual models 1099
5 Maximal accounts 1128
6 Acknowledgements 1138
Bibliography 1138
Counterfactuals, Causation, and Preemption 1144
1 Introduction 1144
2 The logic of counterfactuals 1145
3 The counterfactual theory of causation 1147
4 Causation as influence 1150
5 De facto dependence 1152
6 The blueprint strategy 1156
Bibliography 1159
Logic, Mathematics, and the Natural Sciences 1162
1 Introduction 1162
2 The question of the adequacy of constructive methods for hypothetico-deductivism in natural science 1162
3 Refutation of empirical theories 1163
4 The anti-realist construal of empirical claims that cannot be proved 1167
5 The fate of strictly classical theorems of applied mathematics 1169
6 Meeting an objection of burgess 1173
Bibliography 1178
Default Reasoning 1180
1 Default inference 1180
2 Facing the prospect of error 1181
3 Induction as default reasoning 1183
4 Default reasoning as nonmonotonic 1183
5 Some comforting considerations 1185
Bibliography 1188
Index 1190