Possible Worlds Semantics for Indicative and Counterfactual Conditionals? (eBook)

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Matthias Unterhuber, University of Düsseldorf, Germany.

Preface 11
I Foundational Issues 1
Arguments for Conditional Logics 17
The Framework of My Investigation 18
Indicative Conditionals 18
Counterfactual Conditionals 24
Normic Conditionals 26
The Generic Case 27
The Qualification Problem 29
The Propositional Case 33
The Role of Normic Conditionals 35
Conversational Implicatures 36
The Conditional Logic Project in an Interdisciplinary Context and Default Logics 41
The Conditional Logic Project in an Interdisciplinary Context 41
The Conditional Logic Project 43
The Linguistics of Conditionals Project 43
The Philosophy of Conditionals Project 44
The Psychology of Reasoning Project 45
The Non-Monotonic Logic Project 47
Non-Monotonic Logics, Defaults Logics, and Conditional Logics 48
A Motivation for the Study of Non-Monotonic Logics 48
Reiter Defaults 50
Default Logics, Non-Monotonic Rules, and Inductive Inferences 52
Types of Non-Derivability Conditions and the Rule of Substitution 58
Model Theory, Proof Theory, and Axiomatization of Default Logics 61
Conditional Logics and Default Logics 62
Possible Worlds Semantics and Probabilistic Semantics for Indicative Conditionals: a Survey and a Defense of Possible Worlds Semantics 67
Outline of My Defense of Possible Worlds Semantics for Indicative Conditionals and the Core Idea of Chellas-Segerberg Semantics 69
Possible Worlds Ordering Semantics for Conditionals: D. Lewis' and Kraus et al.'s Semantics and Related Approaches 72
The Ramsey Test, Stalnaker Semantics, and a General Ramsey Test Requirement 84
Ramsey's Original Proposal 85
Stalnaker's Version of the Ramsey Test 87
Stalnaker Models 88
Stalnaker Semantics, Set Selection Semantics, and Chellas-Segerberg Semantics 91
Contrasting Ramsey Test Interpretations of Conditionals and Ordering Semantics 93
Stalnaker Semantics, Conditional Consistency Criteria, and the Principle of Conditional Excluded Middle 96
Bennett's Version of the Ramsey Test 102
A General Ramsey Test Requirement 103
Indicative and Counterfactual Conditionals and Conditional Logics 105
Criteria for Distinguishing Indicative and Counterfactual Conditionals 106
Bridge Principles and Logics for Indicative and Counterfactual Conditionals 110
Subjective and Objective Interpretations of Indicative and Counterfactual Conditionals and the Ramsey Test 114
Fundamental Issues of Probabilistic Approaches to Conditional Logic 117
Subjective and Objective Probabilistic Semantics and the Principle of Total Evidence 118
The Stalnaker Thesis, the Ramsey Test, and Conditional or Non-Conditional Probabilities as Primitive 120
Languages of a Probabilistic Conditional Logic 123
Further Reasons for the Restriction of Languages of Probabilistic Conditional Logics 125
Propositions, NTV (``No Truth Value''), and Conditional Logic Languages 127
Adams' Probabilistic P Systems 131
The P Systems: Systems P, P*, and P+ 132
Threshold Semantics 139
Adams' System P and Schurz's Modification 139
Possible Worlds Semantics for (Indicative) Conditionals and Quasi Truth Value Assignments in Adams' Probabilistic Semantics 148
D. Lewis' Triviality Result and Logics for Indicative Conditionals 155
D. Lewis' Triviality Result 156
Triviality due to Iterations (and Nestings) of Conditional Formulas 161
D. Lewis' Triviality Result, Restrictions of Languages, and Truth Value Accounts of Indicative Conditionals 163
Conclusion 166
Bennett's Gibbardian Stand-Off Argument 166
Bennett's Extended Gibbardian Stand-Off Argument 167
A Criticism of Bennett's Gibbardian Stand-Off Argument 171
Summary 174
Conclusion 175
II Formal Results for Chellas-Segerberg Semantics 175
Formal Framework 179
Why Chellas-Segerberg Semantics? 179
Proof-Theoretic Notions 180
Languages LCL, LCL-, LrCL, LrCL*, and LrrCL 180
Logics 188
Non-Monotonicity 189
Consistency and Maximality 190
A Propositional Basis for Conditional Logics 190
System CK 191
A Further Axiomatization of System CK 193
Model-Theoretic Notions 195
Chellas Frames and Chellas Models 196
Chellas Models and Frames and Kripke Semantics 198
Segerberg Frames and Segerberg Models 202
Validity, Logical Consequence, and Satisfiability 204
Notions of Frame Correspondence 207
Standard and Non-Standard Chellas Models and Segerberg Frames 209
Notions of Soundness and Completeness 210
Frame Correspondence for a Lattice of Conditional Logics 219
Non-Trivial Frame Conditions for a Lattice of Conditional Logics 221
The Notions of Trivial and Non-Trivial Frame Conditions 227
A Translation Procedure from Axiom Schemata to Trivial Frame Conditions 228
A Non-Triviality Criterion 231
Chellas Frame Correspondence Proofs 233
System P 233
Extensions of System P 236
Weak Probability Logic (Threshold Logic) 237
Monotonic Principles 239
Bridge Principles 241
Collapse Conditions Material Implication 243
Traditional Extensions 244
Iteration Principles 246
Soundness and Completeness for a Lattice of Conditional Logics 249
General Overview 249
Focus of the Completeness Proofs 249
Proofs for Segerberg Frame Completeness and Chellas Frames Completeness 251
Singleton Frames for CS Semantics 255
Soundness w.r.t. Classes of Chellas Frames 257
Standard Segerberg Frame Completeness 258
General Principles 258
Canonical Models 259
Canonicity Proofs for Individual Principles 262
Chellas-Segerberg Semantics for Indicative and Counterfactual Conditionals 273
The Basic Chellas-Segerberg Systems (Systems CK and CKR) 276
Objective and Subjective Interpretations of Chellas-Segerberg Semantics 277
Alternative Axiomatizations of System CKR 285
Conditional Logics without Bridge Principles 287
System C 287
System CL 295
System P 295
System R 303
D. Lewis' System V 306
Monotonic Systems without Bridge Principles (Systems CM and M) 314
Conditional Logics with Bridge Principles 322
Adams' System P* 322
D. Lewis' System VC 328
Stalnaker's System S 329
The Material Collapse System MC 331
Summary 336
Concluding Remarks 339
References 343