Philosophy of Logic

Description

The papers presented in this volume examine topics of central interest in contemporary philosophy of logic. They include reflections on the nature of logic and its relevance for philosophy today, and explore in depth developments in informal logic and the relation of informal to symbolic logic, mathematical metatheory and the limiting metatheorems, modal logic, many-valued logic, relevance and paraconsistent logic, free logics, extensional v. intensional logics, the logic of fiction, epistemic logic, formal logical and semantic paradoxes, the concept of truth, the formal theory of entailment, objectual and substitutional interpretation of the quantifiers, infinity and domain constraints, the Löwenheim-Skolem theorem and Skolem paradox, vagueness, modal realism v. actualism, counterfactuals and the logic of causation, applications of logic and mathematics to the physical sciences, logically possible worlds and counterpart semantics, and the legacy of Hilbert’s program and logicism. The handbook is meant to be both a compendium of new work in symbolic logic and an authoritative resource for students and researchers, a book to be consulted for specific information about recent developments in logic and to be read with pleasure for its technical acumen and philosophical insights.

- Written by leading logicians and philosophers
- Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic
- Clear, in-depth expositions of technical detail
- Progressive organization from general considerations to informal to symbolic logic to nonclassical logics
- Presents current work in symbolic logic within a unified framework
- Accessible to students, engaging for experts and professionals
- Insightful philosophical discussions of all aspects of logic
- Useful bibliographies in every chapter

Front Cover 2
Philosophy of Logic 5
Copyright Page 6
General Preface 7
Table of Contents 9
Preface 13
List of Contributors 15
Introduction: Philosophy of Logic Today 19
- Logic and philosophy 19
- In the logic candy store 20
- Understanding the plurality of logics 23
- Future directions in philosophical logic and philosophy of logic 27
What is Logic? 31
- 1 Logic and inference 31
- 2 Deductive inference and information 32

Front Cover 2
Philosophy of Logic 5
Copyright Page 6
General Preface 7
Table of Contents 9
Preface 13
List of Contributors 15
Introduction: Philosophy of Logic Today 19
- Logic and philosophy 19
- In the logic candy store 20
- Understanding the plurality of logics 23
- Future directions in philosophical logic and philosophy of logic 27
What is Logic? 31
- 1 Logic and inference 31
- 2 Deductive inference and information 32
- 3 Analytic inferences 32
- 4 Ampliative inference 34
- 5 Logical vs. extralogical systematization 34
- 6 Syntax vs. semantics 35
- 7 The limits of syntactical approaches 36
- 8 The limits of semantical approaches 37
- 9 Strategic vs. definitory rules 38
- 10 What is first-order logic? 39
- 11 Presuppositions of the received first-order logic 40
- 12 The meaning of quantifiers 41
- 13 Semantical games 43
- 14 The implications of independence-friendly logic: Axiomatizability 44
- 15 Negation and its significance 45
- 16 Higher-order logics 46
- 17 A distinction without ontological difference? 48
- 18 Alternative logics? 49
- 19 Ampliative inference and acquisition of information 50
- Bibliography 52
The Scope and Limits of Logic 59
- 1 The broad sweep 59
- 2 Definitions of logic 60
- 3 Concepts that belong to logic 61
- 4 The syllogism claim 62
- 5 Patterns of reasoning 65
- 6 Reduction to logic 68
- 7 Quine’s ‘scope of logic’ 70
- 8 Logical notions 73
- 9 How we ought to think? 75
- Bibliography 79
Logic in Philosophy 83
- 1 The century that was 83
- 2 A joy-ride through history 86
- 3 Logical form and natural logic: From syllogistics to generalized quantifiers 90
- 4 Reasoning styles: From bolzano to conditional logic and AI 94
- 5 Mechanisms of semantic interpretation: From tarski to ‘dynamics’ 98
- 6 Theories and belief revision: From neo-positivism to computer science 103
- 7 Dynamic logic, communication, action, and games 106
- 8 Further themes 111
- Acknowledgements 112
- 9 Conclusion: Logic and philosophy once more 112
- Bibliography 113
Informal Logic and The Concept of Argument 119
- Technical and everyday senses of “argument” 119
- Argument as discourse supporting a point of view by offering one or more reasons 121
- Arguments as invitations to inference 123
- Extensions: Potential arguments and equivalence classes of arguments 125
- Complex direct arguments 126
- Suppositional arguments 128
- First summary 132
- Competing conceptions of argument 134
- Exclusions from the class of arguments 138
- Postscript 139
- Second summary 139
- Acknowledgements 145
- Bibliography 145
On the Relation of Informal to Symbolic Logic 149
- 1 A strange dichotomy 149
- 2 Toward a definition of formal and informal logic 151
- 3 Formal and informal logic in partnership 154
- 4 Conclusion 170
- Acknowledgements 171
- Bibliography 171
Vagueness and the Logic of Ordinary Language 173
- 1 The epistemic diagnosis 173
- 2 Does logic apply to the ordinary world? 174
- 3 Vagueness in the platonic heavens 175
- 4 Does logic apply to ordinary language? 177
- 5 Vagueness as a pre-logical phenomenon 177
- 6 Does logic only apply loosely? 180
- 7 Truth-value gaps 181
- 8 Logical pluralism 182
- 9 Deviant logic 183
- 10 Many-valued logic 184
- 11 Family resemblance 185
- 12 Prototypes 188
- Bibliography 189
Logic and Semantic Analysis 191
- 1 Introduction 191
- 2 Some classic arguments 193
- 3 Doing semantic analysis by analogy to logic 201
- 4 Doing semantic analysis in logic 211
- 5 Open problems and further reading 219
- Bibliography 220
Justificatory Irrelevance of Formal Semantics 223
- 1 Background and overview 223
- 2 The problem of justifying a formal logic 225
- 3 System E of entailment 226
- 4 On use and interpretation of E-model structures - Ems 228
- 5 On world-line matrices for two and three “points” 236
- Bibliography 244
A Brief History of Truth 245
- Brief introduction 245
- 1 Early views 246
- 2 Middle views 272
- 3 Later views 297
- 4 Brief conclusion 332
- Bibliography 334
Truth and Paradox: A Philosophical Sketch 343
- 1 Aims and structure of this essay 343
- 2 Dtruth and truth 344
- 3 Liars: Broad picture and projects 346
- 4 Paracomplete 347
- 5 Paraconsistent 384
- 6 Dtruth, validity, and truth-preservation 395
- 7 Leaving dtruth: Parameters and contextual shifts 399
- 8 Revision theory 410
- 9 Set-theoretic paradoxes? 414
- 10 Revenge –– At last 416
- 11 Further reading 420
- Acknowledgements 423
- Bibliography 424
Hilbert’s Program Then and Now 429
- 1 Introduction 429
- 2 Hilbert’s program then 430
- 3 Philosophical interpretation of Hilbert’s program 437
- 4 Hilbert’s program now 451
- 5 Conclusion 458
- Acknowledgements 459
- Bibliography 459
Logicism and its Contemporary Legacy 467
- 1 Frege 470
- 2 Context, number and neo-fregean logicism 476
- 3 Russell’s logicism and the “no-class” theory 489
- 4 Carnap’s internal logicism 491
- 5 Contemporary criticisms of logicism 495
- 6 Structuralist alternatives to logicism 501
- 7 Logicism and the theory of types 505
- Bibliography 512
Classical Logic’s Coming of Age 515
- 1 Introduction 515
- 2 The rise to ascendency of first-order logic 516
- 3 Semantic decidability and semantic completeness 518
- 4 Formally undecidable statements, the undefinability of truth, and the existence of nonstandard models 521
- 5 The second incompleteness theorem, and later extensions of Gödel’s results 523
- 6 Consistency proofs in the wake of the second incompleteness theorem 526
- 7 The emergence of recursion theory 528
- 8 The relative consistency of the axiom of choice and the generalized continuum hypothesis 534
- 9 Culmination and codification 537
- Bibliography 537
Infinity 541
- 1 Introduction 541
- 2 Actual infinity 544
- 3 Potential infinity 564
- 4 Physical infinity 582
- Bibliography 597
Löwenheim–Skolem Theorems 605
- 1 Introduction 605
- 2 The Löwenheim-Skolem theorem 607
- 3 Löwenheim-Skolem-Tarski theorems 610
- 4 From “very large” to “very small” 614
- 5 Building cardinalities into the logic 616
- 6 Transfer in general logic systems 619
- 7 General aspects 625
- 8 Transfer in the finite 628
- Acknowledgements 630
- Bibliography 630
The Mathematics of Skolem’s Paradox 633
- 1 Skolem’s Paradox 634
- 2 A quick technical solution 645
- 3 The virtues of quantification 649
- 4 The vices of quantification I 654
- 5 The vices of quantification II 657
- 6 A few concluding remarks 661
- Bibliography 666
Objectual and Substitutional Interpretations of the Quantifiers 667
- Introduction 667
- Quine and ontological commitment 670
- Referential and nonreferential base semantics 672
- The “substitutional strategy” 674
- Kripke on substitutional quantification 681
- The Lycan-Van Inwagen-Fine objection(s) 685
- The problem of too many names 689
- Natural language substitutionalism 691
- Bibliography 691
Many-Valued Logics 693
- 1 Basic ideas 693
- 2 Outline of the history 698
- 3 Basic systems of many-valued logics 701
- 4 Standard and algebraic semantics 704
- 5 Particular three- and four-valued systems 707
- 6 Logics with t-norm based connectives 715
- 7 Residuated implications versus S-implications 717
- 8 Continuous t-norms 718
- 9 The logic of continuous t-norms 720
- 10 The logic of left continuous t-norms 723
- 11 Some generalizations 725
- 12 Pavelka style extensions 727
- 13 Gerla‘s general approach 730
- 14 Some recent applications 731
- Bibliography 735
Relevance Logics 741
- 1 A problem with classical “implication” 743
- 2 First-degree entailment 747
- 3 T - ticket entailment in focus 749
- 4 Other relevance logics: B, L, E and R 792
- Bibliography 803
- Acknowledgements 803
Paraconsistent Logics and Paraconsistency 809
- 1 Introduction 809
- 2 The C-logics 814
- 3 Paraconsistent set theory 848
- 4 Ja&sacute;kowski’s logic 862
- 5 Annotated logics 876
- 6 Developments in paraconsistent logic 888
- 7 Applications 892
- 8 Concluding remarks 920
- Bibliography 921
- Acknowledgments 921
Extensional vs. Intensional Logic 931
- 1 Frege’s bedeutung 931
- 2 Carnap’s extension and extensional logic 936
- 3 Failures of extensionality and Carnap’s intension 942
- 4 Modal logic 944
- 5 Modal predicate logic 946
- 6 Montague’s ‘locally’ intensional logic 948
- 7 ‘Globally’ intensional logic 952
- 8 Quine’s extensionalist program 954
- 9 Beyond intensions 956
- Acknowledgement 958
- Bibliography 958
Logically Possible Worlds and Counterpart Semantics for Modal Logic 961
- Introduction 961
- Part I Worlds without objects 964
  - 1 Basic concepts of modal propositional logic 964
  - 2 Translation into classical logic 965
  - 3 Ontology and duality theory 966
  - 4 Possible worlds as an analytic tool 970
  - 5 Accessibility 972
- Part II The world of objects 976
  - 6 Modal predicate logic 976
  - 7 Counterpart semantics 985
  - 8 Individual concepts 991
  - 9 Objects in counterpart frames 994
  - 10 Dual ontologies, or, the semantical impact of haecceitism 997
  - 11 Metaframes 1001
  - 12 Essence and identity 1002
  - 13 On the status of the modal language 1003
  - Bibliography 1006
- Modal Realism and its Roots in Mathematical Realism 1015
  - Introduction 1015
  - 1 David Lewis’s possible worlds 1016
  - 2 Plantinga’s theory of possible worlds 1023
  - 3 An examination of argument T 1030
  - 4 Concluding comment 1039
  - Acknowledgements 1040
  - Bibliography 1040
- Free Logics 1041
  - 1 The benefits of freedom 1041
  - 2 The empty domain 1043
  - 3 Proof theory 1044
  - 4 Single vs. dual domains 1046
  - 5 Positive semantics 1047
  - 6 Negative semantics 1050
  - 7 Nonvalent semantics 1053
  - 8 Supervaluations 1055
  - 9 Which semantics? 1056
  - 10 Definite descriptions 1057
  - 11 Set theory 1062
  - 12 Definedness logics and programming languages 1064
  - 13 Modal and tense logics 1065
  - 14 Intuitionistic logics 1069
  - 15 Failure of substitutivity of the biconditional 1071
  - 16 An expressive limitation 1072
  - 17 What is a free logic? 1074
  - 18 Conclusion 1075
  - Bibliography 1076
- Fictions and their Logic 1079
  - 1 Logic 1079
  - 2 Targets for a logic of fiction 1084
  - 3 Intuitions 1086
  - 4 Examining the basic conceptual models 1100
  - 5 Maximal accounts 1129
  - Bibliography 1139
  - 6 Acknowledgements 1139
- Counterfactuals, Causation, and Preemption 1145
  - 1 Introduction 1145
  - 2 The logic of counterfactuals 1146
  - 3 The counterfactual theory of causation 1148
  - 4 Causation as influence 1151
  - 5 De facto dependence 1153
  - 6 The blueprint strategy 1157
  - Bibliography 1160
- Logic, Mathematics, and the Natural Sciences 1163
  - 2 The question of the adequacy of constructive methods for hypothetico-deductivism in natural science 1163
  - 1 Introduction 1163
  - 3 Refutation of empirical theories 1164
  - 4 The anti-realist construal of empirical claims that cannot be proved 1168
  - 5 The fate of strictly classical theorems of applied mathematics 1170
  - 6 Meeting an objection of burgess 1174
  - Bibliography 1179

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