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This chapter presents an elaboration on the difficulty about proof beyond reasonable doubt. It is more inclined to hold that a particular conclusion falls short of certainty because there is a particular, specifiable reason for doubting it, than to hold that it is reasonable to doubt the conclusion because it falls short of certainty. Hence a scale of mathematical probability is used for assessing proof beyond reasonable doubt. What is needed instead is a list of all the points that have to be established in relation to each element in the crime. Not that a high statistical probability is necessarily useless; but it must enter into a proof as a fact from which to argue rather than as a measure of the extent to which a conclusion has been established, and its relevance must also be separately established.

This chapter examines the difficulty about corroboration and convergence. It begins by addressing the common structure of testimonial corroboration and circumstantial convergence. The chapter also considers the traditional, Bernoullian analysis. The elucidation that has been most commonly proposed is at least as old as James Bernoulli's Ars Conjectandi. The chapter describes it in the admirably perspicuous form in which it was expounded by George Boole. In addition, the need to take prior probabilities into account is shown. Moreover, a demonstrably adequate analysis of corroboration and convergence in terms of mathematical probability is presented. It then reports the legal inadmissibility of positive prior probabilities. The method of contraposition is elaborated as well.

In most civil cases, the plaintiff's contention consists of several component elements. So the multiplication law for the mathematical probability of a conjunction entails that, if the contention as a whole is to be established on the balance of mathematical probability, there must either be very few separate components in the case or most of them must be established at a very high level of probability. Since this constraint on the complexity of civil cases is unknown to the law, the mathematicist analysis is in grave difficulties here. To point out that such component elements in a complex case are rarely independent of one another is no help. Therefore, a mathematicist might claim that the balance of probability is not to be understood as the balance between the probability of the plaintiff's contention and that of its negation, but as the balance between the probability of the plaintiff's contention and that of some contrary contention by the defendant. However, this would misplace the burden of proof. To regard the balance of probability as the difference between prior and posterior probabilities is open to other objections. To claim that the plaintiff's contention as a whole is not to have its probability evaluated at all is like closing one's eyes to facts one does not like.

This chapter elucidates the logical syntax of inductive probability-gradings. It first presents some logical similarities between inductive and mathematical probability. The inductive probability-gradings conform to quite different principles from those for mathematical probability in regard to contraposition; in regard to the relation between prior and posterior probabilities; in regard to a proposition's conjunction with other propositions; and in regard to its negation. In terms of inductive probability, it is possible to describe a generalized form of reductio ad absurdum argument. The logical structure of inductive probability cannot be mapped on to the calculus of mathematical probability. Indeed, because inductive support does not seem to be additive, inductive probabilities do not seem to be measurable — though they are rankable. Furthermore, the logical syntax of inductive probability may be deployed axiomatically within a modal logic that generalizes on Lewis' system S₄.

This chapter describes the standard of proof in courts of law. There are two main standards for proof of fact in English and American courts. The plaintiff in a civil case must prove on the balance of probabilities, and the prosecutor in a criminal case must prove his conclusion at a level of probability that puts it beyond reasonable doubt. It also addresses the theories about judicial probability. Some philosophers have claimed that it does, some that if such a probability were measurable it would do so, and some that it is not even in principle a mathematical probability. The third of these views is the most defensible, but it has never been properly argued or substantiated hitherto.

This chapter provides an elaboration on the incommensurability of inductive support and mathematical probability. It begins by presenting the argument from the possibility of anomalies. If inductive support-grading is to allow for the existence of anomalies, it cannot depend on the mathematical probabilities involved. A second argument for the incommensurability of inductive support with mathematical probability may be built up on the basis of the conjunction principle for inductive support. If s[H,E] conforms to this principle, the actual value of p_M[H] must be irrelevant to that of s[H,E] unless intolerable constraints are to restrict the mathematical probability of one conjunct on another. Since the actual value of p_M[E,H] must also be irrelevant to that of s[H,E], and s[H,E] cannot possibly be a function of p_M[E] alone, it follows that s[H,E] cannot be a function of the mathematical probabilities involved.

This chapter begins by providing the problem of the detachment conditions for dyadic judgements of probability. The study of criteria for rational belief is very largely the study of the detachment conditions for dyadic judgements of probability. The deductive closure condition and the logical consistency condition present difficulties for any acceptance-rule formulated in terms of mathematical probability. The proposals for dealing with these difficulties that have been put forward by Hintikka and Hilpinen, by Kyburg, by Levi, and by Lehrer, are all, for different reasons, unsatisfactory. But a rule of acceptance formulated in terms of inductive probability does not encounter any of these difficulties. Mathematical probability can provide a basis for decision-theoretic strategies, but not for rational belief.

This chapter investigates the difficulty about negation. Because of the principle that p_M[S] = I − p_M[not-S], the mathematicist analysis implies that in civil cases the Anglo-American system is officially prepared to tolerate a quite substantial mathematical probability that a losing defendant deserved to succeed. There is a limit to the extent that this difficulty can be avoided by supposing a higher threshold for the balance of probability. Nor are the proper amounts of damages held to be proportional to the strength of a winning plaintiff's proof. If there were a legal rule excluding statistical evidence in relation to voluntary acts much of the paradox here would disappear. But it would be unnecessary to suppose such a rule if the outcome of civil litigation could be construed as a victory for case-strength rather than as the division of a determinate quantity of case-merit.

This chapter explores the difficulty about a criterion. It first introduces the inapplicability of Carnapian criteria. No familiar criterion of mathematical probability is applicable to the evaluation of juridical proofs. Statistical criteria have already been shown to be inapplicable. Carnapian criteria require a unanimity about range-measure, which cannot be assumed. To suppose that jurors should evaluate proofs in terms of a coherent betting policy is to ignore the fact that rational men do not bet on issues where the outcome is not discoverable otherwise than from the data on which the odds themselves have to be based. In addition, the point here is not that there is anything intrinsically and universally wrong with evaluating mathematical probabilities in terms of statistical frequencies, range-overlap or betting odds. So the onus is on the mathematicist to propose some other criterion, which is not excluded by any of the special circumstances of judicial proof.