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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 starts by exploring a justly famous paradigm of experimental reasoning about animals — Karl von Frisch's work on bees. An analysis of von Frisch's reasoning about bees' colour-discrimination illustrates that support builds up for a hypothesis when it fails to be falsified in more and more complexly structured tests — where complexity of structure depends on the number of relevant variables manipulated in the test. The results of any such test are essentially replicable, which has important consequences for detachment and the treatment of ‘anomalous’ test-results. The series of relevant variables for testing a generalization has to be defined in a way that will ensure that each variable is non-exhaustive, and independent of every other variable, and it is also necessary to ensure a suitable ordering for the series as a whole. There is also a certain tension between the ontological and epistemological points of view in the philosophy of inductive support. In addition, the subsumption of Mill's canons under the method of relevant variables is explained. It then addresses how the method of relevant variables applies to scientific theories. Furthermore, the chapter elaborates Whewell's consilience, and Lakatos' progressive problem-shift, as inductive criteria. Next, it presents the problem of anomalies.

According to Karl von Frisch's method of reasoning, the conjunction of two generalizations must have the same grade of support as has the less well supported of the two or as both have if they are equally well supported. Also, any substitution-instance of a generalization must have the same grade of support as the generalization, since it is equally resistant to falsification by manipulations of relevant variables. So substitution-instances conform to the same conjunction principle as generalizations. The assumption of evidential replicability is crucial here and bars Carnap's confirmation-measures, or any form of enumerative induction, from applying to experimental reasoning like von Frisch's. Once the correct negation principle for inductive support has been established, it becomes clear how the emergence of mutually contradictory support-assessments can function as a reductio ad absurdum argument for the revision of a list of relevant variables. A support-function for generalizations of a certain category applies not only to those propositions that are constructed out of the basic vocabulary of the category but also to those that have been constructed out of this category when it has been enriched by the addition of terms describing the variants of relevant variables.

This chapter investigates the grading of inductive probability. It first evaluates the relation between inductive support and inductive probability. In the assessment of an inductive probability, favourable and unfavourable circumstances may be balanced off against one another. A tree-structure simile is available. Uncounteracted favourable evidence raises the inductive probability of a proposition, but uncounteracted unfavourable evidence reduces it to zero — which is not at all the same as making the proposition's negation certain. A statement of the inductive probability of S on R may be construed as grading the informativeness (what Keynes called the weight) of R in favour of S. If R states all the relevant evidence known and the inductive probability of S on R is high enough, it may be reasonable to believe that S is true, but one cannot detach here a monadic, unconditional grading for the inductive probability of S.

This chapter offers a description on the foundations of inductive logic. In order to provide a rational reconstruction for the concept of inductive probability, it is necessary first to provide one for the concept of inductive support. But a theory of inductive support should not aim to axiomatize its author's intuitions. Its primary data are the judgements of reputable scientists, not the intuitive deliverances of philosophers. Questions about the nature of inductive support are not easily answered. The issues are highly controversial and there is not even any general agreement about the premisses on which a resolution of the issues should be founded.

This chapter first provides some difficulties in current truth-functional analyses of dispositions. The truth-functional analysis of dispositional statements encounters difficulties that neither Quine's nor Carnap's proposal overcomes. Every dispositional statement encapsulates a statement about an inductive, not a mathematical probability. But statements about inductive support or inductive probability are open to both a nominalist and a realist interpretation. The difference between an Austinian and a Blackstonian interpretation of legal reasoning from precedent is analogous, but adopting an anti-realist position on one such issue does not necessarily commit a philosopher to adopting this position on another, analogous issue. The nominalist interpretation is not superior on grounds of ontological economy, because it involves a principle of plenitude. But it has greater epistemological coherence. So the analysis of dispositions in terms of inductive probabilities does not necessitate any retreat from anti-realism.

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 provides the assessment of judicial proof. It is shown that the concept of inductive probability, as derived from the concept of inductive support for covering generalizations, plays an important part in human reasoning. The inferences about the behaviour of others normally rest on the large stock of rough generalizations about human behaviour that is carried in their heads. So it is possible to construe proof beyond reasonable doubt as proof at a maximum level of inductive probability. Proof of S on the preponderance of evidence may then be construed as proof at a higher grade of inductive probability than that at which not-S is proved; and other standards of proof are also intelligible in these terms. Contextual clues are normally available to determine whether a given statement of probability is to be evaluated in accordance with mathematical or with inductive criteria, though experimental psychologists have not always recognized this. In fact, experimental data confirm the thesis that normal intuitive judgements of probability are often inductive rather than mathematical.

This chapter begins by presenting the prevailing scepticism in the philosophy of science. If a fact that is provable beyond reasonable doubt is inductively certain, the legal assumption that proof beyond reasonable doubt is possible conflicts with the sceptical thesis that knowledge of general truths about the world is impossible. But if it is possible to know that one hypothesis is inductively more reliable than another, it is certainly possible to know also that a hypothesis is fully reliable. Prevalence of the sceptical error is due partly to unawareness of the systematic analogy between the structure of inductive support and the structure of logical truth, partly to a confusion between truth-conditions and justification-conditions, partly to an over-reaction to certain shattering events in the history of science, partly to the mistaken view that a correct assessment of how much one proposition supports another must be regarded as an analytic truth, and partly to the mistaken view that any inductive assessment presupposes certain untestable metaphysical assumptions. When all these points are borne in mind, it becomes clear that on issues of fact proof beyond reasonable doubt, and scientific knowledge, is at least in principle possible.