Laws and Explanations; Theories and Modal Possibilities (Synthese Library Book 410)

The book has two parts: In the first, after a review of some seminal classical accounts of laws and explanations, a new account is proposed for distinguishing between laws and accidental generalizations (LAG). Among the new consequences of this proposal it is proved that any explanation of a contingent generalization shows that the generalization is not accidental. The second part involves __physical__ theories, their modality, and their explanatory power. In particular, it is shown that (1) Each theory has a theoretical implication structure associated with it, such that there are new __physical__ modal operators __on__ these structures and also special modal entities that are __in__ these structures. A special subset of the physical modals, the __nomic__ modals are associated with the laws of theories. (2) The familiar idea that theories always explain laws by deduction of them has to be seriously modified in light of the fact that there are a host of physical theories (including for example, Newtonian Classical mechanics, Hamiltonian, and Lagrangian theory, and probability theory) that we believe are __schematic__ (they do not have any truth value). Nevertheless, we think that there is a kind of non-deductive explanation and generality that they achieve by subsumtion under a schema. Contents 7 Part I: Burnishing the Legacy 10 Chapter 1: Introduction 11 Chapter 2: Hempel ́s Deductive-Nomological Model: In the Beginning 23 Chapter 3: Laws and Corresponding Counterfactuals, - An Untenable Connection 31 Chapter 4: F. Dretske ́s Total Rejection of the Hempel-Model 38 4.1 The Challenge 38 4.2 Laws as Relations 39 4.2.1 Laws as Singular Statements 42 4.3 Laws as Singular Statements that Relate Intensional Terms 43 4.3.1 Are There Many Relations That Are Used in Various Laws? 46 4.3.2 What About Non-equational Laws? 47 Chapter 5: Prelude to Armstrong: A Mathematical Revolution That Inspired F. Ramsey, and Left Russell and Armstrong Unmoved 50 5.1 Extensional and Intensional Relations 52 5.1.1 Nonextensionality and Scientific Laws 62 Chapter 6: D. Armstrong ́s Account of Laws: Identity Lost, Regained, and Lost Again 69 6.1 Advantages and Disadvantages of Armstrong ́s Almost Exclusive Dependence on Universals in His Account of Laws 69 6.2 Universals, Predicates, and Propositional Functions 73 6.3 Examples of Second-Order Laws and Their First-Order Instantiations 75 6.4 Some Formal Properties of the Relation of Necessitation N(F,G) 77 Chapter 7: Laws and Accidental Generalities 81 7.1 The Lotze Uniformity Condition 81 7.2 Observations and Consequences for (LAG) 85 7.3 Laws as Bound with Explanation 89 7.4 Gold Spheres, Rusty Screws, and Rocks with Iron 91 References 93 Part II: Theoretical Scenarios, Schematic Theories and Physical and Nomic Modals 94 Chapter 8: The Explanation of Laws: Two Neglected and Radically Different Theories. One Inspired by D. Hilbert; the Other Insp... 95 8.1 The Campbellian Background 95 8.2 Deductive Systems and Their Laws 96 8.3 Deductive Systems and Theories 98 8.4 The Mill-Ramsey-Braithwaite Account of Laws (MRB) 100 8.5 Deductive Systems and the Explanation of Laws 102 8.6 Ernest Nagel ́s Theory; Campbellian Origins 103 8.7 Explanation and Theories Without Truth-Values 104 8.8 Explanation: Axiomatization and the Hilbert Connection 105 8.9 Explanation: The Schematic Account 108 8.10 Explanation: Whither Campbell ́s Dictionary? 109 8.11 Explanation, Schematic Subsumption, and Factivity 110 8.12 Explanation and Generality 112 8.13 The Model-Theoretic View of Theories - A Disclaimer 114 Chapter 9: David Hilbert ́s Architecture of Theories and Schematic Structuralism 117 9.1 The Architecture of Theories 117 9.2 Hilbertian Concepts 125 9.3 Schematic Physical Theories 127 9.4 Schematic Theories, Schematic Subsumtion, Subsumptive Explanation and Successes 130 Chapter 10: Theories, Theoretical Scenarios, Their Magnitude Vector Spaces and the Modal Physical Possibilities they Provide 134 10.1 Theoretical Scenarios and Laws 134 10.2 The Variety of States of Theories 137 10.3 The Magnitude Vector Space of a Theory (MVS) 139 10.4 A Shift: The Focus on Laws with Background Theories (or Theoretical Scenarios) 141 10.5 The Magnitude Vector Spaces Associated with Theories 142 Chapter 11: The Possibilities That Theories Provide (Physical Modals) and the Possibilities of Laws (Nomic Modals) 146 11.1 Theoretical (Physical) Possibilities 146 11.2 Gentzen Implication Structures, and Their Modals 148 11.3 Necessity and Possibility Operators on the Vector Magnitude Space of Theories 149 11.3.1 Probability Spaces and Modality 149 11.4 Magnitude Vector Spaces, and Their Theoretical (Physical) Possibilities 156 11.5 Laws and Law-Designated Possibilities: The Nomic Possibilities 156 11.5.1 Ohm ́s Law 157 11.5.2 Galileo ́s Law of Falling Bodies 158 11.5.3 A Mach-Style Generalization of (D) 158 11.5.4 A Logical Relation (Gentzen Implication Relation) Between Nomic Possibilities 159 Chapter 12: Schematic Theories, Subsumtion of Laws, and Non-accidental Generalizations 162 12.1 The Incorporation of Laws into Schematic Theories 162 12.1.1 The Newtonian Theory of Mechanics 163 12.1.2 The Lagrangian Theory of Mechanics 168 12.1.3 The Hamiltonian Theory 170 12.1.4 Kolmogorov ́s Schematic Axiomatization of Probability 172 12.1.5 The Kolmogorov Axiomatization of Probability as a Schematic Modal Theory 172 12.1.6 Probabilistic-Truth and the Logical Operators 174 12.1.7 The Modal Character of Kolmogorov ́s Theory of Probability 175 12.2 Subsumtive Explanation, Truth, and Non-accidentality 177 12.3 A Connection Between Explanation and Explanation 177 12.4 Generality Redux. The Generality of Theories Compared to the Laws That They Explain 178 Bibliography 181 Index 184 Front Matter ....Pages i-ix Front Matter ....Pages 1-1 Introduction (Arnold Koslow)....Pages 3-14 Hempel’s Deductive-Nomological Model: In the Beginning ... (Arnold Koslow)....Pages 15-22 Laws and Corresponding Counterfactuals, – An Untenable Connection (Arnold Koslow)....Pages 23-29 F. Dretske’s Total Rejection of the Hempel-Model (Arnold Koslow)....Pages 31-42 Prelude to Armstrong: A Mathematical Revolution That Inspired F. Ramsey, and Left Russell and Armstrong Unmoved (Arnold Koslow)....Pages 43-61 D. Armstrong’s Account of Laws: Identity Lost, Regained, and Lost Again (Arnold Koslow)....Pages 63-74 Laws and Accidental Generalities (Arnold Koslow)....Pages 75-87 Front Matter ....Pages 89-89 The Explanation of Laws: Two Neglected and Radically Different Theories. One Inspired by D. Hilbert; the Other Inspired by F. Ramsey (Arnold Koslow)....Pages 91-112 David Hilbert’s Architecture of Theories and Schematic Structuralism (Arnold Koslow)....Pages 113-129 Theories, Theoretical Scenarios, Their Magnitude Vector Spaces and the Modal Physical Possibilities they Provide (Arnold Koslow)....Pages 131-142 The Possibilities That Theories Provide (Physical Modals) and the Possibilities of Laws (Nomic Modals) (Arnold Koslow)....Pages 143-158 Schematic Theories, Subsumtion of Laws, and Non-accidental Generalizations (Arnold Koslow)....Pages 159-177 Back Matter ....Pages 179-185 "The book has two parts: In the first, after a review of some seminal classical accounts of laws and explanations, a new account is proposed for distinguishing between laws and accidental generalizations (LAG). Among the new consequences of this proposal it is proved that any explanation of a contingent generalization shows that the generalization is not accidental. The second part involves physical theories, their modality, and their explanatory power. In particular, it is shown that (1) Each theory has a theoretical implication structure associated with it, such that there are new physical modal operators on these structures and also special modal entities that are in these structures. A special subset of the physical modals, the nomic modals are associated with the laws of theories. (2) The familiar idea that theories always explain laws by deduction of them has to be seriously modified in light of the fact that there are a host of physical theories (including for example, Newtonian Classical mechanics, Hamiltonian, and Lagrangian theory, and probability theory) that we believe are schematic (they do not have any truth value). Nevertheless, we think that there is a kind of non-deductive explanation and generality that they achieve by subsumtion under a schema."--Back cover