《The probable and the provable》

Introduction1

PART Ⅰ.CAN THERE BE A NON-PASCALIAN CONCEPT OF PROBABILITY?5

1.The Problem for a Polycriterial Theory of Proba-bility5

1.One criterion, or many?5

2.The mathematicist theory9

3.The family-resemblance theory10

2.Probability as the Gradation of Provability13

4.The degree-of-provability theory13

5.Proof-criteria that are general, necessary and exten-sional17

6.Proof-criteria that are general, contingent, and extensional19

7.Proof-criteria that are general, contingent, and non-extensional21

8.Proof-criteria that are singular and either necessary or contingent24

9.Probability-statements as evaluations of inferential soundness27

3.The Completeness Issue33

10.The connection between deductive completeness and mathematical probability33

11.The grading of probabilities by evidential weight36

12.Mathematical and inductive probability39

13.The example of judicial proof43

PART Ⅱ.SIX DIFFICULTIES FOR A PASCALIAN ACCOUNT OF JUDICIAL PROBABILITY49

4.What are the Standards of Proof in Courts of Law?49

14.The difference between the criminal and civil standards49

15.Theories about judicial probability51

16.The nature of the argument against a mathemati-cist theory56

5.The Diffculty about Conjunction58

17.The constraint on complexity in civil cases58

18.The independence issue61

19.Does the balance of probability lie between the plaintiff’s and the defendant’s contentions?62

20.Does the balance of probability consist in the difference between prior and posterior probabili-ties ?63

21.Does the plaintiff’s contention need evaluation as a whole?65

6.The Diffculty about Inference upon Inference68

22.Permissiveness in regard to two- or many-stage proofs68

23.Is the constraint on inference upon inference a purely legal matter?71

7.The Diffculty about Negation74

24.The paradox of the gate-crasher74

25.What is the threshold of proof in civil cases?76

26.When are statistical probabilities admissible evi-dence?78

8.The Diffculty about Proof beyond Reasonable Doubt82

27.Reasons for doubting guilt82

28.The role of statistical probabilities in judicial proof83

9.The Diffculty about a Criterion87

29.The inapplicability of Carnapian criteria87

30.The inapplicability of betting odds89

31.The hiatus to be filled91

10.The Difficulty about Corroboration and Con-vergence93

32.The common structure of testimonial corroboration and circumstantial convergence93

33.The traditional, Bernoullian analysis95

34.The need to take prior probabilities into account97

35.A demonstrably adequate analysis ofcorroboration and convergence in terms of mathematical proba-bility101

36.The legal inadmissibility of positive prior proba-bilities107

37.The method of contraposition114

11.The Case against a Mathematicist Account of Judicial Probability116

38.The impact of accumulated anomalies116

39.The existing standards of assessment118

PART Ⅲ.THE ELEMENTS OF INDUCTIVE PROBABILITY121

12.The Foundations of Inductive Logic121

40.Inductive probability and inductive support121

41.On what foundations should a theory of inductive support be based?123

13.The Grading of Inductive Support129

42.An example of inductive support-assessment in entomology129

43.The importance of evidential replicability133

44.The definition of a series of inductively relevant variables135

45.Ontology versus epistemology in the philosophy of inductive support140

46.The subsumption of Mill’s canons under the method of relevant variables144

47.How the method of relevant variables applies to scientific theories151

48.Whewell’s consilience, and Lakatos’s progressive problem-shift, as inductive criteria157

49.The problem of anomalies162

14.The Logical Syntax of Inductive Support-gradings167

50.The conjunction principle for inductive support167

51.The uniformity principle for inductive support170

52.Some consequences of evidential replicability175

53.The negation principle for inductive support177

54.Some consequences of modifying a generalization in order to ensure reliability182

15.The Incommensurability of Inductive Support and Mathematical Probability187

55.The argument from the possibility of anomalies187

56.The argument from the conjunction principle：rst stage190

57.The argument from the conjunction principle：second stage196

16.The Grading of Inductive Probability199

58.The relation between inductive support and inductive probability199

59.The balancing off of favourable and unfavourable circumstances in the assessment of an inductive probability202

60.The asymmetric effects of favourable and un-favourable evidence207

61.Inductive probability as a grading of evidential informativeness211

62.The presumption of total evidence213

17.The Logical Syntax of Inductive Probability-gradings217

63.Some logical similarities between inductive and mathematical probability217

64.Some logical differences between inductive and mathematical probability219

65.The incommensurability of inductive and mathe-matical probability224

66.A formal axiomatization for the logical syntax of inductive probability229

PART Ⅳ.SOME APPLICATIONS OF INDUCIVE PROBABILITY245

18.The Assessment of Judicial Proof245

67.Are there any uses for the concept of inductive probability?245

68.Proof beyond reasonable doubt247

69.Proof on the preponderance of evidence252

70.Contextual clues to the disambiguation of pro-bability-sentences256

19.An Inductivist Resolution of Six Diffculties for a Pascalian Account of Judicial Proof265

71.The difficulty about conjunction265

72.The difficulty about inference upon inference267

73.The difficulty about negation270

74.The difficulty about proof beyond reasonable doubt272

75.The difficulty about a criterion273

76.The difficulty about corroboration and conver-gence277

20.Criteria of Merit for Explanations of Individual Events282

77.Hempel’s theory of explanation for individual events282

78.Covering laws for both proofs and explanations284

79.The grading of explanatory completeness285

80.The merit of the explanation tends to vary inversely with the generality of the covering law291

21.Statistical Explanation295

81.Hempel’s account of statistical explanation295

82.The need to maximize inductive probability for the appositeness of the reference-class297

83.The need for favourable relevance rather than a high mathematical probability300

84.Salmon’s rejection of the relevance requirement303

85.The need for a propensity interpretation306

22.Criteria of Rational Belief310

86.The problem of the detachment conditions for dyadic judgements of probability310

87.Difficulties for the formulation of an acceptance-rule in terms of mathematical probability312

88.Four unsatisfactory proposals for resolving the difficulties314

89.The advantages of formulating an acceptance-rule in terms of inductive probability318

23.Dispositions325

90.Some difficulties in current truth-functional analyses of dispositions325

91.The inductive probability implicit in a disposi-tional statement327

92.Nominalism versus realism in the philosophy of inductive support331

93.Can one be a nominalist on some issues but not on others?336

94.The nominalist interpretation is epistemologically,not ontologically, superior339

24.An Epistemological Corollary345

95.The prevailing scepticism in the philosophy of science345

96.If inductive knowledge of comparative reliability is possible, so is knowledge of complete reliabityty348

97.Inductive knowledge does not imply knowing that one knows350

98.The prevalence of scepticism is due largely to an inadequate understanding of inductive logic351

Index357